text:
Palliative Pharmacotherapy/Miscellaneous/Helpful Dot Phrases/New Consult Inpatient Pain Management Note Template

The following is an EXAMPLE note template that could be used for comprehensive medication management for an inpatient pain management consult service. This is the author's own personal template and therefore has information in place to remind this scatterbrained author to ask questions that may be very obvious to other practitioners. Please feel free to use and edit this template as you wish. Beware that this note template has been created for CPRS (VA EMR) and is not suitable in its raw form for use in Epic, Cerner, Meditech, etc.

For those unfamiliar with CPRS, the items / text between bars (bars = | |) ''are called TIU (text integration utility) data objects and these pull data from the chart automatically. For example, in this template the first TIU data object encountered is |ADMITTING DIAGNOSIS|, which when used in a template will automatically populate with the selected patient's admitting diagnosis for the current inpatient admission. CPRS TIU data objects vary across VA facilities, though some may be similar. That is to say - if you would like to use this template within CPRS at your facility, some of the TIU data objects may not work properly.''

NOTE: This is an electronic consult/note performed to aid in the treatment of the patient and is based upon a review of the chart. Under some circumstances, this consult may have been requested by internal protocols. The chart was reviewed in the detail reflected in the note and the recommendations were based on the available information and the specific request.

=
=================================================================== SUBJECTIVE

=
=================================================================== --   CONSULT DETAILS    --

-    ADMISSIONS     --- ADMITTING DIAGNOSIS: |ADMITTING DIAGNOSIS|

NFSG ADMISSION HISTORY: --  HPI with a PMH of:
 * PREVIOUS ADMISSIONS|
 * PATIENT FIRST & LAST NAMES| is a |PATIENT AGE| |PATIENT RACE| |PATIENT SEX|

Patient's current pain regimen consists of:

--- PATIENT INTERVIEW: --- Patient seen at bedside today. Patient was amenable to speaking with writer.

Mood: “ “ Affect: [] appropriate [] inappropriate [] congruent [] incongruent [] blunted [] flat  [] normal  [] intense [] labile [] even  [] expansive [] broad [] restricted

NON-VERBAL PAIN CUES: Facial expressions [] frowning, sad or frightened face [] grimacing, wincing, eye tightening or closing [] distorted facial expressions - brow raising/lowering, cheek raising, nose wrinkling, lip corner pulling [] rapid blinking. Vocalisation [] sighing, groaning, moaning [] grunting, screaming, calling out [] aggressive or offensive speech [] noisy breathing [] asking for assistance [] crying out Body movement [] tense posture, guarding, rigid [] fidgeting [] pacing, rocking or repetitive movements [] reduced or restricted movement [] altered gait. Social interaction [] aggressive or disruptive behaviour [] socially inappropriate behaviour [] decreased social interactions [] withdrawn. Autonomic signs [] pallor [] sweating [] rapid breathing (tachypnoea) [] altered breathing [] rapid heart rate (tachycardia) [] hypertension

PAIN DESCRIPTION Onset When did it begin? How long does it typically last? How often does it occur? What were you doing when it started?

Provoking / palliating factors What brings it on? What makes it worse? What makes it better? Quality What does it feel like?

Region & radiation Does your pain radiate? Where does it radiate to? Where does it hurt the most? Where does your pain go from there?

Severity What is the intensity of the pain? Right now? At its worst? Are there any other symptoms that accompany the pain?

Timing & treatment - see below Understanding What do you believe is causing this? How is this affecting your ADLs? How is this affecting your family? Do you have any other concerns?

PAIN IMPACT/FUNCTIONING Sleep: Any trouble falling asleep and/or staying asleep?: Do you wake up during the night due to pain?: Is your sleep restful?: Number of hours per night on average: Diagnosis of sleep apnea?: If no, complete STOP-BANG in assessment If yes, compliant with CPAP?

Mobility/Activity: Current work: General daily activities: Use of mobility aids?:

SOCIAL HISTORY Relationship status: Living situation: Does anyone help with medical care?: Social support: Physical activity: Diet: Alcohol: Caffeine: Nicotine: Cannabis: Non-prescribed opioids: Stimulants: Others:

--- OPIOID MONITORING ---

OMEs over past 24 hours: If using PRN opioids, how long does each dose last?: ____ hours If using long-acting opioids, how long does each dose last?: ____ hours Any scheduled pain medication refusals?: [] No [] Yes:

Current bowel regimen:

Taking bowel regimen as prescribed?: [] Yes [] No: Date of last documented BM: Patient normally has ____ BM(s) every _____ days Nausea/vomiting?: [] Denies [] Endorses:  [] unchanged/at patient's baseline

Date of last fall: Circumstances of last fall: Dizziness?: [] Denies [] Endorses:  [] unchanged/at patient's baseline Sedation?: [] Denies [] Endorses:  [] unchanged/at patient's baseline

Blood pressure:  [] WNL [] at patient's baseline [] elevated [] hypotensive Respiratory rate: [] WNL [] at patient's baseline [] tachypneic [] respiratory depression

-- IF PATIENT ORDERED PATCH --

Skin irritation?: [] Denies  [] Endorses:  [] unchanged/at patient's baseline Patch placement: [] writer verified placement, patch was visualized on ______ (area of body) [] not visualized by writer, but nursing documentation indicates patch remains on patient, located on ______ (area of body) Date patch last change?:

--   PATIENT-REPORTED OUTCOMES / CLINICIAN-ADMINISTERED MEASURES    -- Pain catastrophizing = ____ on _____ PROMIS function = PROMIS pain interference = PSEQ = CSSRS = PHQ-9 = AUDIT-C = ORT = GAD-7 = HAM-D = MMSE = MoCA = Pain Disability Index =

PEG TOOL 1.) Average pain score (see above) 2.) On scale of 0 (no interference) through 10 (extreme/frequent interference), which number best describes how pain has interfered with your enjoyment of    life during the past week? 3.) On a scale of 0 (no interference) through 10 (extreme/frequent interference), which number best describes how pain has interfered with your    general activity during the past week?   4.) Average of scores PEG Score Hx  Date:		Score: Date:		Score:

Defense and Veterans Pain Rating Scale (DVPRS) Date      Avg Pain*      Worst Pain*       Lowest Pain* /10           /10               /10
 * over a specified time frame

Functional Goal(s): What would you like to do that your pain is currently preventing you from doing? Date:    Improved   Same   Worse Date:    Improved   Same   Worse PAIN TREATMENT HISTORY -

Previous Interventional Treatments: [ ] Spinal cord stimulator [ ] Steroid injections [ ] Trigger point injections [ ] RFAs [ ] Surgery [ ] Other:

Previous Non-pharmacological treatment: [ ] PT/OT [ ] Yoga/Tai Chi [ ] Aquatherapy [ ] Acupuncture [ ] Chiropractor [ ] BFA [ ] CBT/Psychotherapy ("regular") [ ] CBT/Psychotherapy (pain-focused) [ ] Mindfulness [ ] MOVE! program [ ] Heating pad [ ] Cold packs [ ] TENS unit [ ] Other e-stim device [ ] Others:

Previous Medication Trials: ANALGESICS/NSAIDS [ ] Aspirin [ ] Acetaminophen [ ] Celecoxib [ ] Diclofenac [ ] Diflunisal [ ] Etodolac [ ] Fenoprofen [ ] Flurbiprofen [ ] Ibuprofen [ ] Indomethacin [ ] Meloxicam [ ] Nabumetone [ ] Naproxen [ ] Oxaprozin [ ] Piroxicam [ ] Salsalate [ ] Sulindac [ ] Tolmetin

OPIOIDS [ ] Codeine [ ] Fentanyl [ ] Hydrocodone [ ] Hydromorphone [ ] Morphine [ ] Methadone [ ] Oxycodone [ ] Oxymorphone [ ] Buprenorphine [ ] Tramadol [ ] Tapentadol [ ] Propoxyphene [ ] Nalbuphine [ ] Levorphanol

TOPICALS [ ] Capsaicin cream/patch [ ] Lidocaine patch/ointment/cream/gel [ ] Diclofenac gel [ ] Menthol/methyl-salicylate cream/patch +/- camphor [ ] Trolamine

ANTICONVULSANTS [ ] Carbamazepine [ ] Gabapentin [ ] Lamotrigine [ ] Levetiracetam [ ] Pregabalin [ ] Topiramate [ ] Valproate/valproic acid/divalproex MUSCLE RELAXANTS []Baclofen []Carisoprodol []Cyclobenzaprine []Metaxalone []Methocarbamol []Tizanidine

ANTIDEPRESSANTS [ ] Amitriptyline [ ] Desvenlafaxine [ ] Duloxetine [ ] Levomilnacipran [ ] Milnacipran [ ] Nortriptyline [ ] Venlafaxine

OTHERS [ ] Propranolol [ ] Verapamil [ ] Clonidine [ ] Calcitonin [ ] Bisphosphonate [ ] Memantine [ ] Ketamine

--

TRIPTANS [ ] Almotriptan [ ] Eletriptan [ ] Rizatriptan [ ] Sumatriptan [ ] Zolmitriptan

CGRP ANTAGONISTS [ ] Erenumab (AIMOVIG) [ ] Fremanezumab (AJOVY) [ ] Galcanezumab (EMGALITY) [ ] Eptinezumab (VYEPTI) [ ] Ubrogepant (UBRELVY) [ ] Rimegopant (NURTEC) [ ] Atogepant (QULIPTA) [ ] Zavegepant (ZAVZPRET)

OTHER HEADACHE MEDS [ ] Aspirin/Butalbital/Caffeine (FIORINAL, CIII) [ ] Acetaminophen/Butalbital/Caffeine (FIORICET) [ ] Ergotamine/dihydroergotamine

=
=================================================================== OBJECTIVE

=
=================================================================== --   ACTIVE PROBLEMS PER CPRS   ---
 * ACTIVE PROBLEMS (1 COLUMN)|

-  MEDICATION PROFILE   --- ALLERGIES/ADRs: |ALLERGIES/ADR| REMOTE ALLERGY/ADR: |RART|

INPATIENT MEDICATION REVIEW
 * DETAILED RECENT MEDS|

OUTPATIENT MEDICATION REVIEW
 * ACTIVE OPT MEDS|

RECENTLY EXPIRED OP MEDS:
 * RECENTLY EXP OP MEDS|

MEDICATION RECONCILIATION: 1.) I have:     [] Reviewed entire outpatient medication list      [] Conducted focused review of outpatient medication list with particular          attention paid to pain and pain-related medications
 * REMOTE ACTIVE MEDICATIONS|

2.) and:     [] the outpatient medication list accurately reflects the medications that          patient is currently taking, including any that may be provided from          non-VA sources, over the counter medications, nutritional or other          supplements. Medications reviewed to identify and address duplicity or          polypharmacy issues.      [] discrepancies were identified and noted above (see med list)

or [] unable to perform medication reconciliation [] med rec not applicable --  RELATED IMAGING   -

-  RELATED SURGERIES

-  PERTINENT CONSULTS/NOTES   -

PDMP  --

---  DRUG SCREENING / TESTING   --- URINE DRUG SCREENING:
 * UA DRUG SCREEN (LAST)|

DRUG TEST GENERAL (CONFIRMATORY):

ALCOHOL METABOLITES:

CDT-PANEL:

GGT:

---   VITALS    --- Age: |PATIENT AGE| y/o; |PATIENT SEX| Weight |PATIENT WEIGHT| Height |PATIENT HEIGHT| BMI: |BMI| IBW: |IBW| SCr  |CREATININE-G,J,D| BP: |BLOOD PRESSURE| Pulse: |PULSE| Temp: |TEMPERATURE| RR: |RESPIRATION| Pain: |PAIN|

PAIN TREND:

---   LABS    - RENAL: Estimated CrCl by Cockcroft-Gault: ~ mL/min based on ___ body weight & SCr of ____

BASIC METABOLIC PANEL:

ELECTROLYTES: - Ca - Mg - Phos

LIVER PROFILE:

PT & INR:

BLOOD COUNTS: - WBC - Plt - Hgb - Hct - MCV - RDW-SD

A1C%:

VITAMINS: - Vitamin D - Vitamin B12:

---   EKG    -- EKG (if pertinent for QTc prolonging meds)

=
=================================================================== ASSESSMENT

=
=================================================================== with a PMH of:
 * PATIENT FIRST & LAST NAMES| is a |PATIENT AGE| |PATIENT RACE| |PATIENT SEX|

Patient’s pain is best described as: [] acute  [] acute-on-chronic  [] chronic [] cancer-related [] not cancer-related [] both cancer- and not cancer-related [] nociceptive [] neuropathic

Current pain regimen includes:

Current limitations to treatment include:

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

There is evidence to support weight loss, smoking cessation, PT, exercise, Pain psychology, and non-opioid medications in treating chronic non-cancer pain.

The use of chronic opioids in non-cancer pain is not recommended. Long-term opioid use or escalation can induce a state of opioid-induced hyperalgesia in which the opioids can increase the perception pain. Additional long-term effects include tolerance, physical dependence, immune dysfunction and hypogonadism. As patients age, issues like cognition, bowel function, sedation, respiratory suppression and falls can become more problematic. Additional situations that increase the risk of opioids include opioid dose, concomitant benzodiazepines, and patient comorbidities that can complicate pain management (medical: COPD, OSA, obesity; mental health: depression, PTSD, insomnia; substance use disorder: alcohol, opioids, tobacco). Functioning will not improve without addressing other comorbidities that can worsen pain and/or pain perception or increase the risks of opioid therapy. For these reasons, pain conditions are most appropriately treated by non-opioid adjuvant medications that have opioid sparing characteristics.

The pain condition this veteran suffers from is best treated with a multidisciplinary approach. This involves an increase in physical activity to prevent de-conditioning and worsening of the pain cycle, psychological counseling (formal and/or informal) to address the co-morbid psychological effects of pain, as well as the use of non-opioid pain medications and interventional strategies. A carefully designed active treatment plan has a greater impact on pain, mobility, function and quality of life. There is emerging evidence that passive treatment strategies can harm patients by exacerbating fears and anxiety about being physically active when in pain, which can prolong recovery. Goals of therapy are objective improvement in function and realistic reduction in pain reports (30% improvement).

--   STOP-BANG Screener for OSA   --- [ ] Do you snore loudly (louder than talking or loud enough to be heard through    closed doors)? [ ] Do you often feel tired, fatigued, or sleepy during the day? [ ] Has anyone ever observed you stop breathing during your sleep? [ ] Do you have or are you being treated for high blood pressure? [ ] BMI > 35kg/m^2 [ ] Age > 50 [ ] Neck circumference > 16 in (40cm) [ ] Male gender

Each YES response = 1 point Low risk: 0 - 2 points		Moderate risk: 3 - 4 points		High risk: 5 – 8

High sensitivity (93%-100%) noted when using STOP-Bang questionnaire to detect moderate to severe and severe sleep disordered breathing in surgical population patients however low specificity noted at original cut-off of 3. Recent studies indicate total scores of 5-8 have higher specificity.

=
=================================================================== RECOMMENDATIONS

=
=================================================================== The provider of record for the controlled substance must document in the medical record the need and intended indication for the controlled substance being prescribed. The provider of record for the controlled substance should either include the necessary documentation in their own progress note or provide such information in an addendum to the CPP’s note

- OPIOIDS

- NON-OPIOID ANALGESICS

- OTHER

- NON-PHARMACOLOGIC > Anti-inflammatory diet > PT/OT > Pain psychology > Mindfulness > Heating pads / packs > Cold packs > Stretches / guided exercises > TENS unit > Chiropractor > MOVE! program > Aquatherapy > CPAP for OSA

- BOWEL REGIMEN

Implementation of recommendations is left to the provider's discretion. Thank you for the consult. **Please re-consult or contact our service if there are any further questions**

=
=================================================================== EDUCATION

=
=================================================================== Rationale for use, dosing instructions, side effects, and precautions of medications reviewed with patient in detail. Patient expressed understanding of the information provided, agreement with our plan of care, and was instructed to call in the event of any drug-related problem.

FUTURE APPOINTMENTS
 * FUTURE APPTS|

Follow-up: will continue to follow peripherally until pain is stabilized

Time spent: 90 min PharmD tool completed

--      ABBREVIATIONS -- OMEs = oral morphine equivalents CSSRS = Columbia Suicide Severity Rating Scale PEG = The Pain, Enjoyment of Life and General Activity PHQ = Patient Health Questionnaire AUDIT-C = Alcohol Use Disorders Identification Test PSEQ = Pain Self-Efficacy Questionnaire ORT = Opioid Risk Tool GAD-7 = General Anxiety Disorder HAM-D = Hamilton Depression Rating Scale


text: User talk:Nick Zoghbi

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Nick Zoghbi


text:
AI Art Generation Handbook/Node Prompting in Stable Diffusion

This type of "programming language" used in ComfyUI may overwhelm especially to the user whom is not familiar to before compared to other local UI (such as Auto1111, Foooocus, Oomost and such...) but if we bite the bullet and get through it, this may become easy.

As you can see, there are square blocks that are connected to one another using multiple coloured lines. In this context, the square blocks are known as "node" or "block diagram" and the multiple coloured lines are known as "wires" or "pipelines"

Control
Once you successfully installing ComfyUI, we learn how to control inside ComfyUI

Anatomy of the nodes
Every nodes have this 3 characteristic : Input Data Types (Red): The data flows into the nodes through this pipelines. Take note that the wire colours are matching to the input data types Output Data Types (Blue): The data flows out of nodes through this pipelines. Take note that the wire colours are matching to the output data types Control Parameter (Green): The results of the calculations performed within this node is largely influenced by the setting values inside control parameters

Basically, how this works is the whole nodes (as shown on the left) is basically a big "black box" where it takes data flow into the nodes and perform abstractions as well as the complex calculations (based on the control parameters within its nodes). After the calculations is done, the data is outputted to the next nodes for other processing.

Node Programming

To learn node programming, we can clear the whole workflow by clicking "Clear" button on the right. As seen on the diagram of the Text-2-Image of how it works, we can recreate the workflow.

Node Programming - Add in new nodes
To add new nodes, we can double click on any empty spaces in canvas and using search functions to search required block diagrams easier:

We add this few important block diagram

Node Programming - Connecting the nodes
We can start connecting the block diagrams by the pipelines.

''Note: Either way is OK but if you wanted to connect many to one nodes, the recommended method is to connect from the block diagrams input (left side) to the block diagrams output (right side) to minimises potential bugs and issues later on. See Connecting CLIP to K-Sampler parts ''

Let's start with the Load Checkpoint block diagrams and K-Sampler block diagrams. Click on the MODEL output (with lavender circle), you can notices that only model input is highlighted meanwhile the rest is grayed out (Meaning it is not connectable).

Another point to take note that, pay attention to the colour of input and output data types (Only both of same colours are able to be connected together)

Now, Load Check Point should be connected to K-Sampler through the MODEL pipelines.

There are CLIP to be connected. As mentioned above, the CLIP is related to Text Encoder, therefore, we can connect both CLIP Block Diagrams. But However, before we started to connect those, rename the Block Diagrams first into more appropriate name to prevent confusions. Right click on the block diagrams and click on "Properties Panel". There should be Title, Mode, Color and Node name for S&R, change the both Title and Node Name for S&R to more self explanatory names likme  as examples.

Note : We want to distinguish which block diagrams is for positive and for negative prompts. Connect the Load Checkpoints CLIP Output to both Positive and Negative Block Diagrams clip input (See yellow pipelines). Lastly connect the CLIP CONDITIONING output to positive and negative K-Sampler input (See orange pipelines)respective to the block diagrams names

You should have the connections as seen in the left screenshot by now

Note: There are potential bugs of connecting output data types CONDITIONING to input data types positive or negative, therefore reverse the sequence to connect from input positive / negative to output CONDITIONING.

As we can see now in Load Checkpoints block diagram, VAE block diagram is still unconnected. We can click on VAE output datatypes (red pipelines) to see which it can connect to, only VAE Decode Block Diagram vae input is highlighted.

Repeat the same in VAE Decode diagram block for sampler input datatypes (pink pipelines) to be connected to K-SAMPLER 's latent output data type

Finally, repeat the same for VAE Decode IMAGE output datatypes (blue pipelines) to be connected to images input datatypes in Save Images Block Diagrams.

You should get the similar diagrams as below and left only one connections between the K-Sampler and Empty Latent Image block diagrams to make it functional.

Connect the LATENT in Empty Latent Image to KSampler 's latent_image to finish the workflow.

Congrats, you finished the basic text to image workflow and have slightly better understanding on how this all works.

Node Programming - Generate Image
Try to click the "Queue Prompt" on the right button on the sidebar. You may not notice at first but the KSampler node is suddenly highlighted in green colour and then, there is green colour bar that starting to go across. It meant that the AI Art Models is working.

To know what does each of the setting meant, go to AI Art Generation Handbook/Stable Diffusion settings to learn more

Save Prompt Names as Image Ouput File Name
As we generate the images, we get the generic names of ComfyUI_0001.png as default, but do you know that with some additional steps , we can save prompt as filenames in ComfyUI ?

Here is how we do it very easily. On the Positive CLIP block diagram, right click on the block diagrams to search for the name of

Use the values that you key in earlier (in this case "Positive CLIP") and type the following into filename_prefix inside Save Image block diagram:

%Positive_CLIP.text% , refer to picture on the left for clarity.

From now onwards, you can see the filename changes to the saved images if the Save Image block diagram is not altered in anyway.

For more customisation on the name, here are the page  where we can get to change the custom names (especially prompt names)  for the ComfyUI output files.

Improve Image Generations Quality
SDXL and above are trained on 1024 x 1024 px images, therefore anything less than that will results in low quality of images as shown per this experiments here.

Therefore, on Empty Latent Image block diagram you can adjust both width and height up to 1024px to get more aesthetic image generations.


text:
User:Cremastra/common.css

.mw-parser-output span.cs1-maint {display: inline;} /* display Citation Style 1 maintenance messages */ .mw-parser-output span.cs1-hidden-error {display: inline;} /* display hidden Citation Style 1 error messages */

title: User:Cremastra/common.css


text: User:Cremastra/sandbox

title: User:Cremastra/sandbox


text:
Cookbook:Sweetfish

Sweetfish or ayu is a small fish with cucumber or melon scented-flesh and sweet flavour. The fish live in freshwater streams in Japan, where they are usually either fried, prepared as sushi, or broiled with salt. They are also eaten in Korea.

title: Cookbook:Sweetfish


text:
Cookbook:Phyllo Dough

Phyllo or filo dough is a variety of extremely thin pastry dough.

title: Cookbook:Phyllo Dough


text:
User talk:Cremastra

Cookbook formatting
Hi @Cremastra! I've fixed up your recent contribution to the Cookbook. In the future, you can use Cookbook:Policy/Ingredient template as a reference for ingredient pages. You can also take a look at ingredient pages from A-C as guidance (I'm working my way through them alphabetically). Let me know if you have any questions. Cheers! —Kittycataclysm (discuss • contribs) 14:04, 19 June 2024 (UTC)

 * Thanks a lot, . Cremastra (discuss • contribs) 14:05, 19 June 2024 (UTC)

Reviewer
Thanks for your contributions. You have recently been promoted to reviewer, meaning you can:
 * Review edits made by others
 * Use rollback
 * Move pages without leaving a redirect

Have a good day, --SHB2000 (discuss • contribs) 22:22, 12 July 2024 (UTC)

 * Thanks. Cremastra (discuss • contribs) 22:26, 12 July 2024 (UTC)

title: User talk:Cremastra


text: Category:Sweetfish recipes

title: Category:Sweetfish recipes


text: Template:Denest

title: Template:Denest


text: Category:Books needing denesting

title: Category:Books needing denesting


text:
Template:Denest/doc

Usage
This template should be added to the main page of books containing excessive/problematic nesting. It adds the book in question to Category:Books needing denesting.

title: Template:Denest/doc


text: User:GreenRedFlag

title: User:GreenRedFlag


text:
Wikijunior:Bugs/Other Intro

There are other small animals commonly called "bugs" that aren't insects or arachnids. They are only loosely related to each other, and in fact, categorizing animals as "bugs" or "not bugs" is very bad way to categorize things. Scientists prefer to look at physical traits animals have in common, and at how their genes are related.

Some of the animals that are usually called "bugs", aside from insects and arachnids, are centipedes, millipedes, woodlice, and earthworms. Centipedes and millipedes look similar, and they're closely related to each other, being both called "myriapods". Centipedes and millipedes are both arthropods, like insects and arachnids. Woodlice are the small arthropods with round shells and many legs you might find living under bricks or logs. They're actually crustaceans, like crabs, shrimp, and lobsters. Finally, earthworms aren't even arthropods at all. Unlike the other creatures in this book, they don't have hard shells to protect them. Earthworms are annelids, a big group of worm-like animals which also includes leeches.

This section is for the "leftover" bugs – the bugs that aren't insects or arachnids. They don't have much in common, because they aren't really one group. They're more like a miscellaneous collection.

title: Wikijunior:Bugs/Other Intro


text: Talk:Roman Culture/Roman Myths/Jupiter

title: Talk:Roman Culture/Roman Myths/Jupiter


text:
User:Cremastra/Spring and autumn

The poetically-named Spring and Autumn period – which takes its name from the Chinese classic text the Spring and Autumn Annals – marked the rise of city-states and set the stage for the Warring States period. The period is generally agreed to begin in 771, and end almost 300 years later in 481 , when the ultimate collapse of the Jin state led to its fission into three factions – known as the Partition of Jin.

Background
The ruling Shang dynasty, with its origins along the Yellow River in east-central China, ruled for some six hundred years, starting the sixteenth century. The Shang are the earliest Chinese dynasty in the archaeological record. They were succeeded and conquered by the Zhou, a people from up the Yellow River.

The rising Zhou began to clash with Shang allies on the fringes of the declining Shang dynasty's control. But when a new Zhou king came to power, he began to menace the Shang more directly, and lead an alliance of anti-Shang states. These preparations culminated in the Battle of Muye, which was the defining event of this conflict: the rebel Zhou fought and defeated the Shang, and seized their capital. With just 45,000 men and a few hundred wagons, the Zhou were initially hugely outnumbered – even though most of the Shang forces were at war to the east, Di Xin of Shang organized some 170,000 troops. But Di Xin made a mistake: many of his fighters were slaves, and he thought that despite low troop morale, his army's superior numbers could, if not defeat, then at least slow down the rebels until reinforcements could arrive. He was wrong. The majority of his Shang troops fled or joined the Zhou, and the few who did were easily overwhelmed by the Zhou forces. The Shang dynasty had been defeated. Some of the Shang officials were incorporated into the Zhou government; Di Xin was not – he committed suicide.

In the following centuries, the Zhou dynasty's centralized power over central China began to decline, and largely-independent feudal states began to rise in power.

Classical sources

 * The Chunqiu (春秋), the Spring and Autumn Annals or more literally, "Springs and Autumns", is a chronological history. It is a rough and frequently terse account that follows the state of Lu across twelve rulers and more than two centuries. It is considered a classic of Confucianism.
 * The Zhou Zhuan (左傳) was a narrative history, and a commentary on the Spring and Autumn Annals, providing more extensive accounts and context to the latter. Its title can be translated as "Zuo's Commentary". Like the Spring and Autumn Annals, it is considered as one of the classics of Confucianism.

States
The states at the start of the Spring and Autumn period could be divided into Zhou and non-Zhou. The Zhou eastern states claimed descent from members and relatives of the Zhou royal family; nearly 150 such states are mentioned in the Zhou zhuan.

Qi
Located on Shandong peninsula – the wide point of land which extends into the Yellow Sea towards Korea – and some of the surrounding land to the north, in present-day Shandong and Hainan provinces, the regional state of Qi (齊; “chi”) was one of the most powerful states. It was founded in the tenth century when the Zhou dynasty overthrew the Shang dynasty.

title: User:Cremastra/Spring and autumn


text:
User:Cremastra/common.js

mw.loader.load('//en.wikipedia.org/w/index.php?title=User:Joeytje50/JWB.js/load.js&action=raw&ctype=text/javascript');

title: User:Cremastra/common.js


text:
Moving objects in retarded gravitational potentials of an expanding spherical shell/Brief historical review

Brief historical review
The finiteness of the propagation speed of gravity and its influence on gravitational forces was originally published by the Austrian astronomer Josef von Hepperger (1855–1928) in 1888 in Vienna.

The German physicist Paul Gerber (1854–1909) published 1898 his paper on "The spatial and temporal expansion of gravity", where he established that the perihelion precession of the planet Mercury is related to the propagation speed of gravity, which is quite close to that of electromagnetic radiation. His formula for the perihelion precession of the planet Mercury wasn't known to Albert Einstein (1879–1955), but six years after Paul Gerber's death, Einstein found an identical formula in his publication "Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie" (English: "Explanation of Mercury's perihelion motion from the general theory of relativity") by applying the laws of General Relativity. However, the contemporary scientists couldn't reproduce the derivation of Paul Gerber for the formula, and furthermore, they stated that some of the prerequisites used by him were wrong.

Shortly before his early death the German astronomer and physicist Karl Schwarzschild (1873–1916) published a paper on "Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie" (English: "On the gravitational field of a sphere of incompressible fluid according to Einstein's theory"), where he described how to compute the smallest possible radius for a sphere with a given mass. He found the radius for a sphere with the mass of the sun to be three kilometres. In recognition of his achievement, the corresponding radius is now called the Schwarzschild radius. It is interesting to notice that Schwarzschild also made important contributions to retarded potentials in electrodynamics already in 1903. According to the electrokinetic potential he claimed:

Es sind in jedem Raumelement die Werte der Dichte und der Geschwindigkeit zu benutzen, welche dort zu einer um die Lichtzeit zurückliegenden Epoche galten.

In each spatial element, the values of density and velocity are to be used, which were valid there at an epoch around light time ago.

The Belgian theologian and physicist Georg Lemaître SJ (1894–1966) is regarded as the founder of the Big Bang theory. In 1931, he introduced the term "atome primitif" (English: "primordial atom") to describe the hot initial state of the universe. Already in 1927 he wrote as the second conclusion of his publication about "a homogeneous universe of constant mass and increasing radius, accounting for the radial velocity of extra-galactic nebulae" with reference to the investigations of Edwin Hubble (1889–1953) in 1926:

Le rayon de l'univers croit sans cesse depuis une valeur asymptotique $$R_0$$ pour $$t = - \infty$$

The radius of the universe increases without limit from an asymptotic value $$R_0$$ for $$t = - \infty$$

In 1933 the Swiss astronomer Fritz Zwicky (1898–1974) obeserved a gravitational anomaly in the Coma galaxy cluster, and he coined the term dark matter (in German: "dunkle Materie") for the cause of this anomaly.

In 1953 the German astrophysicist Erwin Finlay-Freundlich (1885–1964) derived a blackbody temperature for intergalactic space of 2.3 Kelvin according to his theory of tired light. The German-British mathematician and physicist Max Born (1882–1970) immediately recommended taking the problem seriously and pursuing it further.

In 1965 the cosmic microwave background (CMB) was discovered by US-American radio astronomers Arno Allan Penzias (1933–2024) and Robert Woodrow Wilson (born 1936). It has a thermal black body spectrum at a very low temperature of about 2.7 Kelvin. Since the cosmic microwave background was mainly created in the visible area of the electromagnetic spectrum, it must have undergone a strong wavelength extension on its way to the observers. This can happen because of two main reasons:

 * The origin of the radiation moves away from us very quickly, which will cause an increase of the wavelength according to the Doppler effect described by Christian Doppler (1803–1853) in 1842.
 * There is a huge mass behind the origin of the radiation, which will cause an increase of the wavelength according to the gravitation and the relativity principle of Albert Einstein (1879–1955) of 1907.

The redshift factor $$z$$ is defined as a relation between an emitted wavelength $$\lambda_0$$ and an observed wavelength $$\lambda$$ of electromagnetic radiation:

 * $$z = \frac {\Delta \lambda} {\lambda_0} = \frac {\lambda - \lambda_0} {\lambda_0}$$

The redshift of the cosmic microwave background was found to be $$z \approx 1089$$ in 2003, which is an extremely high value. The age of the universe at the time this background radiation was created by hydrogen atoms has been estimated at around 379,000 years.

Observations of distant type Ia supernovae published by both the Supernova Cosmology Project as well as the High-Z Supernova Search Team in 1998 show that the relative expansion of the universe is accelerating. For the analysis the astronomers Saul Perlmutter, Brian P. Schmidt and Adam Riess were awarded the Nobel Prize in Physics in 2011.

From 2001 to 2010 the NASA spacecraft Wilkinson Microwave Anisotropy Probe (WMAP) was investigating the cosmic microwave background. Its measurements led to the current Standard Model of Cosmology. According to this model the universe currently consists of less than 5 percent ordinary baryonic matter; about 24 percent cold dark matter (CDM) that interacts only weakly with ordinary matter and electromagnetic radiation; and more than 70 percent of dark energy that is used to explain the accelerated expansion of the universe. These data were more or less confirmed by the Planck space observatory that was operated by the European Space Agency (ESA) from 2009 to 2013.

In 2013 the Indian researcher Chandrakant Raju proposed to apply the retarded gravitation theory (RGT) to explain the flyby anomaly of spacecrafts in the graviational field of the earth as well as the acceleration of masses in the retarded gravitational fields of spriral galaxies.

In 2016, researchers from the gravitational-wave observatory LIGO reported the first direct measurement of gravitational waves generated by the collision of two black holes (gravitational wave event GW150914). Already in 2017 the Nobel Prize in Physics was awarded for the first direct observation of gravitational waves.

On 17th August 2017 a merging binary neutron star was observed independently and simultanuously by the Advanced LIGO and Virgo detectors (gravitational wave event GW170817) and the Fermi Gamma-ray Burst Monitor as well as the Anticoincidence Shield for the Spectrometer for the International Gamma-Ray Astrophysics Laboratory (gamma ray burst event GRB 170817A). These observations prove that the propagation velocity of gravitational waves must be extremly close to that of electromagnetic waves.

In January 2024 the very young and very far galaxis JADES-GS-z14-0 was found with the Near-Infrared Spectrograph (NIRSpec) of the James Webb Space Telescope (JWST). This galaxy was observed in a state 290 million years after the big bang. Its redshift measured with the well-known Lyman alpha break at a wavelength about 1.8 micometres has a very high value of a good 14, which is a very high value for an astronomical object, but much lower than that of the cosmic microwave background.


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Moving objects in retarded gravitational potentials of an expanding spherical shell/Classical approach

Classical approach
This Wikibook assumes a huge amount of dark matter that is in a huge shell behind the visible part of the universe. This matter is rather cold, and it can be assumed that it is in a in thermal equilibrium. Therefore, the electromagnetic radiation emitted by this shell can be represented by the Planck radiation law. The emission of such a cold source of electromagnetic radiation is not visible and additionally the radiation has experienced a strong redshift due to the relative velocity of the fast expanding emitting shell as well as due to the gravitation of the mass of this shell.

The gravitational force $$F$$ between two masses $$m$$ and $$M$$ in a distance of $$s$$ is given by Newton's law of universal gravitation with the gravitational constant $$G$$:

 * $$F = G \, \frac {m \cdot M} {s^2}$$

In a thought experiment it is possible to investigate the behaviour of a mass $$m$$ that is located within a spherical shell with the mass $$M$$. The approach described in 1687 by Isaak Newton (1643–1727 in Gregorian dates) is called Newton's shell theorem. According to this theorem a mass $$m$$ within a homogeneous and spherically symmetric shell with a mass $$M$$ experiences no net gravitational force. Additionally, this is regardless of the location of the mass $$m$$ within the shell as well as of its velocity.

This behaviour can be easily explained with the following consideration: let a mass $$m$$ be within a homogeneous and spherically symmetric shell with the radius $$r$$ and the areal density $$\rho_a$$. Then the area $$A$$ and the mass $$M$$ of the shell can be expressed as:

 * $$A = 4\pi \cdot r^2$$

 * $$M = \int\limits_{A} \mathrm d M = \int\limits_{A} \rho_a \cdot \mathrm d A = \rho_a \cdot 4\pi \cdot r^2$$

If we take an axis symmetrical double cone where the tips of the two cones are located at the position of the mass $$m$$ within a homogeneous and spherically symmetric shell, we get the situation shown in the adjacent figure.

For the two infinitesimally small area elements $$\mathrm d A_1$$ and $$\mathrm d A_2$$ in the distances $$s_1$$ and $$s_2$$ of a mass $$m$$ with the same infinitesimally small angle elements $$\mathrm d \alpha$$:

 * $$\mathrm d A_1 = s_1^2 \cdot \mathrm d \alpha^2$$

 * $$\mathrm d A_2 = s_2^2 \cdot \mathrm d \alpha^2$$

For the appropriate masses $$\mathrm d M_1$$ and $$\mathrm d M_2$$ of the two area elements:

 * $$\mathrm d M_1 = \rho_a \cdot \mathrm d A_1 = \rho_a \cdot s_1^2 \cdot \mathrm d \alpha^2$$

 * $$\mathrm d M_2 = \rho_a \cdot \mathrm d A_2 = \rho_a \cdot s_2^2 \cdot \mathrm d \alpha^2$$

With Newton's law of universal gravitation, we can express the two infinitesimally small gravitational forces $$\mathrm d F_1$$ and $$\mathrm d F_2$$ due to the masses of the two area elements:

 * $$\mathrm d F_1 = G \, \frac {m \cdot \mathrm d M_1} {s_1^2} = G \, \frac {m \cdot \rho_a \cdot s_1^2 \cdot \mathrm d \alpha^2} {s_1^2} = G \cdot m \cdot \rho_a \cdot \mathrm d \alpha^2$$

 * $$\mathrm d F_2 = G \, \frac {m \cdot \mathrm d M_2} {s_2^2} = G \, \frac {m \cdot \rho_a \cdot s_2^2 \cdot \mathrm d \alpha^2} {s_2^2} = G \cdot m \cdot \rho_a \cdot \mathrm d \alpha^2$$

It is obvious that both forces have the same value, and since they must be exactly directed to opposite directions, they add to zero. In other words: a mass within a spherical shell does not experience any net gravitational force at all.

It is worth noting that this classical approach assumes the infinite propagation speed of gravitational waves, which is applicable for static but not for dynamic situations as well as huge systems.


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Moving objects in retarded gravitational potentials of an expanding spherical shell/Retarded gravitational potentials

Retarded gravitational potentials
Already the German physicist and astronomer Karl Schwarzschild (1873–1916) described retarded potentials for elecrodynamic fields (he still used the term "electrokinetic potential") in 1903. Retarded potentials are a mathematical description of potentials in a field theory in which a field quantity propagates at a finite speed (speed of light) and not instantaneously. They occur in the investigation of time-dependent problems, such as the radiation of electromagnetic waves, but also in the propagation of gravitational waves.

Objects that are moving towards the outer rim of the universe will experience retarded gravitational potentials by the expanding dark matter that can be assumed at the out rim of the universe. Therefore, the appropriate gravitational forces will have delayed effects, and due to the large distances and the finite velocity of gravitational wave propagation they also will be weaker in the direction of the former location of the objects. As a result, the net gravitational force is directed in the direction of movement of these objects, and therefore, all objects that move outwards will be accelerated in the direction of their own movement, which would become observable as an accelerated expansion of the visible universe.

The gravitational potential $$\Phi$$ due to a mass $$M$$ in the distance $$s$$ is given by:

 * $$\Phi(s) = - \frac {G \cdot M} {s}$$

The gravitational force $$F$$ to another mass $$m$$ results as follows:

 * $$F = - m \, \frac {\mathrm d \Phi(s)} {\mathrm d s} = G \, \frac {m \cdot M} {s^2}$$

The time-depending potential $$\Phi(\vec x,t) $$ is the solution of the inhomogeneous wave equation, where $$v(\vec x, t)$$ is the inhomogeneity, and $$c$$ is the speed of the propagation of the waves. For gravitational forces we can consider it as equal to the vacuum speed of electromagnetic waves:

 * $$\frac 1 {c^2} \, \frac {\partial^2 \Phi(\vec x, t)} {\partial t^2} - \Delta\Phi(\vec x, t) = \Box \Phi(\vec x, t) = v(\vec x, t)$$,

where $$\Delta = \nabla^2$$ is the Laplace operator, $$\Box$$ is the D’Alembert operator.

The solution of the inhomogeneous wave equation is called retarded potential, and in three dimensions it can be given as:

 * $$\Phi_{\text{ret}}(\vec x, t) = \frac {1} {4 \pi} \int \frac {v(\vec s, t_{\text{ret}})} {|\vec x - \vec s|} \, \mathrm d^3 s$$

The retardation is to be interpreted in such a way that a source element $$v(\vec x, t_{\text{ret}})$$ at the point $$\vec s$$ and at the time $$t_{\text{ret}}$$ only influences the potential at the distant point of impact at $$\vec x$$ at a later time $$t$$:

 * $$t = t_{\text{ret}} + \frac {|\vec x - \vec s|} {c}$$

 * $$t_{\text{ret}} = t - \frac {|\vec x - \vec s|} {c}$$

$$t_{\text{ret}}$$ is called the retarded time. At the location $$\vec x$$ and at the time $$t$$ the retarded potential only depends on the inhomogeneity $$v$$ in the retarding back cone of the location. This inhomogeneity has a retarded effect to the solution, and it is delayed with the wave velocity $$c$$.

In a shell

In the simplified example in the adjacent figure the source term is the linear mass density $$\lambda_m$$ that is not time-dependant and only exists in the circle of the outer rim with the constant radius $$r$$ and its centre at $$O$$:

 * $$x_O = 0$$

 * $$y_O = 0$$

 * $$x^2 + y^2 = r^2 \rarr \lambda_m = \text{ constans}$$

In all other locations within the plane the linear mass density $$\lambda_m$$ is zero:

 * $$x^2 + y^2 \neq r^2 \rarr \lambda_m = 0$$

All mass elements outside the regarded plane have an symmetrical effect to the mass, and therefore, in these locations the contribution of the mass elements to the potential can be neglected for the determination of inhomogeneity:

 * $$z \ne 0 \rarr \lambda_m = 0$$

Furthermore, the mass element $$\mathrm d M$$ on the homogenous circumference $$C$$ of a circle with the radius $$r$$ is given by:

 * $$\mathrm d M = \lambda_m \cdot \mathrm d C = \lambda_m \cdot r \cdot \mathrm d \alpha$$

 * $$M = \int d M = \int\limits_{0}^{2\pi} \lambda_m \cdot r \cdot \mathrm d \alpha = \lambda_m \cdot 2 \pi \cdot r = \lambda_m \cdot C$$

The cosine formula gives us the relation between the location of the mass point $$m$$ in the horizontal distance $$x$$ of the origin $$\bigcirc$$ of the circle with the radius $$r$$, when the mass element on the circle $$\mathrm d M$$ is in the direction of the angle $$\alpha$$ and the distance $$s$$:

 * $$r^2 = x^2 + s^2 - 2xs \, \cos (\pi - \alpha) = x^2 + s^2 + 2xs \, \cos \alpha$$

In the normalised standard form of the quadratic equation, we get:

 * $$s^2 + 2xs \, \cos \alpha + x^2 - r^2 = 0$$

The solution for the distance $$s$$ is:

 * $$s = \sqrt {x^2 \, \cos^2 \alpha - x^2 + r^2} - x \, \cos \alpha$$

It is obvious that the following simplifications are valid:

 * $$x = 0 \rarr s = r$$
 * $$\alpha = 0 \rarr s = r - x$$
 * $$\alpha = \pm \frac {\pi} {2} \rarr s = \sqrt {x^2 + r^2}$$
 * $$\alpha = \pi \rarr s = r + x$$

The origin of the coordinate system can be also shifted to the mass $$m$$:

 * $$x_O = - x(t)$$

 * $$y_O = 0$$

The retarded time $$t_{\text{ret}}$$ and the retarded potential $$\Phi_{\text{ret}}(\vec x, t)$$ are given as follows, where $$c$$ represents the propagation speed of the potentials:

 * $$t_{\text{ret}} = t - \frac {s} {c}$$

 * $$\Phi_{\text{ret}}(\vec x, t) = \frac {1} {4 \pi} \int\limits_{0}^{2\pi} \frac {v (s, t_{\text{ret}})} {s} \, \mathrm d \alpha$$

Illustration
In a simplified example we only consider the infinitesimally small angles $$\mathrm d \alpha$$ in the origin of a spherical shell with the radius $$r$$ and the areal density $$\rho_A$$.

Symmetrical geometry
If the mass $$m$$ is in the centre of a spherical shell with the radius $$r = s_0$$ we have the following situation:

 * Both angle elements $$\mathrm d \alpha$$ are equal.
 * Both distances $$s_0$$ are equal to $$r$$.
 * Both areal elements $$\mathrm d A_0$$ are equal.
 * Both mass elements $$\mathrm d M_0$$ are equal.
 * The retarded times of the gravitational potentials of both mass elements $$t_{ret} = \frac {s_0} {c} = \frac {r} {c}$$ are equal.

In this situation the mass $$m$$ does not experience any acceleration in the classical approach (see above) or if its velocity is zero.

Moving masses
The situation is changed, if the masses are moving starting at $$t = t_0$$ within a time span of $$\Delta t$$. The mass $$m$$ moves with the velocity $$v_m > 0$$ to the right and the two mass elements $$\mathrm d M_1 \propto \mathrm d A_1$$ and $$\mathrm d M'_1 \propto \mathrm d A'_1$$ with the radial velocity $$v_M > 0$$:

At the time $$t_1$$ the mass has moved with the velocity $$v_m > 0$$ the distance $$\Delta s$$ to the right:

 * $$t_1 = t_0 + \Delta t$$

 * $$\Delta s = v_m \cdot \Delta t$$

The spherical shell has expanded with the velocity $$v_M > 0$$ and gained an increased radius:

 * $$\Delta r = v_M \cdot \Delta t$$

 * $$s_1 = s_0 + \Delta s + \Delta r = s_0 + v_m \cdot \Delta t + v_M \cdot \Delta t = s_0 + (v_M + v_m) \, \Delta t$$

 * $$s'_1 = s_0 - \Delta s + \Delta r = s_0 - v_m \cdot \Delta t + v_M \cdot \Delta t = s_0 + (v_M - v_m) \, \Delta t$$

Therefore:

 * $$s_1 > s'_1$$

This means that the distance of mass element $$\mathrm d M_1$$ to the mass $$m$$ is always greater than the distance of mass element $$\mathrm d M'_1$$ to the mass $$m$$.

For the two retarded times for these distances to the location of $$m$$ we get:

 * $$t_{\text{ret},1} = t_1 - \frac {s_1} {c}$$

 * $$t'_{\text{ret},1} = t_1 - \frac {s'_1} {c}$$

With $$t_0= 0$$ and therefore $$t_1 = \Delta t$$:

 * $$t_{\text{ret},1} = \left( 1 - \frac {v_M + v_m} {c} \right) \cdot \Delta t - \frac {s_0} {c}$$

 * $$t'_{\text{ret},1} = \left( 1 - \frac {v_M - v_m} {c} \right) \cdot \Delta t - \frac {s_0} {c}$$

Therefore:

 * $$t'_{\text{ret},1} > t_{\text{ret},1}$$

This means that the retarded time of mass element $$\mathrm d M'_1$$ is always later than the retarded time of mass element $$\mathrm d M_1$$.

Common case

In the adjacent diagram there are three mass points that move in space. Their speed is given as follows:

 * Outer mass element top (green): $$v_M$$
 * Mass point in between (blue): $$\color{Blue}v_m$$
 * Outer mass element botton (green): $$-v_M$$

Their time-depending location is given by these three functions for their x-coordinates:

 * $$\color{Blue}x_m(t) = v_m \cdot t$$

 * $$\color{OliveGreen}x(t_r) = -v_M \cdot t_r$$

 * $$\color{OliveGreen}x'(t'_r) = v_M \cdot t'_r$$

The time-depending effective distances $$\color{OliveGreen}s(t)$$ and $$\color{OliveGreen}s'(t)$$ between the outer mass elements and the mass point in between them is the difference of the corresponding x-coordinates and linked to the propagation velocity of the interacting waves $$\color{OliveGreen}c$$ as follows:

 * $$\color{OliveGreen}s(t) = x_m(t) - x(t_r) = (t_r - t) \cdot c$$

 * $$\color{OliveGreen}s'(t) = x'(t'_r) - x_m(t) = (t'_r - t) \cdot c$$

As a result, the corresponding retarded times $$t_r$$ and $$t'_r$$ for the two outer mass elemens are:

 * $$t_r = \frac {c - v_m} {c + v_M} \cdot t$$

 * $$t'_r = \frac {c + v_m} {c + v_M} \cdot t$$

And therefore:

 * $$x(t_r) = -v_M \cdot \frac {c - v_m} {c + v_M} \cdot t$$

 * $$x'(t'_r) = v_M \cdot \frac {c + v_m} {c + v_M} \cdot t$$

And:

 * $$\color{OliveGreen}s(t) = \left( v_M \cdot \frac {c - v_m} {c + v_M} + v_m \right) \cdot t$$

 * $$\color{OliveGreen}s'(t) = \left( v_M \cdot \frac {c + v_m} {c + v_M} - v_m \right) \cdot t$$

These are the time-depending effective distances for the gravitational potentials of the outer mass elements at their retarded times that have a simultaneous effect to the mass in between them at the time $$t$$.

For $$t > 0$$ and $$c > v_M > v_m > 0$$ and according to the diagram we can make the following assumption for the comparison of the effective distances:

 * $$t'_r > t_r$$

 * $$\frac {c + v_m} {c + v_M} \cdot t > \frac {c - v_m} {c + v_M} \cdot t$$

 * $$v_m > -v_m$$ quod erat demonstrandum

This means that the retarded time of the upper mass element is always later than the retarded time of the lower mass element.

As well as:

 * $$s(t) > s'(t)$$

 * $$v_M \cdot \frac {c - v_m} {c + v_M} + v_m > v_M \cdot \frac {c + v_m} {c + v_M} - v_m$$

 * $$v_M \cdot \frac {- v_m} {c + v_M} + v_m > v_M \cdot \frac {v_m} {c + v_M} - v_m$$

 * $$2 \cdot v_m > 2 \cdot v_M \cdot \frac {v_m} {c + v_M}$$

 * $$1 > \frac {v_M} {c + v_M}$$ quod erat demonstrandum

Since $$c + v_M$$ is always greater than $$v_M$$, the assumption is proven.

This means that the effective distance of the moving mass in between them to the lower mass element is always greater than to the moving upper mass element. Finally, it can be stated that the absolute value of the retarded gravitational potential at the location of the moving mass in between them is always greater for the upper mass element than for the lower mass element, if both mass elements have the same value $$M$$:

 * $$\Phi(t_r) = - \frac {G \cdot M} {s(t)}$$

 * $$\Phi'(t'_r) = - \frac {G \cdot M} {s'(t)}$$

 * $$|\Phi(t_r)| < |\Phi'(t'_r)|$$

The moving mass $$m$$ experiences the corresponding retarded forces:

 * $$F = - G \, \frac {m \cdot M} {s(t)^2}$$

 * $$F' = + G \, \frac {m \cdot M} {s'(t)^2}$$

The net force to the mass $$m$$ is the sum of both:

 * $$F_{net} = F' + F = G \cdot m \cdot M \cdot \left( \frac {1} {s'(t)^2} - \frac {1} {s(t)^2} \right)$$

For the acceleration $$a$$ of the mass $$m$$ we find:

 * $$a = \frac {F_{net}} {m} = G \cdot M \cdot \left( \frac {1} {s'(t)^2} - \frac {1} {s(t)^2} \right)$$

Since $$s'(t) < s(t)$$ the net force $$F_{net}$$ as well as the acceleration $$a$$ are positive, and the mass experiences an acceleration to positive x-values, i.e. in the direction of its movement.

Special case

Let us have a look at the following special case:

 * $$c = 1$$

 * $$v_M = \frac {1} {2}$$

 * $$v_m = \frac {1} {4}$$

Their time-depending location is given by these three functions for their x-coordinates:

 * $$\color{DarkBlue}x_m(t) = v_m \cdot t = \frac {t} {4}$$

 * $$\color{Red}x(t) = -v_M \cdot t = - \frac {t} {2}$$

 * $$\color{Gray}x'(t) = +v_M \cdot t = + \frac {t} {2}$$

The time-depending effective distances $$\color{Green}s(t)\color{Black}$$ and $$\color{Green}s'(t)\color{Black}$$ between the outer mass elements and the mass point in between them is the difference of the corresponding x-coordinates:

 * $$\color{Green}s(t) = x_m(t) - x(t_r) = \frac {t} {4} + \frac {t_r} {2}\color{Black}$$

 * $$\color{Green}s'(t) = x'(t'_r) - x_m(t) = \frac {t'_r} {2} - \frac {t} {4}\color{Black}$$

As a result, the corresponding retarded times $$\color{Dandelion}t_r$$ and $$\color{CornflowerBlue}t'_r$$ for the two outer mass elemens are:

 * $$\color{Dandelion}t_r = \frac {\frac {3} {4}} {\frac {3} {2}} \cdot t = \frac {1} {2} \cdot t$$

 * $$\color{CornflowerBlue}t'_r = \frac {\frac {5} {4}} {\frac {3} {2}} \cdot t = \frac {5} {6} \cdot t$$

Therefore:

 * $$\color{Dandelion}x(t_r) = -\frac {1} {2} \cdot \frac {\frac {3} {4}} {\frac {3} {2}} \cdot t = -\frac {1} {4} \cdot t$$

 * $$\color{CornflowerBlue}x'(t'_r) = \frac {1} {2} \cdot \frac {\frac {5} {4}} {\frac {3} {2}} \cdot t = \frac {5} {12} \cdot t$$

 * $$\color{Dandelion}s(t) = \left( \frac {1} {2} \cdot \frac {\frac {3} {4}} {\frac {3} {2}} + \frac {1} {4} \right) \cdot t = \frac {1} {2} \cdot t$$

 * $$\color{CornflowerBlue}s'(t) = \left( \frac {1} {2} \cdot \frac {\frac {5} {4}} {\frac {3} {2}} - \frac {1} {4} \right) \cdot t = \frac {1} {6} \cdot t$$

Thought experiment
In a thought experiment we look at the following situation, where $$c$$ is the propagation speed of the gravitational waves. The time line starts at $$t = 0$$, and the effect of the retarded potentials is synchronised with the occurence of the moving mass. The left moving mass element is regarded at $$t = 0$$ and $$t = 1$$, the right moving mass element is regarded at $$t = 0$$ and $$t = 2$$, whilst the moving mass $$m$$ is regarded at $$t = 8$$ and $$t = 11$$, when the retarded gravitational potentials have their effect to the mass.

 * $$v_M = c$$

 * $$v_m = \frac {v_M} {3}$$

 * $$s = c \cdot t$$

In the following diagram the velocity of the waves is normalised and used without unit:

 * $$c = 1$$

And therefore, only for simplification and without any units, too:

 * $$s = t$$

The effective distances $$s_0$$ for the gravitational potential at $$t_0 = 8$$ are both equal:

 * $$s_0 = t_0 - t_{\text{ret,0}} = 8$$

At the time $$t' = 11$$ the mass experiences the retarded potentials on the left-hand side (distance to mass element is $$s_1$$) and at the right-hand side (distance to mass element is $$s_2$$):

 * $$s_1 = t' - t_{\text{ret,1}} = 10$$

 * $$s_2 = t' - t_{\text{ret,2}} = 9$$

All inhomogeneities contribute to the retarded potential at the location of the mass with the value they had at the retarded times $$t_{\text{ret,0}}$$, $$t_{\text{ret,1}}$$ and $$t_{\text{ret,2}}$$:

 * $$t_{\text{ret,0}} = t_0 - \frac {s_0} {c} = 8 - \frac {8} {1} = 0$$, this corresponds to the effective time of the two area elements $$\mathrm d A_0$$.

 * $$t_{\text{ret,1}} = t' - \frac {s_1} {c} = 11 - \frac {10} {1} = 1$$, this corresponds to the effective time of the area element $$\mathrm d A_1$$ on the left.

 * $$t_{\text{ret,2}} = t' - \frac {s_2} {c} = 11 - \frac {9} {1} = 2$$, this corresponds to the effective time of the area element $$\mathrm d A_2$$ on the right.

For a mass $$m$$ moving from the centre of a shell to the right the angle element $$\mathrm d \alpha_1$$ to the left becomes smaller than the original angle element $$\mathrm d \alpha_0$$, and the angle element $$\mathrm d \alpha_2$$ to the right becomes greater than the original angle element $$\mathrm d \alpha_0$$:

 * $$\mathrm d \alpha_2 > \mathrm d \alpha_0 > \mathrm d \alpha_1$$

The following applies to the appropriate mass elements:

 * $$\mathrm d M_0 = \mathrm d M_1 = \mathrm d M_2 = \rho_a \cdot \mathrm d A_0 = \rho_a \cdot 4\pi \cdot s_0^2 \cdot \mathrm d \alpha_0^2$$

For the net force $$\mathrm d F$$ to the mass $$m$$:

 * $$\mathrm d F = \mathrm d F_2 + \mathrm d F_1 = G \cdot m \cdot \mathrm d M_0 \cdot \left( \frac {1} {s_2^2} - \frac {1} {s_1^2} \right)$$

For the acceleration $$\mathrm d a$$ of the mass $$m$$ we find:

 * $$\mathrm d a = \frac {\mathrm d F} {m} = G \cdot \mathrm d M_0 \cdot \left( \frac {1} {s_2^2} - \frac {1} {s_1^2} \right)$$

Since $$s_2 < s_1$$ the net force $$\mathrm d F$$ is positive, and the mass experiences an acceleration $$\mathrm d a$$ to the right, i.e. in the direction of its movement. This result is absolutely inline with the findings above in the section "Common case" above.

Furthermore, it is noteworthy to state that the acceleration $$\mathrm d a$$ is proportional to the areal density $$\rho_a$$ of the expanding shell:

 * $$\mathrm d a \propto \rho_a$$

Nevertheless it should be noted that the areal density is decreasing with the expansion and the increasing radius of the outer shell.


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Moving objects in retarded gravitational potentials of an expanding spherical shell/Gravitational redshift

Schwarzschild radius

The Schwarzschild radius $$r_S$$ of any mass $$M$$ in a Schwarzschild sphere (typically a black hole) is given by:

 * $$r_S = \frac {2 \, G \, M} {c^2}$$

Apart from its mass, the Schwarzschild radius is only depending on two natural constants:

 * Gravitational constant: $$G = 6.67 \cdot 10^{-11} \frac {\text {m}^3} {\text {kg} \, \text {s}^2}$$
 * Speed of light: $$c = 299792458 \frac {\text {m}} {\text {s}}$$

The gravitational redshift z of photons emitted to the opposite direction of the Schwarzschild sphere and with a distance to the centre of the sphere $$r$$can be computed by the following formula:

 * $$z = \frac {1} {\sqrt {1 - \frac {2 \, G \, M} {r \, c^2}}} - 1 = \frac {1} {\sqrt {1 - \frac {r_S} {r}}} - 1$$

The redshift of very far objects such as the galaxy JADES.GS.z14-0 has a value of more than 14, and this is much larger than expected. The possible high speed of such a galaxy is not sufficient for such a high value. The distance of this galaxy is given by 13.5 billion light-years, and its age is assumed to be 290 million years after Big Bang.

The cosmic microwave background even has a redshift value of 1089, which is extremely high. It is associated with the first hydrogen atoms that occurred some 380,000 years after the Big Bang.

For comparison:

The Schwarzschild radius of the universe can be computed by its mass, too:

 * $$r_{S, universe} = \frac {2 \, G \, M_{universe}} {c^2} \approx 4.4 \cdot 10^{26} \text {m} \approx 46.6 \cdot 10^{9} \text {ly}$$

In a shell

However, a gravitational redshift can not only occur outside of spheres, but also within in a hollow spherical cap. To estimate its gravitational redshift, the effective mass $$M_{eff}$$ of such a cap can be integrated for any point within the cap. The corresponding effect can be described by a Schwarzschild sphere with the Schwarzschild radius.

For very fast-moving objects we can assume that they only experience the retarded gravitational potentials of the mass elements in front of them, since the backward potentials are even much more retarded, and therefore, contribute only weakly to the net gravity. In the following section only the arc of the shell cap is considered. The gravitational forces rectangular to the direction of movement are very small and can be neglected. This also holds for the gravitational forces behind the moving mass $$m$$. If the distance $$d$$ between the mass $$m$$ and the shell is given, we can estimate the fraction of the retarded forces from the opposite part of the shell in the distance $$2 \, R - d$$, where $$R$$ is radius of the universe (cf. previous section "Retarded gravitational potentials"):

 * $$F_{right} \propto \frac {1} {d^2}$$

 * $$F_{left} \propto \frac {1} {(2 \, R - d)^2}$$

If we assume that the distance is a fraction of the radius

 * $$d = \frac {R} {n}$$

then the ratio of the two forces has the following relation:

 * $$\frac {F_{right}} {F_{left}} = \left( \frac {2 \, R - \frac {R} {n}} {\frac {R} {n}} \right)^2 = (2 \, n - 1)^2$$

This means if the distance $$d$$ is a tenth of the radius $$R$$ of the universe, the error by neglecting the left force would be less than 0.3 percent. This distance is corresponding to a redshift of about 7.5.

In the diagram on the left the ratio of the two forces is plotted against the redshift which is observed at corresponding distances. The high values of redshift at far distant objects seem to be dominated by the gravitaional redshift, whereas at shorter distances the redshift is dominated by the Doppler effect.

Computation

The bold arc on the right-hand side of the diagram is representing all mass elements of the outer shell of the universe with the linear mass density $$\lambda_M$$ in front of the mass $$m$$ that is moving to the right with high velocity. The outer shell is consisting of dark matter (mainly hydrogen), and in the central area of the universe this dark matter might only be visible due to the cosmic microwave background.

We use the following constant values for the estimates derived from these premises:

 * Light-year: $$ 1 \, \text {ly} = 9.46 \cdot 10^{15} \text {m}$$
 * Hubble length: $$R = 1.36 \cdot 10^{26} \text {m}$$ (equals 14,4 billion light-years)
 * Mass of the universe: $$M_{universe} = 2.97 \cdot 10^{53} \text {kg}$$:

The mass of this arc $$M_{arc}$$ can be computed by integrating the arc between the angles $$-\alpha_R$$ and $$+\alpha_R$$ with its linear mass density $$\lambda_M$$:

 * $$\alpha_R = \arcsin \frac {x} {R}$$

 * $$M_{arc} = \int_{-\alpha_R}^{+\alpha_R} \mathrm {dM} = \int_{-\alpha_R}^{+\alpha_R} R \, \lambda_M \, \mathrm {d\alpha} = 2 \, \int_{0}^{\alpha_R} R \, \lambda_M \, \mathrm {d\alpha} = 2 \, R \, \lambda_M \, \alpha_R$$

The mass of the whole shell $$M_S$$ is given by integrating a complete circle:

 * $$M_S = \int_{-\pi}^{+\pi} \mathrm {dM} = 2 \pi \, R \, \lambda_M$$

The vertical components of the gravitational forces are symetrical, and therefore, their net effect is zero.

The Pythagorean theorem gives the following result for the half chord of the circle:

 * $$x = \sqrt {2 \, R \, d - d^2}$$

If the distance $$d$$ between the mass $$m$$ and the outer shell is given, we can compute the distances of the mass $$m$$ to any infinitesimal mass element $$\mathrm {dM}$$ on the arc depending on the angle $$\alpha$$:

 * $$\alpha = \arcsin \frac {h} {R}$$

 * $$s(\alpha) = \sqrt {2R^2 - 2 \, R \, d + d^2 - 2 \, R \, (R - d) \, cos \, \alpha} \, \ge \, d$$

The angle $$\beta$$ as seen from the moving mass m to the infinitesimal mass element $$ \mathrm {dM}$$ on the arc can be derived by applying the law of sines and is given by the following expression:

 * $$\beta = \arcsin \left( \frac {R} {s} \sin \, \alpha \right) = \arcsin \frac {h} {s}$$

With the auxiliary sagitta $$e$$ we get:

 * $$e = R \, \left( 1 - \cos \alpha \right)$$

 * $$\cos \beta = \frac {d - e} {s}$$

The horizontal force of this arc $$F_{hor}$$ that is accelerating the mass $$m$$ in horizontal direction can be achieved by integrating the arc with the correction factor $$\cos \beta$$:

 * $$F_{hor} = G \, m \int_{-\alpha_R}^{+\alpha_R} \frac {\cos \beta} {s^2} \, \mathrm {dM} = G \, m \, R \, \lambda_M \, \int_{-\alpha_R}^{+\alpha_R} \frac {\cos \beta} {s^2} \, \mathrm {d\alpha}$$

With the effective gravitational force $$F_{hor}$$ that acts on the mass $$m$$ by the gravitational force of the virtual effective mass $$M_{eff}$$ that is located in the intersection point of the trajectory of the mass $$m$$ with the outer shell, we have to consider the varying distances $$s$$ between $$m$$ and the infinitesimal mass elements $$\mathrm {dM}$$ along the arc:

 * $$F_{hor} = G \, m \frac {M_{eff}} {d^2} = G \, m \, R \, \lambda_M \, \int_{-\alpha_R}^{+\alpha_R} \frac {\cos \beta} {s^2} \, \mathrm {d\alpha}$$

We finally get the effective mass $$M_{eff}$$ that is acting via gravitational forces on the mass $$m$$ in the distance $$d$$:

 * $$M_{eff} = R \, \lambda_M \, d^2 \int_{-\alpha_R}^{+\alpha_R} \frac {\cos \beta} {s^2} \, \mathrm {d\alpha} = \frac {M_S \, d^2} {2 \pi} \int_{-\alpha_R}^{+\alpha_R} \frac {\cos \beta} {s^2} \, \mathrm {d\alpha}$$

Schwarzschild distance
The Schwarzschild distance $$d_S$$ of this effective mass $$M_{eff}$$ is equal to the Schwarzschild radius of a sphere with the effective mass:

 * $$d_S = \frac {2 \, G \, M_{eff}} {c^2}$$

The gravitational redshift $$z$$ of photons emitted to the centre of the universe that is caused by the effective mass in the distance $$d$$ of the outer shell is:

 * $$z = \frac {1} {\sqrt {1 - \frac {2 \, G \, M_{eff}} {d \, c^2}}} - 1 = \frac {1} {\sqrt {1 - \frac {d_S} {d}}} - 1$$

The equations above can be solved for the Schwarzschild distance $$d_S$$ of every shell with the mass $$M_S$$. With the following condition we can get the solutions with $$d_S = d$$, where $$z (d_S) = \infty$$:

 * $$2 \, G \, M_{eff} (M_S, d_S) = d_S \, c^2$$

Therefore $$d_S$$ can be determined for a given $$\lambda_M$$:

 * $$d_S (\lambda_M) = \frac {2 \, G} {c^2} M_{eff} (M_S, d_S) = \frac {2 \, G \, R \, \lambda_M \, d_S^2} {c^2} \int_{-\alpha_R}^{+\alpha_R} \frac {d_S - R \, (1 - \cos \alpha)} {{\left( 2R^2 - 2 \, R \, d_S + d_S^2 - 2 \, R \, (R - d_S) \, \cos \, \alpha \right)}^{\frac {3} {2}}} \, \mathrm {d\alpha}$$

The Schwarzschild distance $$d_S$$ can be interpreted as the distance between the surface of the Hubble sphere with the Hubble radius $$R$$ and the inner Schwarzschild sphere of the black shell, which is the visible limit of the universe. The Hubble radius $$R$$ would be the distance between an observer in the centre of the universe and all objects that are receding from him at the speed of light. It can be expressed by the Hubble time $$t_H$$:

 * $$R = t_H \cdot c = 14.4 \cdot 10^{9} \, \text{s} \cdot {c} = 14.4 \cdot 10^{9} \, \text{ly} = 1.36 \cdot 10^{26} \, \text{m}$$

Results
It is assumed that the outer shell is expanding, spherical and consists of dark matter. The effective mass of the outer shell is computed only considering the geometrical region in front of the object. This assumption is based on the fact that the retarded gravitational potentials from the opposite border of the black shell can be neglected. The following diagrams show the solution in two different representations over the relation of the mass of the invisible universe surrounding the visible universe in a spherical shell in units of the mass of the visible universe $$q_M$$:

 * The Schwarzschild distance $$d_S$$ between the visible border of the universe and the invisible outer shell of the universe.
 * The radius of the visible universe $$R_v = R - d_S$$.

The overall mass of the invisible black shell $$M_S$$ can be expressed in relation to the mass of the visible universe $$M_{universe}$$:

 * $$q_M = \frac {M_S} {M_{universe}}$$

→ See appendix: table with results for different values of $\lambda_M$ numerically computed by a Java program.

In this model the mass $$M_S$$ of the invisible outer black shell continuously increases by absorbing mass from the visible universe, due to the net forces of the retarded gravitational potentials and the corresponding acceleration in direction of the black shell. More and more matter is moving from the visible universe behind the event horizon, where it becomes invisible and unaccessible, but leaves its gravitational action to the visible part of the universe. Furthermore, the mass of the black shell is in the same magnitude as the mass of the visible universe even in the earlier cosmical ages.

It is very noteworthy to recapitulate that the age of the cosmic microwave background with its afterglow light pattern is about 380,000 years (the corresponding Schwarzschild distance belongs to a shell mass $$M_S = 2.8769 \cdot 10^{53} \, \text {kg} = 0.96865 \cdot M$$, Schwarzschild distance $$d_S = 377 \cdot 10^{3} \, \text {ly} = 3.56 \cdot 10^{21} \, \text {m}= 2.62 \cdot 10^{-5} \cdot R$$), exactly where the radius of the visible universe had significantly begun to decrease in relation to the Hubble radius (see right diagram). The so-called "dark ages" begun, and they lasted for several hundreds of million years.

The age of the oldest known galaxy JADES.GS.z14-0 represents the youngest stars that emit light, and therefore, the end of the dark ages. Its age is about 290 million years (the corresponding Schwarzschild distance belongs to a shell mass $$M_S = 2.92 \cdot 10^{53} \, \text {kg} = 0,984 \cdot M$$, Schwarzschild distance $$d_S = 293 \cdot 10^{6} \, \text {ly} = 2.78 \cdot 10^{24} \, \text {m}= 0.020 \cdot R$$), where the conversion from dark matter to "dark energy" during the so-called "dark ages" was finished.


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Moving objects in retarded gravitational potentials of an expanding spherical shell/Conclusion

Conclusion
The considerations described above show that retarded gravitational potentials can contribute to the explanation of the accelerated expansion of the visible part of the universe, if the outer spherical black shell of the universe would consist of expanding dark matter with a huge mass. However, this would also hold in a steady state universe and even if the outer dark shell of the universe would not move at all. But in the latter case any moving mass in that shell would reach the shell in a finite time.

It can also be considered that the effect of the retarded gravitational potentials will accelerate moving objects the more, the faster they move, and therefore, the closer they are to the spherical shell of dark matter. Furthermore, also objects within smaller spherical shells that not only are surrounded by dark matter, but also by visible objects will experience the retarded gravitational potentials of these objects, and therefore, will experience an additional acceleration which is directed outwards.

Computing the corresponding and permanently increasing Schwarzschild distances results in two remarkable points in time where we today can observe the cosmic microwave background at a cosmic age of 380,000 years as well as the most distant known galaxy at a cosmic age of 290 million years. Cosmologists call the period in between as the dark ages, where dark matter was massively transformed into "dark energy". So far, the nature of "dark energy" is unclear, but the invisible black shell at the outer rim of our universe could contribute to find an explanation.

Due to Newton's third law (actio = reactio) the shell experiences the same retarded gravitational forces as the moving masses within the shell. These forces will cause a deceleration of the shell which is the stronger the closer the masses within the shell are to it.

The high velocity of visible objects near the outer dark rim of the universe as well as the huge mass of the dark matter in front of them seem to be the reason for the extreme redshift of their light that can be observed. The observation that this redshift is greater than expected for distant objects led to the assumption that the further away these objects are, the more they are accelerated. Another reason for this could be found in the effects of the retarded gravitational potentials of a massive dark outer shell. Furthermore, the additional gravitational redshift leads to a higher value than expected only by the relativistic Doppler effect. This is in accordance with the observations of very far and young galaxies such as JADES-GS-z14-0 or of the cosmic microwave background (CMB).

However, the computation of the accelerations of moving objects due to retarded gravitational forces becomes extremely expensive, if the outer shell is neither homogeneous, nor spherical or symmetrical. Furthermore, relativistic effects (including the transverse Doppler effect) could cause time dilatation or mass increase that would have to be considered, too. Finally, is would be necessary to consider the loss of mass in the visible universe over the time, since a significant amount of visible matter may have traversed the event horizon that is caused by the huge mass of the outer shell of the universe.


text:
Moving objects in retarded gravitational potentials of an expanding spherical shell/Conclusion/

The considerations described above show that retarded gravitational potentials can contribute to the explanation of the accelerated expansion of the visible part of the universe, if the outer spherical shell of the universe would consist of expanding dark matter with a huge mass. However, this would also hold in a steady state universe and even if the outer dark shell of the universe would not move at all. But in the latter case any moving mass in that shell would reach the shell in a finite time.

It can also be considered that the effect of the retarded gravitational potentials will accelerate moving objects the more, the faster they move, and therefore, the closer they are to the spherical shell of dark matter. Furthermore, also objects within smaller spherical shells that not only are surrounded by expanding dark matter, but also by expanding visible objects will experience the retarded gravitational potentials of these visible objects, and therefore, will experience an additional acceleration which is directed outwards.

Computing the corresponding Schwarzschild distances results in two remarable points in time where we today can observe the cosmic microwave background at a cosmic age of 380,000 years as well as the farest known galaxy at a cosmic age of 290 million years.

Due to Newton's third law (actio = reactio) the shell experiences the same retarded gravitational forces as the moving masses within the shell. These forces will cause a deceleration of the shell which is the stronger the closer the masses within the shell are to it.

The high velocity of visible objects near the outer dark rim of the universe as well as the huge mass of the expanding dark matter in front of them seem to be the reason for the extreme redshift of their light that can be observed. The observation that this redshift is greater than expected for distant objects led to the assumption that the further away these objects are, the more they are accelerated. One reason for this could be found in the effects of the retarded gravitational potentials of a massive dark outer shell. Furthermore, the additional gravitational redshift leads to a higher value than expected only by the relativistic Doppler effect, which is in accordance with the observations of very far and young galaxies such as JADES-GS-z14-0.

However, the computation of the accelerations of moving objects due to retarded forces becomes extremely expensive, if the outer shell is neither homogeneous, nor spherical or symmetrical. Furthermore, relativistic effects could cause time dilatation and mass increase that would have to be considered, too. Finally, is would be necessary to consider the loss of mass in the visible universe over the time, since visible matter may traverse the event horizon that is caused by the huge mass of the outer shell of the universe.


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Moving objects in retarded gravitational potentials of an expanding spherical shell/Preface

Summary

 * Retarded gravitational potentials of a very large spherical shell with mass.
 * Acceleration forces on objects with mass within such a shell.
 * Concept of the Schwarzschild distance from the outer invisible surface of such a shell to the visible universe ("black shell").
 * Gravitational redshift due to the mass of a shell.
 * Evolution of the Schwarzschild distance due to the continuously increasing mass of the black shell.

Abstract
The principles of retarded gravitational potentials are presented and explained. The application of retarded gravitational potentials to an expanding spherical shell of matter with a large mass (a "black shell") leads to acceleration forces on moving objects within such a huge shell.

The concept of a Schwarzschild distance from the inner surface of a hollow spherical cap is introduced. It can be used to determine equivalent spherical masses which can provide information about the evolution of the universe.

The unexpected high values of the cosmological redshift that are observed at extremely far astronomical objects including the cosmic microwave background could be related to the strong asymmetrical effect of the gravitation of the outer black shell of the universe. Due to the integrative effect of the retarded gravitational potentials of a concave shaped spherical black shell the gravitational forces of masses within the Schwarzschild distance are so enormous that the light from any atoms can no longer reach us. Exactly at the boundary to our visible universe, the gravitational redshift is infinite, just like on the surface of a convex shaped spherical black hole.

The evolvement of the Schwarzschild distance due to the continuously increasing mass of the outer invisible shell of our universe has two inflection points around the very points in time and space, where the cosmic microwave background as well as the youngest galaxies can be observed today. The strongly increasing Schwarzschild distance between these two points in time goes hand in hand with the postulated massive transformation of dark matter to "dark energy". Therefore, the effect of retarded gravitational potentials could also contribute to the explanation of the accelerated expansion of the universe, which often is related to the effect of "dark energy".

It seems that the hypothesis of relevant dark matter that continuously moves beyond the event horizon of our visible universe could open the gates for further interesting investigations and findings.


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Moving objects in retarded gravitational potentials of an expanding spherical shell/Printable version

=Appendix=


text:
Category:Book:Moving objects in retarded gravitational potentials of an expanding spherical shell

This category contains pages that are part of the Moving objects in retarded gravitational potentials of an expanding spherical shell book.


text:
Simple Guide to Minecraft/Enchanting

The enchanting process in Minecraft is one that can be seen as complicated. However, this page will help you crush the confusion that stands in between the clear window of level 30 enchanting. Enchanting is a simple thing that takes little practice (though practice does help) to master, even small youtube videos on this subject are very helpful and could answer many question that this article might fail to bring to subject.

To start you will need a few late game items. One of the most noticeable on this list being the use of a Diamond Pickaxe. See the full list below:

 * An Enchanting Table (Enchantment Table in Minecraft: Bedrock Edition) Crafted with 4 obsidian (thus the Diamond Pickaxe), 2 diamonds, and one book.
 * 0-15 Bookshelves (Depends on enchantment level, see Enchantment Levels section for further information.)
 * Lapis Lazuli 1-3 (Preferably a stack of 64)

Gather your materials (listed above) to begin, if you don't have 15 bookshelves, skip the second step, if you do, continue as normal.

The first step is to place down you're enchanting table/enchantment table on a flat space, it can be inside a house, on a flat turf, of floating in the middle of the End.

The second step is to place your 15 bookshelves in the configuration shown below (Important note: make sure there are no blocks obscuring the view between the bookshelves and the enchanting table):

And the third and final step is to place your Lapis Lazuli and your item in the enchanting table and click the enchant level you want. And there you go, you have enchanted for your first time.

title: Simple Guide to Minecraft/Enchanting


text: User talk:149.88.20.92

title: User talk:149.88.20.92


text:
AI Art Generation Handbook/IPAdapter

IP Adapter (Image Prompt Adapter) is Stable Diffusion extension for using images as prompts. They functions as to copy the style, composition, or a face from the reference image.

It is pioneered by Tencent (WeChat developer) AI Lab as a way to achieve image prompt capability for the pre-trained text-to-image diffusion models.

Here are some of the Local UI that already utilizes the IP Adapter

 * Auto1111

 * SD.Next

 * ComfyUI

title: AI Art Generation Handbook/IPAdapter


text:
Wikibooks:Reading room/Administrative Assistance/Archives/2024/May

Special:Contributions/156.193.229.151
Possible w:WP:LTA/GRP case, please check their edits. MathXplore (discuss • contribs) 07:39, 1 May 2024 (UTC)

 * Yes, that is them. Nonetheless, ✅ by Kittycataclysm. --SHB2000 (discuss • contribs) 22:16, 1 May 2024 (UTC)

Special:Contributions/Saeedghafary1
Possible non-English spam, please check their edits. MathXplore (discuss • contribs) 07:50, 8 May 2024 (UTC)

 * Blocked for spamming—thank you! —Kittycataclysm (discuss • contribs) 12:08, 8 May 2024 (UTC)

Religion of Islam and Special:Contributions/Michael_STP (Alert of possible block/lock evasion of Special:Contributions/Germany_Poul_Ah)
May I have your attention here? I noticed that the Religion of Islam (one of the many deleted GPA creations) has been reinstated (was there a discussion about this?). If the deleted revisions closely match with the current one, we may need to consider this as a possible block/lock evasion. I have checked the creator's global edits. They are based in fawiki and here, as same as GPA. Please see Reading room/Administrative Assistance/Archives/2023/December (reported by User:Kittycataclysm) and Reading room/Administrative Assistance/Archives/2023/October (reported by me) for similar incidents. Thank you for your attention. MathXplore (discuss • contribs) 14:03, 9 May 2024 (UTC)
 * CheckUser confirms this is a sock of the same person. I will block / lock / delete. MarcGarver (discuss • contribs) 16:30, 9 May 2024 (UTC)


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AI Art Generation Handbook/IPAdapter/ComfyUI

Note: We are focusing more on IPAdapter for SDXL models here:

GO to Your_Installed_Directory/ComfyUI/custom_nodes/ and on the address bar, type   and inside terminal , type

Go to the official huggingface SDXL to download the IP Adaptor Encoder (clip_vision) first at link below :

https://huggingface.co/h94/IP-Adapter/tree/main/sdxl_models/image_encoder

You should find a model clip_vision file named: model.safetensors and download it to directory Your_Installed_Directory/ComfyUI/models/clip_vision and renamed it exactly to

Go back up one level https://huggingface.co/h94/IP-Adapter/tree/main/sdxl_models and download all the models with .safertensors and .bins file type into this directory Your_Installed_Directory/ComfyUI/models/ipadapter (Create folder ipadapter if not present)

It should be: 1) ip-adapter-plus-face_sdxl_vit-h.safetensors

2) ip-adapter-plus_sdxl_vit-h.safetensors

3) ip-adapter_sdxl.safetensors

4) ip-adapter_sdxl_vit-h.safetensors

For the Unified Loader to works, the .bins files (Take note of FaceID not Plus version) are also needed to be downloaded from here

https://huggingface.co/h94/IP-Adapter-FaceID/tree/main

into this directory Your_Installed_Directory/ComfyUI/models/ipadapter

5) ip-adapter-faceid-portrait_sdxl.bin

6) ip-adapter-faceid_sdxl.bin

7) ip-adapter-faceid-plusv2_sdxl.bin

8) ip-adapter-faceid-portrait_sdxl_unnorm.bin

Finally, we need to download the Lora for the FaceID to work

Your_Installed_Directory/ComfyUI/models/loras

9) ip-adapter-faceid_sdxl_lora.safetensors

10) ip-adapter-faceid-plusv2_sdxl_lora.safetensors


text:
Wikijunior talk:Biology/Kingdoms

Why 5 kingdoms
Is protista no more?

Rozzychan (discuss • contribs) 17:43, 21 June 2024 (UTC)

 * I added link to protista into the page. As of this date, Wikispecies includes protista as a kingdom.
 * Reference: Wikispecies Main page. https://species.wikimedia.org/wiki/Main_Page Rozzychan (discuss • contribs) 08:21, 22 June 2024 (UTC)

title: Wikijunior talk:Biology/Kingdoms


text: Category:Recipes using bread

title: Category:Recipes using bread


text: Category:Recipes for bread

title: Category:Recipes for bread


text: Category:Recipes for flatbread

title: Category:Recipes for flatbread


text: Yoruba/loan Words

title: Yoruba/loan Words


text:
Yoruba/Vocabulary

Roots
Most verbal roots are monosyllabic of the phonological shape CV(N), for example: dá (to create), dán (to polish), pọ́n (to be red). Verbal roots that do not seem to follow this pattern are mostly former compounds in which a syllable has been elided. For example: nlá (to be large), originally a compound of ní (to have) + lá (to be big) and súfèé (to whistle), originally a compound of sú (to eject wind) + òfé or ìfé (a blowing). Vowels serve as nominalizing prefixes that turn a verb into a noun form.

Nominal roots are mostly disyllabic, for example: abà (crib, barn), ara (body), ibà (fever). Monosyllabic and even trisyllabic roots do occur but they are less common.

title: Yoruba/Vocabulary


text:
Grand Theft Auto: Vice City/Vehicles

The following is a complete categorized listing of vehicles which can be driven, riden on, or controlled in Grand Theft Auto: Vice city.

Car Categories

 * 1) /Sports/
 * 2) /4WD and utility/
 * 3) /Four door cars/
 * 4) /Two-door cars/
 * 5) /Van/
 * 6) /Industrial trucks/
 * 7) /Public Service/
 * 8) /Government and emergency/
 * 9) /Miscellaneous/
 * 10) /Aircraft/
 * 11) /Motorcycles/
 * 12) /Boats/
 * 13) /RC Vehicles/

title: Grand Theft Auto: Vice City/Vehicles


text:
Grand Theft Auto: Vice City/Vehicles/Sports

 * Banshee
 * Cheetah
 * Comet
 * Deluxo
 * Infernus
 * Stinger


text:
Grand Theft Auto: Vice City/Vehicles/Four door cars

 * Admiral
 * Glendale
 * Greenwood
 * Oceanic
 * Perennial
 * Regina
 * Sentinel
 * Sentinel XS
 * Stretch
 * Washington


text:
Grand Theft Auto: Vice City/Vehicles/Two-door cars

 * Blista Compact
 * Cuban Hermes
 * Esperanto
 * Hermes
 * Idaho
 * Manana
 * Phoenix
 * Sabre
 * Sabre Turbo
 * Stallion
 * Virgo
 * Voodoo


text:
Grand Theft Auto: Vice City/Vehicles/Van

 * Burrito
 * Gang Burrito
 * Moonbeam
 * Mr Whoopee
 * Pony
 * Rumpo
 * Securicar
 * Top Fun


text:
Grand Theft Auto: Vice City/Vehicles/Industrial trucks

 * Benson
 * Boxville
 * Flatbed
 * Mule
 * Linerunner
 * Packer
 * Spand Express
 * Yankee


text:
Grand Theft Auto: Vice City/Vehicles/Public Service

 * Bus
 * Cabbie
 * Coach
 * Kaufman Cab
 * Taxi
 * Trashmaster
 * Zebra Cab


text:
Grand Theft Auto: Vice City/Vehicles/Government and emergency

 * Ambulance
 * Fire Truck
 * Patriot
 * Barracks OL
 * Police
 * Enforcer
 * FBI Rancher
 * FBI Washington
 * Rhino
 * VCPD Cheetah


text:
Grand Theft Auto: Vice City/Vehicles/Miscellaneous

 * Baggage Handler
 * Bloodring Banger (Glendale)
 * Bloodring Banger (Oceanic)
 * Caddy
 * Hotring Racer
 * Hotring Racer (hotrina)
 * Hotring Racer (hotrinb)
 * Love Fist
 * Romero's Hearse


text:
Grand Theft Auto: Vice City/Vehicles/Aircraft

 * Hunter
 * Maverick
 * Police Maverick
 * Sea Sparrow
 * Skimmer
 * Sparrow
 * VCN Maverick


text:
Grand Theft Auto: Vice City/Vehicles/Motorcycles

 * Angel

 * Faggio
 * Freeway
 * PCJ 600
 * Pizza Boy
 * Sanchez


text:
Grand Theft Auto: Vice City/Vehicles/Boats

 * Coast Guard
 * Cuban Jetmax
 * Dinghy
 * Marquis
 * Predator
 * Reefer
 * Rio
 * Speeder
 * Squalo
 * Tropic


text: Yoruba/common words and phrases

title: Yoruba/common words and phrases


text:
Grand Theft Auto: Liberty City Stories/Dead Meat

Prerequisites

 * Missions: Grease Sucho
 * Location: Cipriani Restaurant, St.Marks District, Portland

Walkthrough
A disappointed Ma informs Toni that she's put a hit on Toni, with some hit men on the way to kill him.

Reward

 * $0


text:
Grand Theft Auto: Liberty City Stories/No Son of Mine

Prerequisites

 * Missions: Dead Meat
 * Location: Cipriani Restaurant, St.Marks District, Portland

Walkthrough
Giovanni Casa has stopped paying protection money to the Cipriani’s after Toni pictured him in the mission Snappy Dresser, so Ma tells Toni to sort the situation out.

Reward

 * $500


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The Legend of Zelda: Link's Awakening/Walkthrough/Angler's Tunnel

Angler's Tunnel is the fourth dungeon in "The Legend of Zelda: Link's Awakening." This dungeon is set in a water-themed environment and is known for its challenging puzzles and aquatic enemies. Angler's Tunnel is a notable dungeon for its aquatic theme and the introduction of new mechanics related to water traversal.

Location

 * Access: Located in Tal Tal Heights, Angler's Tunnel. To enter Angler's Tunnel, Link needs the Angler Key, which he finds after defeating the Lanmola in the Yarna Desert. The entrance to the dungeon is in the eastern part of Koholint Island, in Tal Tal Heights. Link must use the key to unlock the waterfall blocking the dungeon entrance.
 * Entrance: The entrance is hidden underwater, and Link must swim to reach it.

Key Items

 * Flippers: The key item needed to navigate the dungeon's water-filled rooms.
 * Angler Key: Required to unlock the entrance to the dungeon, found in the Yarna Desert.

Dungeon Layout

 * Flooded Areas: Much of the dungeon is submerged, requiring Link to frequently swim to progress. The Flippers, which allow Link to swim, are obtained in this dungeon and are crucial for navigation.
 * Water Level Mechanisms: Some rooms have mechanisms that change water levels, adding a layer of complexity to navigation and puzzle-solving. These often involve hitting switches or manipulating objects to raise or lower the water.
 * Enemies: The dungeon features a mix of land-based and aquatic enemies, including familiar Zelda foes and new water-themed adversaries.
 * Puzzles: The puzzles in Angler's Tunnel are intricate and often involve water mechanics, such as swimming through narrow passages, moving blocks while submerged, and hitting switches underwater.

Rewards

 * Surf Harp: The instrument obtained after defeating the Angler Fish. It's one of the eight Instruments of the Sirens needed to awaken the Wind Fish


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Dialect/North-West Yoruba (NWY)/Yewa

Yewa, are a subgroup of the Yoruba people and mostly inhabit the eastern area of Ogun West Senatorial District, Ogun State, in south-west Nigeria, Africa. In 1995, the group's name was changed to Yewa after the Yewa River, the river (odo) they foraged towards. The name of this river is derived from the Yoruba goddess Yewa.

title: Dialect/North-West Yoruba (NWY)/Yewa


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The Legend of Zelda: Link's Awakening/Walkthrough/Catfish's Maw

Catfish's Maw is the fifth dungeon in "The Legend of Zelda: Link's Awakening." This dungeon is known for its complex layout and the presence of multiple mini-boss battles, which adds a unique challenge compared to other dungeons in the game. Catfish's Maw in "The Legend of Zelda: Link's Awakening" is a dungeon that emphasizes exploration, combat, and the strategic use of the Hookshot.

Location and Access

 * Underwater Entrance: Catfish's Maw is located in Martha's Bay. Link must swim underwater to find and enter the dungeon through the mouth of a giant catfish statue.
 * Requirement: The Flippers, obtained in the previous dungeon (Angler's Tunnel), are needed to dive underwater and access the dungeon.

Dungeon Layout

 * Complex Navigation: The dungeon features a multi-layered, labyrinthine layout with numerous rooms and passageways that often require backtracking and careful exploration.
 * Puzzles and Obstacles: The puzzles in Catfish's Maw often involve moving blocks, hitting switches, and using the Hookshot to traverse gaps or pull items closer.

Key Item

 * Hookshot: The primary item found in Catfish's Maw. It allows Link to:
 * Grapple across gaps.
 * Pull enemies and objects towards him.
 * Hit distant switches.
 * Cross large bodies of water and chasms.The Hookshot significantly expands Link's mobility and combat options, making it one of the most important tools in his arsenal.

Post-Dungeon

 * Wind Marimba: Upon defeating Slime Eel, Link receives the Wind Marimba, one of the eight Instruments of the Sirens needed to awaken the Wind Fish.


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Bodybuilding and Weight Training/Dumbbell Flys

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Dumbbell Flys

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Main Muscles Used:

Chest

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Description:

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Dumbbell Flys, also known as Dumbbell Flyes, are a strength training exercise specifically designed to target the pectoral muscles, which are the primary muscles of the chest. This exercise is typically performed lying on a flat bench, though variations can include an incline or decline bench to target different areas of the chest. To perform Dumbbell Flys, you start by lying flat on the bench with your feet firmly planted on the ground for stability. Hold a dumbbell in each hand, with your palms facing each other and your arms extended straight above your chest. A crucial aspect of this exercise is maintaining a slight bend in the elbows throughout the movement to reduce strain on the joints.

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Safety Tips

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 * Weight Selection: Start with lighter weights to master the form and gradually increase the weight.
 * Control: Perform the exercise slowly and with control to maximize muscle engagement and reduce injury risk.
 * Elbow Position: Keep a slight bend in your elbows throughout the exercise to protect your joints.
 * Range of Motion: Avoid lowering the weights too far below your chest level to prevent shoulder strain.
 * Spotter: Consider having a spotter, especially when using heavier weights, to assist and ensure safety.

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Mistakes to Avoid

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 * Straight Arms: Keeping arms completely straight can put undue stress on the elbow joints.
 * Overstretching: Lowering the weights too far can lead to shoulder injuries.
 * Rapid Movements: Fast, uncontrolled movements can reduce the effectiveness of the exercise and increase injury risk.
 * Using Momentum: Swinging the weights or using momentum rather than controlled muscle movement can diminish results.


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Dialect/Central Yoruba (CY)/Igbomina

The Ìgbómìnà (also colloquially Igboona or Ogboona) are a subgroup of the Yoruba ethnic group, which originates from the north central and southwest Nigeria. They speak a dialect called Ìgbómìnà or Igbonna, classified among the Central Yoruba of the three major Yoruba dialectical areas. The Ìgbómìnà spread across what is now southern Kwara State and northern Osun State. Peripheral areas of the dialectical region have some similarities to the adjoining Ekiti, Ijesha and Oyo dialects. Idoba is on Owode Ofaro Land

title: Dialect/Central Yoruba (CY)/Igbomina


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Dialect/Central Yoruba (CY)/Ijesha

Ilesa (Yoruba: Iléṣà) is a historic city located in Osun State, southwest Nigeria; it is also the name of a historic kingdom (also known as Ijesha) centred on that town. The state is ruled by a monarch bearing the title of the Owa Obokun Adimula of Ijesaland. The state of Ilesa consists of Ilesa itself and a number of smaller surrounding cities.

title: Dialect/Central Yoruba (CY)/Ijesha


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Dialect/Central Yoruba (CY)/Ifẹ

Ifẹ̀ (Yoruba: Ifẹ̀, Ilé-Ifẹ̀) is an ancient Yoruba city in south-western Nigeria, founded approximately between the 10th century and 6th century B.C.E. The city is located in present-day Osun State. Ifẹ̀ is about 218 kilometers northeast of Lagos with a population of over 500,000 people, which is the highest in Osun State according to population census of 2006.

title: Dialect/Central Yoruba (CY)/Ifẹ


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Dialect/Central Yoruba (CY)/Ekiti

The Ekiti people are one of the largest historical subgroups of the larger Yoruba people of West Africa, located in Nigeria. They are classified as a Central Yoruba group, alongside the Ijesha, Igbomina, Yagba and Ifes. Ekiti State is populated exclusively by Ekiti people; however, it is but a segment of the historic territorial domain of Ekiti-speaking groups, which historically included towns in Ondo State such as Akure (the current capital and largest city of Ondo State), Ilara-Mokin, Ijare, and Igbara-oke. Ogbagi, Irun, Ese, Oyin, Igasi, Afin and Eriti in the Akoko region, as well as some towns in Kwara State, are also culturally Ekiti, although belong in other states today.

title: Dialect/Central Yoruba (CY)/Ekiti


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Dialect/Central Yoruba (CY)/Ẹfọn

Efon is a Local Government Area of Ekiti State, Nigeria. Its headquarters are in the town of Efon-Alaaye. This local government area was established primarily to bring the local population into the federal government's awareness and to open up opportunities for equitable development.It has an area of 232 km2 and a population of 86,941 at the 2006 census.The postal code of the area is 362

title: Dialect/Central Yoruba (CY)/Ẹfọn


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Dialect/Central Yoruba (CY)/Akoko

The Akoko are a large Yoruba cultural sub-group in the Northeastern part of Yorubaland. The area spans from Ondo State to Edo State in southwest Nigeria. The Akokos as a subgroup make up 20.3% of the population of Ondo State, and 5.7% of the population of Edo State. Out of the present 18 Local Government Councils it constitutes four; Akoko North-East, Akoko North-West, Akoko South-East and Akoko South-West, as well as the Akoko Edo LGA of Edo State. The Adekunle Ajasin University, a state owned university with a capacity for about 20,000 tertiary education students and more than 50 departments in seven faculties is located in Akungba-Akoko. A state specialist hospital is situated at Ikare Akoko, while community general hospitals are located in Oka-Akoko and Ipe-Akoko.

title: Dialect/Central Yoruba (CY)/Akoko


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Dialect/South-West Yoruba (SWY)/Ketu

Ketu is the name of a Yoruba subgroup, historical kingdom and region straddling parts of what is now southeastern Republic of Benin and parts of southwest Nigeria. The chief town and traditional capital of the area was the town of Kétou (Kétu), which is considered to be one of the oldest capitals of the Yoruba-speaking people, tracing its establishment to a settlement founded by a descendant of Oduduwa (also known as; Odùduwà, Oòduà and Eleduwa) called Sopasan or Soipasan.

title: Dialect/South-West Yoruba (SWY)/Ketu