I want a new MBP but...the x1600 sux!
Discussion in 'MacBook Pro' started by macjonny1, Mar 26, 2007.
Actually the x1600 is pretty good, and a great choice a year ago. As the title says, I really want to buy a MBP but I can't justify to myself buying a computer with a graphics card that was introduced over a year ago. I am thinking this has to be upgraded at the next revision coming up. The x1600 would really be fine for now, but I want my computer to be somewhat future proof for a little while. I've read reports that some think that newer cards draw more power or are too hot....but I really have a hard time believing that it won't be upgraded.
Are the demo laptops as thin as a MBP?
There is usually some ways to go between samples and high volume production.
Quite true but hopefully there have been some improvements since January. Intel's Santa Rosa platform is set to launch with 8 and X2xxx Series support.
the GPU will be updated some time in the future, and im 99% sure it'll be changed in the next rev. What do you plan to do that you need a better card than the X1600 for?
There is actually nothing I would do *now* that I would need more than an x1600, in fact it would blow away what I am using now. Currently I play WoW on integrated graphics (Intel 950). My idea is that I want to get as much mileage out of a computer as I can because invariably something will be updated software-wise that will take advantage of the new GPUs. If you buy a computer with a just-upgraded GPU then you will maximize that. If you buy it with a GPU over a year old, then you are already that far behind the curve. I don't buy computers often so I just want to get the most benefit for the longest time.
If I want to sacrifice performance for the sake of "thin" I will buy one of those Sony Vaio UPMC or whatever they call it these days.
...welcome to the waiting game.
the GPU will be updated some time in the future, and im 99% sure it'll be changed in the next rev.
If you are a serious gamer, why are you considering a laptop? In fact, honestly, why are you even considering a mac?
Don't get me wrong, I have games that I play on my MBP, but it's not my first priority and I plan on building a desktop system if I ever get into real gaming.
me too. I sold my 12" powerbook (man was that stupid) about 9 months ago and have been waiting for next revision mbp ever since it'll probably be june till it shows up...hopefully with all sorts of new, neat stuff-such as an upgraded video card.
I am in that situation as well, i am waiting for a new revision before i buy. I will not spend that amount of money on something that is over one years old in technology. If you did and something new came out relatively soon you would really kick yourself anyway and the value of your computer would have slashed quite a bit in value in that such a short time.
I sold my MBP that i was given as a new replacement and i am soon going to be without a Mac laptop to use. I am at this minute using a laptop that has to go back to Apple later on today. I am thinking about buying a G4 iBook or Powerbook just to tide me over so i can use something while i wait.
Even when Apple introduce something there is still a usually about a months wait until you actually receive it. So all in all we maybe waiting for some time.
I may put up with my mothers Toshiba laptop which i don't like using until new models come out and save myself a bit of money or i may buy a relatively cheap G4 and use that in the stop gap.


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text:
Institutional Portals
an international database for artistic research
Home Institutional Portals Journals Researchers
KC Research Portal
About this portal
Master students at the Royal Conservatoire use the online Research Catalogue for the communication with their supervisor, for the development and formulation of their research proposal, for their work-in-progress, and for the final documentation and publication of their research.
contact person(s): Henk Borgdorff , Kathryn Cok , Koncon Master Coordinator , Casper Schipper
url: https://www.researchcatalogue.net/view/517228/517229
3. Internal publication Research published in this issue are only for internal circulation within the Royal Conservatoire, The Hague.
2. Royal Conservatoire Investigations Royal Conservatoire Investigations
1. Master Research Projects All research in KC
Size Matters: Historical String Lengths and Possible Tunings of the Chitarrone or Theorbo (2021) author(s): Gao Menglin published in: KC Research Portal
What was considered the "normal" size for the Italian theorbo or chitarrone upon its creation in the 17th century, and why should 21st century players like us embrace the historical size? This research tackles the question from several different angles: description in historical texts, surviving instruments, the optimal string gauge and tension, and tone comparison. The study finds the vast majority of surviving chitarroni to have a stopped string length around 88-89cm, with some exceeding 90cm and very few falling short of 80cm. This sparks the question of whether smaller instruments required a different tuning. Upon comparing with lute family instruments from the same era, the logical conclusion is that smaller instruments with string lengths around 76cm most probably did not need to have a double re-entrant tuning which is the standard tuning for all solo chitarrone music. When comparing a large instrument (95cm) and a small instrument (76cm, with single re-entrant tuning), one can hear that they perform very different roles as continuo instruments, and are not mutually replaceable. As modern-day lutenists, one should really embrace the large size if one wishes to play the chitarrone, and develop a technique to suit the instrument, instead of scaling down the instrument to suit the technique. Menglin Gao has been a recorder player, a countertenor, and now a theorbist, studying at the Royal Conservatoire of the Hague. He is the founder and director of the Shanghai Camerata, an early music ensemble based in Shanghai, China.
open exposition comments (1)
One in Norway by One of Them: Vlogging for Classical Musicians (2021) author(s): Ursula Skaug published in: KC Research Portal
The internet is a promising place for musicians; one can promote and perform music for free, with possibly endless reach. Especially online video and streaming are booming spaces with many opportunities. Online music videos are often either recordings of a live performance, or they are an "online performance" from a musician who acts as if they are part of a regular live concert with an audience. However, the practices of regular concerts and online video are very different in nature. Perhaps online video requires a different way of engaging with the musical material within the video format. What if classical musicians created videos that emphasized the more casual and chatty style of the online platforms? Through an iterative process I researched how classical musicians can use the vlog-style format with classical music and music theatre. In these vlogs I draw inspiration from the book "Three in Norway by Two of Them" and combine it with music by Julius Röntgen and Edvard Grieg. From this process I drew some conclusions and came up with a process for other musicians to try out this video format.
Exploring Expressive Lied Performance: Re-enacting Lotte Lehmann's Pre-World War II Lied Performances (2021) author(s): Marlina Deasy Hartanto published in: KC Research Portal
After the second world war, the musical performance style remains changing from intensely expressive into clean and litteral. The exaggeration of expression through tempo, rhythm, portamento, ornamentation were suddenly gone and became utterly unrealistic and old-fashion. The cleanliness of the performance style brought singers to put more focus only to the score and text. Realizing how much the gap between two different styles, I have only partially understood the half picture of the style. My thesis will focus on the re-enactment study of Lotte Lehmann's recording in 1930s of An die Musik by Schubert and Ich grolle nicht by Schumann.
Integration and development of jazz drum soloing (2021) author(s): Borut Rampih published in: KC Research Portal
The desire to improve my drum soloing and the lack of in-depth material and examples on the subject led me to research drum soloing. Evaluating my playing, I found my approach to drum soloing was based heavily on playing "licks," copying other drummer's vocabulary literally with no clear idea of my own sound and identity with problems associated with this approach, such as forgetting phrases, feeling uncomfortable in unknown musical situations, etc. In my research, I tried to answer the question of how to integrate the playing of the master drummers of the bebop era into my own vocabulary and how modern drummers that I admire developed the classic language to fit their own style. I chose some of the most notable drummers of the bebop era, namely Max Roach, Philly Joe Jones, and Roy Haynes, transcribed their solos, and analyzed their playing. Through their playing, I discovered fundamental concepts of drumming like the Call and response concept, the rudimental concept, the melodic concept, and tools and methods that help with technique, musicianship, and creativity on the drum set.
The Arabic Maqam on Double Bass (2021) author(s): Remy Dielemans published in: KC Research Portal
After participating in a project, Remy awakened his interest in Arabic traditional music. Being trained as a jazz, pop and classical bassist, he found that this music left him clueless, but mesmerized. That was when he decided to learn more about this musical genre, and dedicated his Master's degree to the subject. In his research, Remy aimed to find ways to incorporate elements from traditional Arabic music into his own musical practice as a Jazz double bass player. To reach this goal, Remy researched both rhythmic and melodic aspects of the Arabic music tradition. The first step was to understand the theory and the melodic and rhythmic information involved, encountering challenging topics like quartertones and complex rhythms and song structures. Secondly he searched to find ways to translate the melodic and rhythmic elements on to the double bass, creating several exercises to gain familiarity with the material. The third step of the research was about applying his findings into his artistic practice. By composing new tunes and arranging existing songs, both from the traditional Arabic world as from the jazz genre, Remy aimed to let the discoveries of his research sink in in his musical creativity. Remy's presentation will take you through his journey discovering this music. He will guide you through the three steps of his discoveries, looking at the melodic elements of the Arabic musical tradition, and how he implemented these into his own musical practice. Master Research exposition by Remy Dielemans Jazz Double Bass, Royal Conservatoire the Hague Supervised by: Loes Rusch Master Circle Leader: Wim Vos
What is the word (2021) author(s): Renee Jonker published in: KC Research Portal
What than is music? – Music is language.' Composer Anton Webern was quite outspoken in 1932 : 'A human being wants to express thoughts in this language, but not a thought that can be transferred into an idea, but only into another musical thought.' Almost sixty years later composer Wolfgang Rihm isn't sure whether music is a language but states in his speech 'Was 'sagt' Musik/ What does music 'say'?:' if music is saying something, than the first what is addressed to us is: speak! Music wants to make us speak. That's what music says!' That raises the question whether music can be referential or carry meaning just as language can. A question that has made many speak and filled libraries of studies. Cognitive psychologist Aniruddh D. Patel writes in 2008: 'A natural place to turn for help in defining musical meaning is the work of music theorists and philosophers of aesthetics'. After summing up a dozen publications on the topic since Webern made his statement, Patel reports: 'No consensus has emerged from these writings for a definition of musical meaning.' There is a lot of evidence in linguistics that qualities attributed to musicality contribute to language being the carrier of meaning. So what about the other way around? Can language help to understand what is experienced as meaningful in music? Language itself is an indicator that qualities inherent to language are often given to music. In German the word Interpret is used for instrumentalists performing music. The Dutch language has the word zeggingskracht that attributes power to music. 'Zegging-' stems from the verb zeggen (to say), 'kracht' means power. Zeggingskracht was one of the three criteria to assess the work of composers by the Fund for the Creation of Music (Fonds voor de Scheppende Toonkunst ) in The Netherlands. When the power to speak is inhibited and people stammer or lose their speech suffering from aphasia, it has been said that "only music, can do the calling." And it's almost a cliché to say that music can express what can't be given words. Music is not a language but often sounds like one. What do musicians that are 'speaking' that music have to say about meaning in music, singing, performing or creating what composer Louis Andriessen describes as 'talkative' music? Or the stammering that composer György Kurtág calls his mother language? How do musicians give words to those moments when their music does the talking? What is the word is the last text that Samuel Beckett completed at the end of his life when through a stroke he periodically suffered from the disability to finds words, commonly diagnosed as aphasia. The Irish author inspired many to explore the zeggingskracht of music. Precise as he was, Beckett left out a question mark in the title, both in the original French version of Comment dire and in his English translation. That the title of Samuel Beckett's last text is not posing a question but may provide us with an answer, is the hypothesis of this research.
Society for Artistic Research
Copyright Concerns
Current RC Developments
The Research Catalogue server is operated at
KTH's IT-department

redpajama_set_name: RedPajamaCommonCrawl


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The overall industrial style that the architecture firm focused on is emphasized by the use of rough wood, untreated concrete core walls and bold rusted metalwork as main materials for the new headquarters. "The space is intelligently designed so that it mimics the environment the Windward company undergoes its activity in. We wanted to shift our design strategy so that the space would not reflect a luxurious yacht design but that of the harsh industrial ports." explains the architect and founder of Roy David Architecture, Roy David.
The design strategy is well rooted in the planning of the space. Having a circular layout, the space presented a lot of challenges that the architecture team had to solve in a small time frame of just two months. As part of the overall design process the company wanted their layout to be open space. This strategy created a series of unique technical approaches designed in collaboration with the execution company, Shin Angel. From specially calculated open acoustic barriers to a supra-structure of 14 new rusted metal columns, each design element was added so that the companies brief was met fully.


text:
Q: Display First DIV beneath Second DIV say I have...

How can the end-user view div B on top of div A on their browser?
I'm trying to do this with CSS without editing the html.

A: You can use flex-box and order to acheive what you want

body {
  display: flex;
  flex-wrap:wrap;
}

#A {
  width: 100%;
  height:50px;
  background: red;
  color:white;
  order: 2;
}

#B {
  width: 100%;
  height:50px;
  background: black;
  color:white;
  order: 1;
}

A

B

A: You need to add display:flex; and flex-direction:column-reverse; to the parent of your two divs.
body{
    display:flex;
    flex-direction:column-reverse;
}

Or you can choose div's order manually with order property:
body {
  display: flex;
}

#A {
  order: 2;
}

#B {
  order: 1;
}

A: Use CSS3 flex to change the positioning of flex elements and this can be done using order property,

The CSS order property specifies the order used to lay out flex items
  in their flex container.

#box{
 display:flex;
 width:100%;
 flex-direction:column;
}
.A{
 order:2;
  background:#111;
  color:#fff;
}
.B{
 order:1;
  background:#111;
  color:#fff;
}

A

B

A: A way to do this in CSS:
The container must have display:flex attribute
Then :
#A{
   order:2;
}

#B{
   order:1;
}

You can also achieve this with jQuery
$('#B:parent').each(function () {
    $(this).insertBefore($(this).prev('#A'));
});

A: It feels unclean to do it this way but here you go (no container element needed)

#A {
  display: table-footer-group;
}

#B {
  display: table-header-group;
}

A

B

redpajama_set_name: RedPajamaStackExchange


text:
Lowering taxes, more R&D will stablise consumer electronics sector, say players
The government should consider lowering tax rate on consumer electronics and ensure a stable GST tax slab in the Union Budget 2022-23 as this would encourage consumers to buy higher ASP (average selling point) products and stabilise the sector shaken by the pandemic, industry players said ahead of the Union Budget 2022-23.
Stressing that the industry needs a stable GST tax slab to boost local manufacturing and MSMEs sectors, Avneet Singh Marwah, CEO, Super Plastronics Pvt Ltd (SPPL), a Kodak brand licensee, said that the electronics products should not exceed the 18 per cent slab.
"India can become the world's third-biggest market for televisions, leading to an estimated growth of 15 per cent in the market size, increasing to up to 16 million units annually. We would like to urge the government not to change custom duties, as the industry is moving towards a stable condition," he said.
The India Cellular & Electronics Association (ICEA) has also recommended a reduction in the GST rate on colour televisions, which, at 28 per cent for televisions above 32 inches (and 18 per cent otherwise), is inexplicably high.
"We emphasise that the 43-inch model is now considered the average for the Indian market. We would recommend the standard GST rate of 18 per cent for all television sets. A massive grey market has built up. Any revenue loss will be more than made up through increased production, sale, and the reduced grey market," it said in a paper, titled 'Major Recommendations for Union Budget 2022-23.'
"The extremely high GST rate (28 per cent) is stifling the industry. Therefore, we request that the GST rates be rationalised to 18 per cent for both fixed speed air conditioners and inverter-based air conditioners too," the ICEA added.
Hailing the government for introducing a $10 billion PLI (performance-linked scheme) for display panels and semiconductor chips, they said that they would also request Finance Minister Nirmala Sitharaman to have fixed timelines for these projects.
Pallavi Singh, Vice President, SPPL, India brand licensee of Westinghouse TV, said that given the current semiconductor shortage in the world, the government should aid the potential sector and afford schemes under the National Electronics Policy (NEP) 2019.
"Those aiming to manufacture in India and contributing towards the growth of the economy can be provided aid in the form of subsidies. The NPE can also be expanded to include and aid those already manufacturing consumer electronics in India as opposed to importing them," she said.
Since there are no open cell manufacturers in India and the viability of setting up a plant is far from certain, the Indian government should take this into consideration and reduce the custom duty on open cells to 0 per cent from the current 5 per cent, the industry players demanded.
They demanded that with the pandemic causing a major hit to the retail sector, the government should introduce schemes and rebates to help rebuild this sector.
"The government should fund a scheme for tech adoption, and government purchases of high-end IT products should be encouraged/prioritised to help create more demand," said Sandeep Lodha, Co-founder at Netweb Technologies.
"There has to be special incentive for companies investing in R&D and there must be government incentives and facilitated collaborations between the companies and premium R&D facilities," Lodha added.


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text:
ELTON JOHN NEWS
Elton John and band performed in Porto on July 14, 2016
Fan report by Julio, from La Coruña, Spain.
It was an excellent summer evening. People began slowly to enter the Festival Mares Vivas venue in Portugal to be located at the top front of the stage. After a brief performances of Foy Vance and Kelis, Elton and the band appeared promptly at 21:00.
Elton was in a good mood. He congratulated Portugal for its victory in the European Football Championship and said he loved Porto, and is thinking to back on tour soon.
I really liked that he played two songs from the "Wonderful Crazy Night" album, especially "A Good Heart", where he made a special emotion.
The "Rocket Man" intro was amazing, with pieces from "Sixty Years On" and "Take Me To The Pilot". In addition, the new version of "Candle In The Wind" starting with Elton solo piano and the band joining, was wonderful.
Notably, "I'm Still Standing" and "Goodbye Yellow Brick Road" have new videos for he background screen.
A magnificent show. Elton still standing!
Following is the complete set list of the Porto show on July 14, 2016:
Benny And The Jets
Philadelphia Freedom
All The Young Girls Love Alice
Your Sister Can't Twist (But She Can Rock And Roll)
Saturday Nights Alright for Fighting


text:
Q: Assigning a numeric value to a string through calling a function in Python 3 I am trying to make a blackjack simulator in Python 3 and I am trying to define a function that will assign a numeric value to the card it deals. I believe I am making a mistake when calling my function.
{
import random
import CardVal

cards = [
'AS', '2S', '3S', '4S', '5S', '6S', '7S', '8S', '9S', 'TS', 'JS', 'QS', 'KS',
'AC', '2C', '3C', '4C', '5C', '6C', '7C', '8C', '9C', 'TC', 'JC', 'QC', 'KC',
'AD', '2D', '3D', '4D', '5D', '6D', '7D', '8D', '9D', 'TD', 'JD', 'QD', 'KD',
'AH', '2H', '3H', '4H', '5H', '6H', '7H', '8H', '9H', 'TH', 'JH', 'QH', 'KH'
]
decks = 6 * cards

player = []
dealer = []

for i in range(2):
    player.append(decks.pop(random.randrange(len(decks) - 1)))
    dealer.append(decks.pop(random.randrange(len(decks) - 1)))

print('Players cards:', player)
print('Dealers card:', dealer[1])

total = 0
for i in range(len(player)):
    CardVal.card_value(player[i])

    total += player[i]
}

And here is the function I am calling
{
def card_value(card):
    if '2' in card:
        card = 2
    elif '3' in card:
        card = 3
    elif '4' in card:
        card = 4
    elif '5' in card:
        card = 5
    elif '6' in card:
        card = 6
    elif '7' in card:
        card = 7
    elif '8' in card:
        card = 8
    elif '9' in card:
        card = 9
    elif 'T' in card or 'J' in card or 'Q' in card or 'K' in card:
        card = 10
    elif 'A' in card:
        card = 11
}

A: Just use a dictionary:
def card_value(card):
    values = {'T': 10, 'J': 10, 'Q': 10, 'K': 10, 'A': 11}
    if card[0].isdigit():
        return int(card[0])
    else:
        return values[card[0]]


text:
\section{Introduction}

We were very much surprised when we learned during the 8$^{\text{th}}$
Marcel Grossmann Meeting in Jerusalem \cite{Piran} that Kaniel \& Itin
\cite{kani97} were able to propose a gravitational model which looks
viable at a first sight even if it had neither an Einstein-Hilbert
type of Lagrangian nor the Schwarzschild metric as an exact solution.
Their gravitational potential is represented by a quartet of 1-forms
$\vartheta^{\hat{0}},\vartheta^{\hat{1}},\vartheta^{\hat{2}},
\vartheta^{\hat{3}}$ or, for short, by $\vartheta^\alpha$, which
constitutes the coframe field of spacetime. Their vacuum field
equation is simply the wave equation with an additional `massive'
contribution depending on some scalar field $\lambda(x)$:
\begin{equation}\label{WAVE}
  \left[\square+\lambda(x)\right]\vartheta^\alpha=0\,.\end{equation}
They show that the Yilmaz-Rosen metric \cite{yilmaz58,rose73} solves
the \emph{tracefree} part of \eqref{WAVE} exactly.

Let us be a bit more specific: The Yilmaz-Rosen metric, in isotropic
coordinates, is given by
\begin{equation}\label{YILMAZ} g=e^{-\frac{2m}{r}}\,dt^2-e^{\frac{2m}{r}}\,
  \left(dx^2 +dy^2 +dz^2\right)\;,\end{equation} where
\(r^2:=x^2+y^2+z^2\). If we introduce an orthonormal coframe,
\begin{equation}\label{ORTHONORMAL}
  g=o_{\alpha\beta}\,\vartheta^\alpha\otimes\vartheta^\beta\qquad
  \text{with} \qquad o_{\alpha\beta} = \text{diag}(+1, -1, -1,
  -1)\;,\end{equation} then the following coframe, up to arbitrary
local Lorentz transformations, represents the Yilmaz-Rosen metric:
\begin{equation}\label{COFRAME}
  \vartheta^{\hat{t}}=e^{-\frac{m}{r}}\,dt\;,\quad
  \vartheta^{\hat{x}}=e^{\frac{m}{r}}\,dx \;,\quad
  \vartheta^{\hat{y}}=e^{\frac{m}{r}}\,dy \;,\quad \text{and}\quad
  \vartheta^{\hat{z}}=e^{\frac{m}{r}}\,dz\;. \end{equation} The
tracefree part of \eqref{WAVE} will be determined in Sec.\
\ref{sec:44} and turns out to be
\begin{equation}\label{TRACEFREE}\left[
    \square - \frac{1}{4} (e_\beta \rfloor \square
    \vartheta^\beta)\right]\vartheta^\alpha=0\,.\end{equation} The
coframe \eqref{COFRAME} solves the tracefree field equation
\eqref{TRACEFREE} exactly. We have verified this by means of our
computer algebra program {\tt kaniti.exi} displayed in the appendix in
Sec.\ \ref{sec:ca}.

Kaniel \& Itin tried to derive the field equation \eqref{WAVE} from a
suitable Lagrangian. For that purpose they had to assume specifically
that the variation $\delta\vartheta^\alpha$ of the coframe
$\vartheta^\alpha$ commutes with the Hodge star:
$^\star\delta\vartheta^\alpha=\delta\,^\star\vartheta^\alpha$.
However, such a commutativity is only valid for internal Yang-Mills
fields. It is violated for the coframe and the metric. Therefore the
Kaniel-Itin model is based on somewhat shaky foundations.

In the light of the results mentioned so far, the following questions
come to mind: (i) What is the source on the right hand side of the
field equation \eqref{WAVE}? (ii) Can the Yilmaz-Rosen metric also be
adjusted to the trace part of \eqref{WAVE} and, more generally, to a
possible source term on the right hand side of \eqref{WAVE}? (iii) Is
there a consistent variational principle available which would allow
to derive \eqref{WAVE}, including a source term, from a suitable
Lagrangian?  (iv) What is the (geometrical?)  meaning of the
constrained variations of Kaniel \& Itin?

The purpose of this article is to try to answer these questions.
Moreover, along our way, we will discuss some unclear points on the
commutativity of variation and Hodge star which led to some (so far
uncorrected) mistakes in the literature.---

In Sec.\ \ref{sec:model-KI} we provide some background material on how
to derive wave equations of the general type \eqref{WAVE} from
Lagrangians in Maxwell's theory and in theories of other internal
fields. Here and in the following \emph{internal} fields are those
which do not depend on the spacetime geometry (in contrast to
$\vartheta^\alpha$ and $g_{\alpha\beta}$). In this way we are able to
understand how the Lagrangian of Kaniel-Itin comes up in the first
place. But since they identify the \emph{gravitational potential} with
the \emph{coframe}, we run into trouble from the point of view of
finding a suitable Lagrangian.

Any reasonable gauge approach to gravity contains in some way the
gauging of the translation group. The simplest gauge theories of
gravity are teleparallel theories with only the translation group as
gauge group.  They already require the knowledge of how to vary the
Hodge dual of forms. In a teleparallel theory, spacetime can be
described by an \emph{orthonormal} coframe \(\vartheta^\alpha\) as the
only gravitational field variable, which is interpreted as
translational gauge potential, see \cite{gron96}. And this gauge
potential was used by Kaniel-Itin in their model. Accordingly, their
Lagrangian is a special teleparallelism Lagrangian with the additional
postulate of constrained variations. The commutativity of \(\delta\)
and \({}^\star\) is, in general, \emph{not} fulfilled for gauge
theories of {\it external\/} (or spacetime) groups, i.e., for
gravitational gauge theories. In this case it is important to know the
commutator $\delta{}^\star-{}^\star\delta$ of the variation $\delta$
and the Hodge star $^\star$.

Therefore, in Sec.\ \ref{sec:hodge} we will derive the master formula
\eqref{eq:var-hodge} for $\delta{}^\star-{}^\star\delta$. We will
include general variations of the components \(g_{\alpha\beta}\) of
the Riemannian metric $g$ besides those of a (not necessarily
orthonormal) coframe \(\vartheta^\alpha\). If we \emph{insist}, in
accordance with the Kaniel-Itin postulate, on commutativity of
$\delta$ and $^\star$, then the variations $\delta g_{\alpha\beta}$ of
the components of the metric are no longer independent and can be
expressed in terms of the variation $\delta\vartheta^\alpha$ of the
coframe, see \eqref{constraint}.

In Sec.\ \ref{sec:tele} we give a short overview of teleparallelism
theories and the relevant qua\-dra\-tic Lagrangians. We will discuss
the viable set of Lagrangians and display the results in Table 1. We
will show that the KI-Lagrangian, for arbitrary variations, is not
viable. Some errors in the literature (see Schweitzer et al.\
\cite{straumann1,straumann2}) are rectified.

In Sec.\ \ref{sec:KI}, we evaluate the model of Kanin \&
Itin \cite{kani97}. The field equation of the constrained variational
principle is the \emph{antisymmetric part} of a wave equation
for $\vartheta^\alpha$, in contrast to the full wave equation as
claimed by Kaniel and Itin.

The Yilmaz-Rosen metric, found by Yilmaz \cite[Eqs.(18) and
(20)]{yilmaz58} in 1958 as a solution in the context of a scalar field
theory of gravitation, also turned out to be a solution of the
bi-metric theory of gravitation of Rosen \cite{rose73}; cf.\ also
\cite{rose74,yilmaz76}.  And, in the Kaniel-Itin model, it solves the
tracefree wave equation.  In Sec.\ \ref{sec:YR-sol} we compare the
Yilmaz-Rosen with the Schwarzschild metric and give, in Sec.\
\ref{YR-mot}, a motivation for the emergence of the Yilmaz-Rosen
metric. Finally, we investigate the implications that would arise if
the Yilmaz-Rosen metric is considered to be a solution of the field
equation of Kaniel \& Itin including its trace. In Sec.\ 6 we collect
our arguments.

\section{Prolegomena to the Kaniel-Itin model}\label{sec:model-KI}

\subsection{Maxwell's theory and the wave equation}
The kinetic part of the Lagrangian of a Yang-Mills theory is
conventionally built from the first derivative of the gauge potential
$A$ and the corresponding Hodge dual. For an internal gauge group,
such as for the $U(1)$ or the $SU(2)$, the gauge potential $A$ is
\emph{independent} of the metric $g$ or the coframe $\vartheta^\alpha$
of the underlying spacetime manifold. Then the variation \(\delta\) of
$A$ commutes with the Hodge star operator \({}^\star\). Let us
illustrate this for Maxwell's theory, i.e., for $U(1)$-gauge theory in
Minkowski spacetime.

The Maxwell Lagrangian is given by\footnote{We are using the calculus
  of exterior differential forms, cf.\ \cite{Choquet,Thirring}. Our
  conventions are fixed in \cite{hehl95}.}
\begin{equation} \label{U1Lagrangian}L_{\rm
    Max}=\frac{1}{2}\,dA\wedge\ensuremath{{}^\star} dA\,.
\end{equation} The variation of the 1-form \(A\) is
independent of the variations \(\delta\vartheta^\alpha\) or \(\delta
g_{\alpha\beta}\); furthermore, it commutes with the exterior
derivative, since the variation is defined in this way.  Therefore,
with the \emph{co}derivative $d^\dagger:=-\ensuremath{{}^\star}
d\ensuremath{{}^\star}$, we find
\begin{equation}\delta
L_{\rm Max}= d(\delta A\wedge\ensuremath{{}^\star} d A)-\delta A\wedge
\ensuremath{{}^\star}
d^\dagger d A\,.\end{equation}
Thus the vacuum field equation reads:
\begin{equation} -\ensuremath{{}^\star}
  d^\dagger d A=0\,.\label{eq:MaxEL}\end{equation} Additionally, we
take the Lorentz condition
\begin{equation} \label{Lorentz}
d^\dagger A=0 \end{equation}
as a gauge
condition. Then, introducing the d'Alembertian
\begin{equation}\square:= d^\dagger d+d
  d^\dagger=-{}^\star d{}^\star d-d{}^\star d{}^\star\,,\end{equation}
the vacuum field equation can be rewritten as
\begin{equation}
- \square\,\ensuremath{{}^\star} A=0\,. \label{eq:wave-max} \end{equation}

One could try to derive (\ref{eq:wave-max}) directly by supplementing
(\ref{U1Lagrangian}) with a suitable Lagrangian. The choice
\begin{equation} L_{\rm Max}^\dagger:=\frac{1}{2}\,d^\dagger A\wedge
  \ensuremath{{}^\star} d^\dagger A\,
\label{eq:lmax-dagger}\end{equation}
looks suggestive. It leads to the field equation
\begin{equation}\label{Maxadjoint}
 -\ensuremath{{}^\star}
  d\, d^\dagger A=0\,,
\end{equation}which should be compared with \eqref{eq:MaxEL}. Consequently
the sum of the Lagrangians \eqref{U1Lagrangian} and
\eqref{eq:lmax-dagger}, enriched by a matter Lagrangian,
\begin{equation}\label{Alt}
  L_{\rm Max}+L_{\rm Max}^\dagger+ L_{\rm mat}
  =\frac{1}{2}\left(dA\wedge\ensuremath{{}^\star} dA +d^\dagger
    A\wedge\ensuremath{{}^\star} d^\dagger A\right)+ L_{\rm mat}
\end{equation}
would yield directly the wave equation:\begin{equation}\label{Altwave}
  \square\,{}^\star A=\frac{\delta L_{\rm mat}}{\delta A}\,.\end{equation}
However, the Lagrangian (\ref{eq:lmax-dagger}) is not gauge-invariant:
For the regauging by means of the arbitrary function \(f\),
\begin{equation} A \longrightarrow A + df\;,
\label{eq:maxdagger-gauge}\end{equation}
one finds\footnote{Incidentally, as pointed out by Obukhov, the
  Lagrangian \eqref{eq:lmax-dagger} represents an example for a
  Lagrangian of a gauge theory of \(p\)-forms, see \cite{yuri82a,
    yuri82b}. Then, instead of \eqref{eq:maxdagger-gauge}, one has \(A
  \rightarrow A + d^{\dagger}\phi\) as gauge transformation, since
  \(d^{\dagger}d^{\dagger}=0\). And the new ``Lorentz condition'' is
  \(dA=0\).}
\begin{equation}\label{gaugevariance}
  L_{\rm Max}^\dagger\longrightarrow L_{\rm Max}^\dagger +
  \ensuremath{{}^\star} d^{\dagger} df \wedge \left( d^{\dagger} A +
    \frac{1}{2} d^{\dagger} df \right) \;.
\end{equation} Accordingly, the Lagrangian \eqref{Alt} has to be rejected.
We can obtain the wave equation \eqref{eq:wave-max} only in the
\emph{special} gauge \eqref{Lorentz}, \emph{after} the derivation of
the field equation by means of the variational principle with the
Lagrangian \eqref{U1Lagrangian}.

\subsection{A quartet of massive one-form fields}

If we used massive fields, then we would have no difficulties with
lack of gauge invariance, because the mass term is not gauge invariant
anyway. Since we want to study gravity \`a la Kaniel-Itin, we start
with a quartet of 1-form fields $k^I$, where $I$ is an {internal}
index with $I=\check{0},\check{1},\check{2},\check{3}$. We again
derive a wave type equation as in the last subsection, but we now add
a massive term for each of the four fields:
\begin{equation}\label{k-lagrangian}
  L_k=\frac{1}{2}\left(dk^I\wedge{}^\star dk^I +\;d^\dagger
    k^I\wedge{}^\star d^\dagger k^I-m_{(I)}\,k^I\wedge{}^\star
    k^I\right)+L_{\text{mat}} \,.\end{equation} We vary with respect to
$k^I$ and find as the Euler-Lagrange equation:
\begin{equation}\label{k-fieldeq}
  \left(\square +m_{(I)}\right){}^\star k^I= \frac{\delta
    L_{\text{mat}}}{\delta k^I} \,.\end{equation} One could also think
of an additional Higgs-type (or `cosmological') term.  Then we would
have
\begin{equation}\label{k-lagrangian'}\begin{split}
  L_{k'}=\frac{1}{2}\left(dk^I\wedge{}^\star dk^I\right.&\left. +\;d^\dagger
    k^I\wedge{}^\star d^\dagger
    k^I-m_{(I)}\,k^I\wedge{}^\star
    k^I\right)\\&-\frac{\lambda}{4!}\,{\epsilon}_{IJKL}\,k^I\wedge
  k^J\wedge k^K\wedge k^L+L_{\text{mat}} \,,\end{split}\end{equation}
and, as field equation,
\begin{equation}\label{k-fieldeq'}
  \left(\square +m_{(I)}\right){}^\star
  k^I+\frac{\lambda}{3!}\,\epsilon_{IJKL}\, k^J\wedge k^K\wedge k^L
  =\frac{\delta L_{\text{mat}}}{\delta k^I} \,.\end{equation}

This is as near as we can approach the field equation \eqref{WAVE}.
Since currents are 3-forms, we take the Hodge dual of \eqref{WAVE} and
remember $^\star\,\square=\square\,^\star$. Furthermore we put a
source term on its right hand side. In gravitational theory, this can
be only the matter current $\Sigma_\alpha$, representing the
energy-momentum flux of matter. Then the completed Kaniel-Itin field
equation reads:
\begin{equation}\label{KI-fieldeq}\left[\square +\lambda(x)\right]{}^\star
  \vartheta_\alpha=\frac{\delta L_{\text{mat}}}
  {\delta\vartheta^\alpha} =:\Sigma_\alpha\,.\end{equation} Since
$\lambda(x)$ is a \emph{function}, it cannot be identified with some
constant mass $m_{(I)}$. Also an interpretation of $\lambda(x)$ as a
cosmological constant is obviously meaningless. Therefore the
equations \eqref{k-fieldeq'} and \eqref{KI-fieldeq} have to be
carefully distinguished. In future, we will refer to \eqref{KI-fieldeq}
as the (completed) Kaniel-Itin field equation. Eq.\eqref{KI-fieldeq}
represents the heart of their theory.

\subsection{Relation to teleparallel theories}\label{sec:KI+tele}

Kaniel-Itin proposed in \cite[Eq.(8)]{kani97} the following Lagrangian
for the derivation of the vacuum version of \eqref{KI-fieldeq}:
\begin{equation}
  \stackrel{\text{orig}}{V_{\rm{KI}}} \,=\, \frac{1}{2}\left[
    d\vartheta^\alpha \wedge \ensuremath{{}^\star} d\vartheta_\alpha -
    d^\dagger \vartheta^\alpha\wedge\ensuremath{{}^\star} d^\dagger
    \vartheta_\alpha+\lambda(x)(\vartheta^\alpha\wedge{}^\star
    \vartheta_\alpha-4\eta)\right]\,.
\end{equation}
We used our notation here. We want to correct this Lagrangian in two
respects: (i) If we compare \eqref{k-lagrangian'} with
\eqref{k-fieldeq'}, it is clear that we should \emph{add} the first
two terms, instead of subtracting them. (ii) The Lagrange multiplier
term is a hoax since the expression multiplying $\lambda(x)$
\emph{vanishes} identically: We have quite generally
$\vartheta^\alpha\wedge\eta_\beta=\delta^\alpha_\beta\eta$, with
$\eta_\beta:={}^\star\vartheta_\beta$, and the trace of this equation
proves our contention. Taking care of both objections, we will call
\begin{equation} V_{\rm{KI}} \,=\,
  \frac{1}{2}\left( d\vartheta^\alpha \wedge \ensuremath{{}^\star}
    d\vartheta_\alpha + d^\dagger
    \vartheta^\alpha\wedge\ensuremath{{}^\star} d^\dagger
    \vartheta_\alpha\right)\,\label{KI-lagrangian} \end{equation} the
(corrected) Kaniel-Itin Lagrangian. If we recall that we were able to
derive \eqref{k-fieldeq'} from \eqref{k-lagrangian'} only because the
$k^I$ was an internal field, the variation of which commutes with the
star, then it becomes clear that Kaniel-Itin field equation
{\em (\ref{KI-fieldeq})}, for $\Sigma_\alpha=0$, \emph{is not the Euler-Lagrange
  equation of} {\em (\ref{KI-lagrangian})}.

In the Kaniel-Itin model, the Hodge star {no longer commutes} with the
variation $\delta\vartheta^\alpha$ since their gravitational potential
$\vartheta^\alpha$ is inseparably connected to the spacetime manifold.
A postulate of Kaniel-Itin to the opposite, see \cite[statement
between Eqs.(7) and (8)]{kani97}, is without foundation, at least from
a geometrical point of view. Consequently, we will give up this
postulate. The Kaniel-Itin Lagrangian belongs to the so-called
teleparallelism models of gravity. We will come back to this in Sec.\
\ref{sec:tele}.

The addition of the adjoint piece in \eqref{KI-lagrangian} does {not}
break the translational invariance since
\(d\ensuremath{{}^\star}\vartheta^\alpha\), and thus
\(d^{\dagger}\vartheta^\alpha\), can be rewritten in terms of
\(d\vartheta^\alpha\) (shown here for an orthonormal coframe):
\begin{equation} d\,\ensuremath{{}^\star} \vartheta^\alpha =
  d\eta^\alpha = d\vartheta_\beta \wedge \eta^{\alpha\beta}
\label{eq:d-eta}\;. \end{equation}
An analogous procedure is \emph{not} available in the Maxwellian case
for $d{}^\star A$, that is, $d{}^\star A$ cannot be expressed in terms
of $dA$. Therefore $L_{\rm Max}^\dagger$ is not gauge invariant, see
\eqref{gaugevariance}, and has to be rejected as a decent Lagrangian.

\section{Variation of the Hodge dual of a form}\label{sec:hodge}
\subsection{The master formula}
In order to install the \emph{variation} $\delta$ as a derivation, we
demand that it fulfills an even Leibniz rule,\label{sec:rules}
\begin{equation}
\delta(\omega_1 \wedge \omega_2) = \delta\omega_1 \wedge \omega_2 +
\omega_1 \wedge \delta\omega_2 \;,\label{eq:var-leibniz}
\end{equation}
where \(\omega_1\) and \(\omega_2\) are arbitrary exterior
differential forms.  The Leibniz rule is even, because the variation
does not change the degree of the form.
In contrast to this, the interior product \(v\rfloor\) (here $v$ is a
vector) and the exterior derivative \(d\) decrease or increase,
respectively, the degree of the form by one and fulfill an \emph{odd}
Leibniz rule.

Furthermore, we need a relation between the variation and the exterior
derivative. According to the definition of the variation, they simply
commute:
\begin{equation} [d,\delta] = 0\;. \label{eq:var-comm} \end{equation}
Let us now turn to the Hodge star operator, see \cite{hehl95}. It maps
a $p$--form
$\psi=\frac{1}{p!}\,\psi_{\alpha_1\cdots\alpha_p}\vartheta^{\alpha_1}
\wedge\cdots\wedge\vartheta^{\alpha_p}$ into an $(n-p)$--form
$^\star\psi\,$; here $n$ is the dimension of the manifold, i.e., in our
case $n=4$. In terms of components we have
\begin{multline}\label{Hodgestar}
    ^\star\psi:=  \frac{1}{(n-p)!\, p!}\, \sqrt{|\det g_{\mu\nu}|}\,
    g^{\alpha_1\gamma_1}\cdots g^{\alpha_p\gamma_p}\,\times\\
     \epsilon_{\alpha_1\cdots\alpha_p\beta_1\cdots\beta_{n-p}}\,
    \psi_{\gamma_1\cdots\gamma_p}\,
    \vartheta^{\beta_1}\!\wedge\cdots\wedge\vartheta^{\beta_{n-p}}\,,
\end{multline}
where $\epsilon$ is the Levi-Civita symbol. Besides the
$\vartheta$-basis $\Bigl\{{ 1},\vartheta^{\alpha_1},
\vartheta^{\alpha_1}\wedge\vartheta^{\alpha_2},\dots,
\vartheta^{\alpha_1}\wedge\vartheta^{\alpha_2}\wedge\dots
\wedge\vartheta^{\alpha_n}\Bigr\}$, having the Hodge star at our
disposal, we may define the so-called $\eta$-basis:
\begin{equation}\label{etabasis}\begin{split}&\Bigl\{\eta,\eta^{\alpha_1},
  \eta^{\alpha_1\alpha_2},\dots,\eta^{\alpha_1\alpha_2\cdots
    \alpha_n}\Bigr\}:=\\ &  \qquad\Bigl\{{}^\star{ 1},{}^\star
  \vartheta^{\alpha_1},
  {}^\star(\vartheta^{\alpha_1}\wedge\vartheta^{\alpha_2}),\dots,
  {}^\star(\vartheta^{\alpha_1}\wedge\vartheta^{\alpha_2}\wedge\dots
  \wedge\vartheta^{\alpha_n})\Bigr\}\,.\end{split}
\end{equation}

Now we can derive the desired expression for
\(\delta\ensuremath{{}^\star}\phi\), for an arbitrary \(p\)-form
\(\phi\). We can saturate the $(n-p)$-form $^\star\phi$ with coframes
$\vartheta^\beta$ such as to arrive at the $n$-form
\newcommand{\ptetr}{\ensuremath{\vartheta^{\beta_1}\wedge \dots
\wedge \vartheta^{\beta_p}}
\begin{equation}\label{vorher}
  \ptetr \wedge \ensuremath{{}^\star} \phi
  \stackrel{\eqref{eq:hodge-wedge}}{=} \phi \wedge
  \ensuremath{{}^\star} \left(\ptetr\right)
  \stackrel{\eqref{eq:eta-def}}{=} \phi \wedge
  \eta^{\beta_1\dots\beta_p} \;.\end{equation} We vary \eqref{vorher}.
Then the even Leibniz rule \eqref{eq:var-leibniz} for the variation
leads to
\begin{multline}\label{varphi
    \delta \left( \ptetr \right
\wedge \ensuremath{{}^\star} \phi +
    \ptetr \wedge \delta \ensuremath{{}^\star} \phi = \\
\delta \phi
    \wedge \eta^{\beta_1\dots\beta_p} + \phi \wedge \delta
    \eta^{\beta_1\dots\beta_p} \;
\end{multline} Thus,
apparently, we know how to vary the Hodge star, provided we know how
to vary the $\eta$-basis. The variation of the $(n-p)$-form
$\eta^{\alpha_1\cdots\alpha_p}$ is computed in the appendix in Sec.\
\ref{sec:2.2}. It turns out to be
\begin{equation}
\label{eq:var-eta} \delta \eta^{\beta_1\dots\beta_p} =
\delta\vartheta^\mu \wedge \left( e_\mu \rfloor
  \eta^{\beta_1\dots\beta_p} \right) + \delta
g_{\kappa\lambda}\left( \vartheta^{(\kappa |} \wedge
  \eta^{\beta_1 \dots \beta_p | \lambda)} - \frac{1}{2} \,
   g^{\kappa\lambda} \eta^{\beta_1\dots\beta_p} \right) \,.
\end{equation}
Incidentally, for the special choice of an orthonormal tetrad, we have
\(\delta g_{\alpha\beta}=0\), and the last two terms vanish; in
particular, we then have \(\delta \eta^{\beta_1\dots\beta_n}=0\). But we
will not introduce this specialization at the present stage.

If we resolve (\ref{varphi}) with respect to $\delta
\ensuremath{{}^\star} \phi$, then, after some intermediary algebra,
see Sec.\ \ref{deduc33} of the appendix, we find for arbitrary
\(p\)-forms \(\phi\) the master formula:
\begin{equation}
  \boxed{\begin{split} \left(\delta \ensuremath{{}^\star}
        -\ensuremath{{}^\star} \delta\right) \phi & = \;\delta
      \vartheta^\alpha \wedge \left(e_\alpha \rfloor
        \ensuremath{{}^\star} \phi \right) - \ensuremath{{}^\star}
      \big[ \delta \vartheta^\alpha \wedge \left( e_\alpha \rfloor
        \phi \right)\big] \label{eq:var-hodge} \\ & + \delta
      g_{\alpha\beta}\Big[\vartheta^{(\alpha}\! \wedge\
      ( e^{\beta)
 \rfloor \ensuremath{{}^\star} \phi ) -
      \frac{1}{2}\,g^{\alpha\beta}\;
      \ensuremath{{}^\star} \phi\Big] \;.
\end{split}}
\end{equation}
Again, for {orthonormal} (co)frames, the terms in the second line
vanish since \(\delta g_{\alpha\beta}=0\).

\subsection{Constrained variations \`a la Yang-Mills}\label{sec:constraint}
\textbf{Proposition}: \emph{The condition
\begin{equation}
\delta\ensuremath{{}^\star}\phi=\ensuremath{{}^\star}\delta\phi
\label{eq:comm}\end{equation}
for an arbitrary \(p\)-form \(\phi\) is equivalent to
the following relation between the variation of the metric and the coframe:
\begin{equation}\label{constraint}
  \delta g_{\alpha\beta}=-2g_{\gamma (\alpha}\,e_{\beta
    )}\rfloor\delta\vartheta^\gamma =-2\omega_{(\alpha\beta)}\,,\qquad
  \text{with}\qquad
  \delta\vartheta^\gamma=\omega_\delta{}^\gamma\vartheta^\delta
  \,.\end{equation} Therefore, for an orthonormal coframe, the allowed
variations are of the Lorentz type, i.e.,
$\omega_{(\alpha\beta)}\equiv 0.$}

To prove\footnote{It was Yuri Obukhov who suggested essential parts of
  this proof to us.} this equivalence we first assume that
\eqref{eq:comm} is valid. We apply this constrained variation to the
volume $n$-form \(\eta:={}^\star 1\), defined in \eqref{etabasis},
\begin{equation} \delta\eta = \delta(\ensuremath{{}^\star} 1) =
  \ensuremath{{}^\star}(\delta 1) \equiv 0\;,\label{P1}
\end{equation}
since the constant 1 is not varied.
In turn, for the identity
\begin{equation} \vartheta^\alpha\wedge\eta_\beta = \delta^\alpha_\beta\,\eta
\label{P2}\end{equation}
we find:
\begin{equation} \delta\vartheta^\alpha\wedge\eta_\beta +
\vartheta^\alpha\wedge\delta\eta_\beta = 0\;.\label{P3}
\end{equation}
The commutation rule \eqref{eq:comm} applied to \(\vartheta_\beta\)
yields
\begin{equation} \delta\eta_\beta = \delta(\ensuremath{{}^\star}
  \vartheta_\beta)
  =\ensuremath{{}^\star}(\delta\vartheta_\beta)\;.\end{equation} Thus,
by exterior multiplication with $\vartheta^\alpha$ we arrive at
\begin{equation}
  \vartheta^\alpha\wedge\delta\eta_\beta =
  \vartheta^\alpha\wedge\ensuremath{{}^\star} (\delta\vartheta_\beta)=
  \delta\vartheta_\beta\wedge\ensuremath{{}^\star}\vartheta^\alpha=
  \delta(g_{\beta\gamma}\vartheta^\gamma)\wedge\eta^\alpha\;.\end{equation}
On substitution into \eqref{P3}, we find
\begin{equation} \delta\vartheta^\alpha\wedge\eta_\beta +g_{\beta\gamma}\,
  \delta\vartheta^\gamma\wedge\eta^\alpha+\delta
  g_{\beta\gamma}\,\vartheta^\gamma\wedge \eta^\alpha =0 \label{P4}
\end{equation}or, since $\eta_\beta=e_\beta\rfloor\eta$,
\begin{equation}\label{strain}
  \delta g_{\alpha\beta}=-2g_{\gamma(\alpha}\,e_{\beta)}\rfloor\delta
  \vartheta^\gamma\,.\end{equation} The 1-form
$\delta\vartheta^\alpha$ can be expanded with respect to the coframe:
\begin{equation} \label{eq:var-exp} \delta \vartheta^\gamma =
  \omega_\delta{}^\gamma \vartheta^\delta \,.\end{equation} We insert
\eqref{eq:var-exp} into \eqref{strain}. Then we find
\begin{equation} \label{previous}
  \delta g_{\alpha\beta}=-2 \omega_{(\alpha\beta)} \,.
\end{equation}

To investigate the reverse part of the proposition, we apply the
general rule \eqref{eq:var-hodge} for the variations of Hodge dual
forms and use \eqref{eq:var-exp} and \eqref{previous}:
\begin{equation}
  \begin{split} \left(\delta \ensuremath{{}^\star}
      -\ensuremath{{}^\star} \delta\right) \phi & =
    \;\omega^{\beta\alpha}\Big[\vartheta_\beta \wedge \left(e_\alpha
      \rfloor \ensuremath{{}^\star} \phi \right)-
    \ensuremath{{}^\star} \big[\vartheta_\beta \wedge \left( e_\alpha
      \rfloor \phi \right)\big]\Big] \label{eq:var-hodge-sub} \\
    &\quad -2 \omega^{(\beta\alpha)}\,\vartheta_{\alpha}\! \wedge\!  (
    e_{\beta} \rfloor \ensuremath{{}^\star} \phi
    )+\omega_\gamma{}^\gamma\; \ensuremath{{}^\star} \phi\\&=
    \;\omega^{\beta\alpha}\Big[-\vartheta_\alpha \wedge \left(e_\beta
      \rfloor \ensuremath{{}^\star} \phi \right)-
    \ensuremath{{}^\star} \big[\vartheta_\beta \wedge \left( e_\alpha
      \rfloor \phi \right)\big]+g_{\alpha\beta}\,\ensuremath{{}^\star}
    \phi\Big] \\&=
    \;\omega^{\beta\alpha}\Big[e_\beta\rfloor(\vartheta_\alpha \wedge
    \ensuremath{{}^\star} \phi )- \ensuremath{{}^\star}
    \big[\vartheta_\beta \wedge \left( e_\alpha \rfloor \phi
    \right)\big]\Big]\\ & \stackrel{\text{\eqref{eq:hodge-coframe1},
        \eqref{eq:hodge-coframe3}}}{=}
    \omega^{\beta\alpha}\,e_\beta\rfloor{}^\star(e_\alpha\rfloor\phi)\Big[
    \frac{1}{(-1)^{p-1}}-(-1)^{p-1}\Big]= 0\;.
\end{split}
\end{equation}
Thus the proposition is proved.

\section{Teleparallelism theories of gravity}\label{sec:tele}

A Minkowski space is invariant under rigid translations. In order to
make a manifold locally translation invariant, one can introduce a
gauge potential by means of a dynamical coframe \(\vartheta^\alpha\),
see \cite{gron97,gron96,egg1,eggannals,haynew,Thirring,Wallner}. Such
a spacetime carries a torsion, but no curvature: It is a so-called
\emph{Weitzenb\"ock spacetime}. Then, by picking a suitable frame, the
connection \(\Gamma_\alpha{}^\beta\) can always globally be
transformed to zero.

Therefore, in a teleparallel theory, the coframe \(\vartheta^\alpha\)
is the basic gravitational field variable. Furthermore, let a metric
\(g\) be given of Minkowskian signature, i.~e.~of index 3 (number of
negative eigenvalues of the metric). For the rest of this paper we
choose the coframe to be \emph{orthonormal}, \(g := o_{\alpha\beta}\,
\vartheta^\alpha \otimes \vartheta^\beta\), with \(o_{\alpha\beta} =
\text{diag}(+1, -1, -1, -1)\), and raise and lower the frame indices
by means of \(o_{\alpha\beta}\).

\subsection{The Rumpf Lagrangians}
According to Rumpf \cite{rumpf78}, a general quadratic Lagrangian for
the coframe \(\vartheta^\alpha\) can be expanded in terms of the
\emph{gauge-invariant} translational Lagrangians ($\ell=$ Planck
length, $\Lambda=$ cosmological constant $=\rho_0/2$):
\begin{equation}
  V = \frac{1}{2\ell^2}\sum_{K=0}^4 \rho_{K} \; {}^{[K]}V\;,
  \label{eq:decomp-rumpf}
\end{equation}
with
\begin{subequations}
\label{eq:rumpf}
\begin{align}
  \,{}^{[0]}V &= \frac{1}{4} \, \vartheta^\alpha \wedge
  \ensuremath{{}^\star} \vartheta_\alpha = \eta \;,\\ \,{}^{[1]}V &=
  d\vartheta^\alpha \wedge \ensuremath{{}^\star} d\vartheta_\alpha
  \qquad \text{(pure Yang-Mills type)}\;, \\\, {}^{[2]}V &=
  \Big(d\vartheta_\alpha \wedge \vartheta^\alpha \Big) \wedge
  \ensuremath{{}^\star} \left( d\vartheta_\beta \wedge \vartheta^\beta
  \right) \;, \\\, {}^{[3]}V & = \left(d\vartheta^\alpha \wedge
    \vartheta^\beta \right) \wedge \ensuremath{{}^\star} \left(
    d\vartheta_\alpha \wedge \vartheta_\beta \right) =
  d\vartheta^\alpha \wedge \vartheta^\beta \wedge \left( e_\beta
    \rfloor \ensuremath{{}^\star} d\vartheta_\alpha \right) = 2 \,
  {}^{[1]}V \;, \\\, {}^{[4]}V &= \left(d\vartheta_\alpha \wedge
    \vartheta^\beta \right) \wedge \ensuremath{{}^\star} \Big(
  d\vartheta_\beta \wedge \vartheta^\alpha \Big) \;.
\end{align}
\end{subequations}
Since \({}^{[3]}V =2 \,{}^{[1]}V\), we can always put \(\rho_{3}=0\).

\subsection{The irreducible Lagrangians}
Alternatively, the field strength \(d\vartheta^\alpha\) can be
decomposed into three pieces which transform irreducibly under the
Lorentz group:
\begin{equation}
d\vartheta^\alpha\,=\,{}^{(1)}d\vartheta^\alpha +
{}^{(2)}d\vartheta^\alpha + {}^{(3)}d\vartheta^\alpha
\;. \label{eq:irred-decomp}
\end{equation}
Here we defined (in parentheses we are mentioning the
corresponding names of our computer algebra programs):
\begin{subequations}
\label{eq:irred}
\begin{align}
  {}^{(1)}d\vartheta^\alpha &:= d\vartheta^\alpha -
  {}^{(2)}d\vartheta^\alpha-{}^{(3)}d\vartheta^\alpha &&
  \texttt{(tentor),}\\ {}^{(2)}d\vartheta^\alpha &:= {1\over
    3}\vartheta^\alpha\wedge(e_\beta\rfloor d\vartheta^\beta) &&
  \texttt{(trator),}\\ {}^{(3)}d\vartheta^\alpha &:= {1\over 3}
  e_\alpha \rfloor \left(\vartheta^\beta\wedge d \vartheta_\beta\right
  ) && \texttt{(axitor).}
\end{align}
\end{subequations}
In terms of the numbers of components involved, we have the
decomposition \(24 = 16 \oplus 4 \oplus 4\).

Then we can write
\begin{equation} V =\frac{1}{2\ell^2}\left[a_0\,
    \eta+\sum_{I=1}^3 a_I \left( d\vartheta^\alpha \wedge
      \ensuremath{{}^\star} \, {}^{(I)}d\vartheta_\alpha \right)
  \right]\;. \label{eq:decomp-irred}
\end{equation}

We substitute \eqref{eq:irred-decomp} and \eqref{eq:irred} into
\eqref{eq:rumpf}. Then a comparison between \eqref{eq:decomp-rumpf}
and \eqref{eq:decomp-irred} yields
\begin{align}
\label{eq:rho-a} \rho_{1} & = \frac{1}{3} \, \left(a_2 + 2 a_1 \right)  \;,&
\rho_{2} & = \frac{1}{3} \, \left(a_3 - a_1 \right) \;,&
\rho_{4} & = \frac{1}{3} \, \left(a_1 - a_2 \right) \;,\\
\label{eq:a-rho} a_1 & = \rho_{1} + \rho_{4} \;,&
a_2 & = \rho_{1} - 2 \rho_{4} \;, &
a_3 & = \rho_{1} + 3 \rho_{2} + \rho_{4} \;,
\end{align}
and, additionally \(a_0=\rho_0=2\Lambda\).
These relations were checked by means of a computer algebra program, see
Sec.\ \ref{sec:ca} of the appendix.

\subsection{Field equation}
The field equation of a general translation invariant Lagrangian reads
\cite{gron97}
\begin{equation} dH_\alpha - E_\alpha = \Sigma_\alpha
\;,\label{eq:field-orig}\end{equation}
with \begin{equation} H_\alpha : = -\frac{\partial V}{\partial
d\vartheta^\alpha}\;,  \quad E_\alpha
:= \frac{\partial V}{\partial \vartheta^\alpha}\;, \quad
\text{and} \qquad \Sigma_\alpha := \frac{\delta L_{\text{mat}}}{\delta
\vartheta^\alpha} \label{eq:definis}\;. \end{equation}
In \eqref{eq:definis}, the partial derivatives are implicitly defined by
means of the variation of the Lagrangian:
\begin{equation} \delta V = \delta \vartheta^\alpha \wedge
\frac{\partial V}{\partial
\vartheta^\alpha} + \delta\, d\vartheta^\alpha \wedge \frac{\partial V}{\partial
d\vartheta^\alpha} \;. \end{equation}

If we use the abbreviations \eqref{eq:definis}, we don't need our
master formula for the computation of the field equation.
Alternatively, one can take the Lagrangian \eqref{eq:decomp-rumpf}
together with \eqref{eq:rumpf} and vary the resulting expression by
using \eqref{eq:var-hodge} with $\delta g_{\alpha\beta}=0$. This
yields the explicit form of the field equation \eqref{eq:field-orig},
cf.\ Kopczy\'nski \cite{Wojtek}\footnote{Kopczy\'nski denoted the
  Rumpf Lagrangians by $K$. We have the following translation rules:
  $K^1={}^{[4]}V$, $K^2={}^{[2]}V$, $K^3={}^{[1]}V$. In
  \eqref{genLag}, the second derivatives of the coframe are exactly
  the same (for $\ell^2=1$) as those in the corresponding three
  equations of Kopczy\'nski \cite[top of p.\ 503]{Wojtek}.}:
\begin{equation}\label{genLag}
\begin{split}
  - 2\,&\ell^2\Sigma_\alpha = 2\rho_1 \; d\,{}^\star d\vartheta_\alpha
  - 2\rho_2\; \vartheta_\alpha \wedge d \;{}^\star\left(
    d\vartheta^\beta\wedge \vartheta_\beta\right) - 2\rho_4\;
  \vartheta_\beta \wedge d \; {}^\star \left( \vartheta_\alpha \wedge
    d \vartheta^\beta \right) \\ & + \rho_1 \Big[ e_\alpha \rfloor
  \left( d\vartheta^\beta \wedge {}^\star d\vartheta_\beta \right) - 2
  \left( e_\alpha \rfloor d\vartheta^\beta \right) \wedge {}^\star
  d\vartheta_\beta\Big] \\ & + \rho_2\;\Big[2 d\vartheta_\alpha \wedge
  {}^\star \left( d\vartheta^\beta \wedge \vartheta_\beta \right) \\&
 \quad + e_\alpha \rfloor \left( d\vartheta^\gamma \wedge
    \vartheta_\gamma \wedge {}^\star \left( d\vartheta^\beta \wedge
      \vartheta_\beta\right) \right) - 2\left( e_\alpha \rfloor
    d\vartheta^\beta\right) \wedge \vartheta_\beta \wedge {}^\star
  \left( d\vartheta^\gamma \wedge \vartheta_\gamma\right)\Big] \\ & +
  \rho_4\Big[2 d\vartheta_\beta \wedge {}^\star \left( \vartheta_\alpha
    \wedge d \vartheta^\beta\right)\\&\quad + e_\alpha \rfloor
  \left( \vartheta_\gamma \wedge d\vartheta^\beta \wedge {}^\star
    \left( d\vartheta^\gamma \wedge \vartheta_\beta \right)\right) - 2
  \left( e_\alpha \rfloor d\vartheta^\beta\right) \wedge
  \vartheta_\gamma \wedge {}^\star \left( d\vartheta^\gamma \wedge
    \vartheta_\beta \right)\Big] \,.
\end{split}
\end{equation}
In the first line of this equation we displayed the leading terms
containing second derivatives of the coframe. In the remaining terms
there enter only first derivatives.

\subsection{Decomposition of the Kaniel-Itin Lagrangian}
In order to better recognize the structure of the KI-Lagrangian, one
can decompose it in its irreducible pieces as well as into the Rumpf
Lagrangians.\label{sec:KI-decomp}

The first term of the KI-Lagrangian \eqref{KI-lagrangian} is exactly
the Yang-Mills type Lagrangian \({}^{[1]}V\), cf.\ with
\eqref{eq:rumpf}.  The term \(d^{\dagger} \vartheta^\alpha \wedge
\ensuremath{{}^\star} d^{\dagger} \vartheta_\alpha =
-d\ensuremath{{}^\star} \vartheta^\alpha \wedge \ensuremath{{}^\star}
d\ensuremath{{}^\star} \vartheta_\alpha\) is left for scrutiny. Using
formula \eqref{eq:d-eta} and other rules, see, e.g., \cite{hehl95} and
Sec.\ \ref{formulae} for details, we find
\begin{equation}
\begin{split}
  -d\,\ensuremath{{}^\star} \vartheta^\alpha \wedge
  \ensuremath{{}^\star} d\, \ensuremath{{}^\star} \vartheta_\alpha & =
  -d\eta^\alpha \wedge \ensuremath{{}^\star} d\eta_\alpha =
  -d\vartheta_\beta \wedge \eta^{\alpha\beta} \wedge
  \ensuremath{{}^\star} \left( d\vartheta^\gamma \wedge
    \eta_{\alpha\gamma} \right) \\ & = -\vartheta^\alpha \wedge
  \vartheta^\beta \wedge \ensuremath{{}^\star} d\vartheta_\beta \wedge
  \ensuremath{{}^\star} \left( \vartheta_\alpha \wedge
    \vartheta_\gamma \wedge \ensuremath{{}^\star} d\vartheta^\gamma
  \right) \\ & = \vartheta^\alpha \wedge \vartheta^\beta \wedge
  \ensuremath{{}^\star} d\vartheta_\beta \wedge \ensuremath{{}^\star}
  \left[ \vartheta_\alpha \wedge \ensuremath{{}^\star} \left(e_\gamma
      \rfloor d\vartheta^\gamma \right)\right] \\ & = \vartheta^\beta
  \wedge \ensuremath{{}^\star} d\vartheta_\beta \wedge
  \vartheta^\alpha \wedge \left[ e_\alpha \rfloor \left( e_\gamma
      \rfloor d\vartheta^\gamma \right)\right] \\ & =
  \ensuremath{{}^\star} d\vartheta_\beta \wedge \vartheta^\beta \wedge
  \left( e_\gamma \rfloor d\vartheta^\gamma\right) =
  \ensuremath{{}^\star} \left[ \vartheta^\beta \wedge \left( e_\gamma
      \rfloor d\vartheta^\gamma\right) \right] \wedge
  d\vartheta_\beta\\ & = - d\vartheta_\beta \wedge
  \ensuremath{{}^\star} \left[ e_\gamma \rfloor \left( \vartheta^\beta
      \wedge d\vartheta^\gamma\right)\right] + d\vartheta_\gamma
  \wedge \ensuremath{{}^\star} d\vartheta^\gamma \\ &= -
  d\vartheta_\alpha \wedge \vartheta_\beta \wedge
  \ensuremath{{}^\star} \left( \vartheta^\alpha \wedge
    d\vartheta^\beta \right) + d\vartheta^\alpha \wedge
  \ensuremath{{}^\star} d\vartheta_\alpha = {}^{[1]}V - {}^{[4]}V\;.
\end{split}
\end{equation}
Accordingly, the KI-Lagrangian \eqref{KI-lagrangian} can be rewritten
as
\begin{equation}
\label{eq:base-rumpf}
V_{\text{KI}} = \frac{1}{2\ell^2}\Big[2\, d\vartheta^\alpha \wedge
\ensuremath{{}^\star} d\vartheta_\alpha - \left(d\vartheta_\alpha
\wedge \vartheta^\beta \right) \wedge \ensuremath{{}^\star}
\Big( d\vartheta_\beta \wedge \vartheta^\alpha \Big)\Big]\;,
\end{equation}
and we can read off the \(\rho_K\) coefficients as follows:
\begin{equation}\label{KIcoeff}
\rho_{1} = 1+1 = 2\;, \quad
\rho_{2} = 0 \;, \quad
\rho_{4} = - 1 \;.
\end{equation}
By \emph{subtracting} the adjoint term we would have
\(\rho_1=\rho_2=0\) and \(\rho_4=1\), i.e., we would get the von der
Heyde Lagrangian \cite{heyde76}.

The coefficients of the decomposition into irreducible pieces, by
using \eqref{eq:a-rho}, turn out to be
\begin{equation}
  a_1 = 1 \;, \quad a_2=4 \;, \quad a_3 = 1\label{eq:coeff-ki}
\end{equation}
(in the von der Heyde case, we have \(a_1=1\), \(a_2= -2\), \(a_3=1\)).
Accordingly, the KI-Lagrangian can be rewritten in the form
\begin{equation}
V_{\text{KI}}= -\frac{1}{2}\, d\vartheta^\alpha\wedge H_\alpha \;,
\label{eq:base-irred} \end{equation}
with the translational ``excitation''
\begin{equation}
  H_\alpha = -\frac{1}{\ell^2} \,\ensuremath{{}^\star}
  \!\!\left(a_1\,{}^{(1)}d\vartheta_\alpha+
    a_2\,{}^{(2)}d\vartheta_\alpha+a_3\,{}^{(3)}d\vartheta_\alpha\right)
\end{equation}
and the coefficients \eqref{eq:coeff-ki}.

The Lagrangian \(V_{\text{KI}}\) is \emph{not}
locally Lorentz invariant.  Rather, a locally Lorentz invariant theory results
from the following choice of the parameters:
\begin{equation}
a_1\, =\, 1\;, \quad a_2\,=\, -2 \;,  \quad a_3 = -{1\over 2}\;.
\label{eq:coeff-gr}
\end{equation}
This represents the teleparallel
\emph{equivalent} of Einstein's general relativity.

\subsection{Viable Lagrangians}
The form of a general quadratic Lagrangian was displayed in Eqs.\
(\ref{eq:decomp-irred}) and (\ref{eq:decomp-rumpf}).  Various choices
of parameters $a_I$ or $\rho_K$ correspond to various teleparallel
theories of gravity. We call a specific Lagrangian \emph{viable} if it
leads to a theory which fulfills the following conditions: (i) It has
the correct Newtonian approximation. (ii) It agrees with the first
post-Newtonian approximation of general relativity. (iii) It has the
Schwarzschild metric as exact solution in the case of spherical
symmetry.

The question which parameters yield a viable Lagrangian has already
been discussed in the literature, see for example \cite{Nitsch,
  straumann2, Wojtek,Thirring}. The result is that we have viable
Lagrangians for $a_1=1\,,\;a_2=-2\,,\;a_3=\text{arbitrary}$ or
$\rho_4=1\,,\;\rho_1=0\,,\;\rho_2=\text{arbitrary}$. The arbitrary
$a_3$ or $\rho_2$ pieces, respectively, represents the axial square
contribution $A\wedge{}^\star A $ of the torsion, with
$A:=\frac{1}{3}\,\vartheta^\beta\wedge d\vartheta_\beta$ and
${}^{(3)}d\vartheta^\alpha=e_\alpha\rfloor A$. Deviations between
viable theories due to different axial pieces only show up in fifth
order of the post-Newtonian approximation \cite{Nitsch}.  Therefore,
on a phenomenological level, all viable teleparallel theories are
indistinguishable. In Table 1 we have listed some quadratic torsion
Lagrangians.

It follows from the end of the last subsection, see also
\eqref{eq:coeff-gr}, that only the teleparallel equivalent of
Einstein's general relativity is both viable and locally Lorentz
invariant. It has not yet been clearly answered if in this context
local Lorentz invariance is obligatory or merely an aesthetic feature.

\begin{table}[htb]
\begin{center}
\begin{tabular}{c|rrrrrr}
  & $\text{GR}_\parallel$ \cite{gron96}& vdH \cite{heyde76} & viable
  & YM &YM$^\dagger$& $\text{KI}$ \cite{kani97}\\ \hline $a_1$ &
  1 & 1 & 1 & 1 & 0 & 1\\ $a_2$ & $-2$ & $-2$ &$-2$  & 1 &3 & 4 \\
  $a_3$ & $-\frac{1}{2}$ & 1 &\text{arb.}  & 1& 0& 1\\ \hline $\rho_1$
  & 0 & 0 & 0 & 1 &1& 2\\ $\rho_2$ & $-\frac{1}{2}$ & 0 & \text{arb.} & 0 &0&
  0\\ $\rho_4$ & 1 & 1 & 1 & 0&$-1$&$-1$
\end{tabular}
\end{center}
\caption{\it This table lists the $a_I$ and the $\rho_K$ coefficients for
  different teleparallel Lagrangians. $\text{GR}_\parallel$, spelled
  out in the first column, represents a viable gravitational model,
  the same is true for the von der Heyde case. Obviously, the Kaniel-Itin
  Lagrangian, in the   framework of the conventional variational procedure, is
  not viable.   We have $\text{KI}=\text{YM}+\text{YM}^\dagger$ and
  $\text{vdH}= \text{YM}-\text{YM}^\dagger$.}
\end{table}

\subsection{Schweizer-Straumann-Wipf amended}

Schweizer and Straumann \cite{straumann1} and Schweizer, Straumann,
and Wipf \cite{straumann2} investigated the von der Heyde
teleparallelism Lagrangian \cite{heyde76}, see also \cite{Nitsch}.  In
particular they showed that, to first post-Newtonian order, the von
der Heyde theory predicts the same gravitational radiation loss as
general relativity. However, they assumed in some places the incorrect
commutation rule \(\delta\, {}^\star = {}^\star \delta\). Therefore
some equations in these articles must be corrected. We stress that the
corrections do not influence their overall results, though.

We present the corrected formulas of the article by Schweizer,
Straumann, and Wipf \cite{straumann2} in the numbering used there, but
in our notation. The correcting terms are printed in bold.

The wrong variation first shows up in the explicit expressions of
the canonical energy-momentum tensors derived from the teleparallel
version of the Hilbert-Einstein Lagrangian, \(\epsilon_\alpha^E\), and the
difference between the von der Heyde Lagrangian and the teleparallel
version of the Hilbert-Einstein Lagrangian,
\(\Delta \epsilon_\alpha\):
\begin{align}
  \tag*{(2.12),\cite{straumann2}} \epsilon_\alpha^E & = - d\left[
    \vartheta^\beta \wedge \ensuremath{{}^\star} \left(
      d\vartheta_\beta \wedge \vartheta_\alpha \right)\right] -
  d\vartheta^\beta \wedge \ensuremath{{}^\star} \left(
    d\vartheta_\alpha \wedge \vartheta_\beta \right) \\ & \quad \notag
  + \frac{1}{2} d \left\{ \vartheta_\alpha \wedge
    \ensuremath{{}^\star} \left( d\vartheta^\beta\wedge
      \vartheta_\beta \right) \right\} + \frac{1}{2} d\vartheta_\alpha
  \wedge \ensuremath{{}^\star} \left( d\vartheta^\beta\wedge
    \vartheta_\beta \right) \\ & \notag \quad \qquad \qquad
  \boldsymbol{+ \frac{1}{2} e_\alpha \rfloor \left( \vartheta_\gamma
      \wedge d\vartheta^\beta \right) \wedge \ensuremath{{}^\star}
    \left( d\vartheta^\gamma \wedge \vartheta_\beta \right)} \\ &
  \notag \quad \qquad \qquad \boldsymbol{+ \frac{1}{2}
    d\vartheta^\gamma \wedge \vartheta_\beta \wedge e_\alpha \rfloor
    \ensuremath{{}^\star} \left( \vartheta_\gamma \wedge
      d\vartheta^\beta \right)} \\ & \notag \quad \qquad \qquad
  \boldsymbol{- \frac{1}{4} e_\alpha \rfloor \left( d\vartheta^\beta
      \wedge \vartheta_\beta \right) \wedge \ensuremath{{}^\star}
    \left( d\vartheta^\gamma \wedge \vartheta_\gamma \right)} \\ &
  \notag \quad \qquad \qquad \boldsymbol{ - \frac{1}{4}
    d\vartheta^\beta \wedge \vartheta_\beta \wedge e_\alpha \rfloor
    \ensuremath{{}^\star} \left( d\vartheta^\gamma \wedge
      \vartheta_\gamma \right)}\;, \\ \tag*{(2.13),\cite{straumann2}}
  \Delta \epsilon_\alpha & = d \vartheta_\alpha \wedge
  \ensuremath{{}^\star} \left( d\vartheta^\beta\wedge \vartheta_\beta
  \right) - \frac{1}{2} \vartheta_\alpha \wedge d\,
  \ensuremath{{}^\star} \left( d\vartheta^\beta\wedge \vartheta_\beta
  \right) \\ & \notag \quad \qquad \qquad \boldsymbol{- \frac{1}{4}
    e_\alpha \rfloor \left( d\vartheta^\beta \wedge \vartheta_\beta
    \right) \wedge \ensuremath{{}^\star} \left( d\vartheta^\gamma
      \wedge \vartheta_\gamma \right)} \\ & \notag \quad \qquad \qquad
  \boldsymbol{-\frac{1}{4} d\vartheta^\beta \wedge \vartheta_\beta
    \wedge e_\alpha \rfloor \ensuremath{{}^\star} \left(
      d\vartheta^\gamma \wedge \vartheta_\gamma \right)}\;.
\end{align}
Since the additional terms do not influence the antisymmetric part of
\(\Delta \epsilon_\alpha\), we only need to correct the symmetric
part:
\begin{align}
  \tag*{(3.1),\cite{straumann2}} \Delta \epsilon_\alpha^s & = -
  \frac{1}{2}\; \ensuremath{{}^\star} \, \bigg[ \left(
    d\vartheta_\alpha \wedge \vartheta^\beta + d\vartheta^\beta \wedge
    \vartheta_\alpha \right) \wedge \ensuremath{{}^\star} \left(
    d\vartheta^\gamma \wedge \vartheta_\gamma\right) \\ & \notag
  \qquad \qquad \qquad \boldsymbol{ -\frac{1}{2} \vartheta^\beta
    \wedge d\vartheta^\gamma \wedge \vartheta_\gamma \wedge e_\alpha
    \rfloor \ensuremath{{}^\star} \left( d\vartheta^\delta \wedge
      \vartheta_\delta\right)} \\ & \notag \qquad \qquad \qquad \qquad
  \qquad \left. \boldsymbol{ + \frac{1}{2} d\vartheta^\gamma \wedge
      \vartheta_\gamma \wedge \ensuremath{{}^\star} \left(
        d\vartheta^\delta \wedge \vartheta_\delta\right) \,
      \delta^\beta_\alpha} \right] \,\eta_\beta\;. \\\intertext{Hence
    we find the corrected field equation as}
  \tag*{(3.3),\cite{straumann2}} \epsilon_\alpha^E & - \frac{(\lambda
    -1)}{2} \; \ensuremath{{}^\star} \, \bigg[ \left(
    d\vartheta_\alpha \wedge \vartheta^\beta + d\vartheta^\beta \wedge
    \vartheta_\alpha \right) \wedge \ensuremath{{}^\star} \left(
    d\vartheta^\gamma \wedge \vartheta_\gamma\right) \\ & \notag
  \qquad \qquad \qquad \boldsymbol{ -\frac{1}{2} \vartheta^\beta
    \wedge d\vartheta^\gamma \wedge \vartheta_\gamma \wedge e_\alpha
    \rfloor \ensuremath{{}^\star} \left( d\vartheta^\delta \wedge
      \vartheta_\delta\right)} \\ & \notag \qquad \qquad \qquad \qquad
  \qquad \left. \boldsymbol{ + \frac{1}{2} d\vartheta^\gamma \wedge
      \vartheta_\gamma \wedge \ensuremath{{}^\star} \left(
        d\vartheta^\delta \wedge \vartheta_\delta\right) \,
      \delta^\beta_\alpha} \right] \,\eta_\beta = - t^\alpha \;.
\end{align}

{In our units, we have for the $\lambda$-parameter of Schweitzer et
  al.  $\lambda=-2\rho_2=\frac{2}{3}\,(1-a_3)$.} Like the terms
discussed in \cite{straumann2}, the additional terms are at least
quadratic in \(\phi_{\alpha\beta}\) (which is the symmetric part of
\(\Phi_{\alpha\beta}\) in the expansion \(\vartheta^\alpha = dx^\alpha
+ \Phi^{\alpha}{}_\beta\, dx^\beta\)), such that the arguments of
\cite[\S3]{straumann2} remain unchanged.

The formulas \cite[(4.1)]{straumann2} and \cite[(4.6)]{straumann2}
have to be corrected similarly as the last two equations, replacing
\(d\vartheta^\gamma\) by \(T^\gamma\). Since these explicit
expressions are not used in the remainder of \cite[\S4]{straumann2},
the conclusions remain valid therein.

Finally, the quantities \(A^{\mu\nu}\), \(B^{\mu\nu}\), and
\(C^{\mu\nu}\) (formulas \cite[(5.4b)--(5.4d)]{straumann2}) in the
expansion of the quadratic and higher-order terms
\begin{equation} \tag*{(5.4a),\cite{straumann2}} \epsilon_Q^\alpha =
  \Delta\epsilon_s^\alpha + \left\{ A^{\beta\alpha} + B^{\beta\alpha}
    + C^{\beta\alpha}\right\} \end{equation} would need corrections.
Since these quantities are not used explicitly in \cite{straumann2},
we did not display the exact expressions here.

\section{Kaniel-Itin examined}\label{sec:KI}
\subsection{Lagrangian}
We come back to the Kaniel-Itin Lagrangian \eqref{KI-lagrangian}:
\begin{equation} V_{\rm{KI}} \,=\,
  \frac{1}{2\ell^2}\left( d\vartheta^\alpha \wedge
    \ensuremath{{}^\star} d\vartheta_\alpha + d^\dagger
    \vartheta^\alpha\wedge\ensuremath{{}^\star} d^\dagger
    \vartheta_\alpha\right)\,.\label{base} \end{equation} Its
Euler-Lagrange equation can be read off from \eqref{genLag} by
substituting the coefficients \eqref{KIcoeff}:
\begin{equation}\label{genLag'}
\begin{split}
  - 2&\ell^2\,\Sigma_\alpha = 2\; \vartheta_\beta \wedge d \; {}^\star
  \left( \vartheta_\alpha \wedge d \vartheta^\beta \right)+4 \;
  d\,{}^\star d\vartheta_\alpha \\ & + 2 e_\alpha \rfloor \left(
    d\vartheta^\beta \wedge {}^\star d\vartheta_\beta \right) - 4
  \left( e_\alpha \rfloor d\vartheta^\beta \right) \wedge {}^\star
  d\vartheta_\beta - 2 d\vartheta_\beta \wedge {}^\star
  \left( \vartheta_\alpha \wedge d \vartheta^\beta\right)\\& -
  e_\alpha \rfloor \left[ \vartheta_\gamma \wedge d\vartheta^\beta
    \wedge {}^\star \left( d\vartheta^\gamma \wedge \vartheta_\beta
    \right)\right] + 2 \left( e_\alpha \rfloor d\vartheta^\beta\right)
  \wedge \vartheta_\gamma \wedge {}^\star \left( d\vartheta^\gamma
    \wedge \vartheta_\beta \right) \;.
\end{split}
\end{equation}
In the first line, we displayed the second derivatives of the
gravitational potential. As we already saw in Table 1, this field
equation is not viable.

If we use the \emph{the constrained variations} in \eqref{base}, then
we commute $\delta$ and the star $^\star$ and find by simple algebra:
\begin{equation}\label{last}\ell^2\,\delta\left(V_{\rm{KI}}
    + L_{\rm{mat}}\right)
  =\delta\vartheta^\alpha\wedge\left(-\square\eta_\alpha+
   \ell^2\,\Sigma_\alpha\right) =0\,.\end{equation} Since the
variations are constrained, we have to turn to our proposition and to
use \eqref{constraint}:
$\delta\vartheta^\alpha=\omega_\beta{}^\alpha\, \vartheta^\beta$, with
$\omega_{(\alpha\beta)}=0$. On substitution in \eqref{last}, the field
equation turns out to be proportional to the \emph{antisymmetric} part
of the wave equation,
\begin{equation}
  \vartheta_{[\alpha} \wedge \square\, \eta_{\beta]} =
  \ell^2\,\vartheta_{[\alpha} \wedge \Sigma_{\beta]}
\label{eq:ki-correct}\;, \end{equation}
rather than to the wave equation itself. Accordingly, the constrained
variations also lead to a dead end and we have to turn back our
attention again to the KI-field equation \eqref{KI-fieldeq}.

\subsection{Decomposition of the field equation}

Let us split the full wave equation into its different pieces. For
that purpose, we most conveniently start from the decomposition of the
energy-momentum current\label{sec:44} as a covector-valued 3-form with
16 independent components:
\begin{equation}\label{cut}
  \Sigma_\alpha=\>\overset{\frown}{\Sigma\kern-1.1em\nearrow}_\alpha +
  \frac{1}{2} \vartheta_\alpha \wedge \left(e_\gamma \rfloor
    \Sigma^\gamma \right)+ \frac{1}{4} e_\alpha \rfloor \left(
    \vartheta^\gamma \wedge \Sigma_\gamma \right)\;. \end{equation} Here
$\overset{\frown}{\Sigma\kern-1.1em\nearrow}_\alpha$ is its
\emph{symmetric traceless} part, the second term on the right hand
side its \emph{antisymmetric} and the last term its \emph{trace part},
see \cite[eq.(5.1.15)]{hehl95}.

We come into better known territory if we decompose $\Sigma_\alpha$
with respect to the $\eta$-basis:
\begin{equation}
  \Sigma_\alpha=T^\beta{}_\alpha\,\eta_\beta \;. \end{equation} The
$T^\beta{}_\alpha$'s are the components of the energy-momentum tensor.
We can `saturate' the 3-form $\Sigma_\beta$ by means of the 1-form
$\vartheta^\alpha$:
\begin{multline} \ensuremath{{}^\star} \left( \vartheta^\alpha \wedge
    \Sigma_\beta \right) = \ensuremath{{}^\star} \left(
    \vartheta^\alpha \wedge T^\gamma{}_\beta \;\eta_\gamma \right) =
  T^\gamma{}_\beta \; \ensuremath{{}^\star} \left(\vartheta^\alpha
    \wedge \eta_\gamma \right) \\ = T^\alpha{}_\beta \;
  \ensuremath{{}^\star} \eta = T^\alpha{}_\beta
  \;\ensuremath{{}^\star}\ensuremath{{}^\star} 1 = -
  T^\alpha{}_\beta\; \end{multline} or
\begin{equation}
  T_{\alpha\beta}=\ensuremath{{}^\star}\left(\Sigma_\beta
    \wedge\vartheta_\alpha \right)=e_\alpha\rfloor{}^\star\Sigma_\beta
\label{eq:energy-tensor}\;.
\end{equation}
The analog of \eqref{cut} is, of course, the following splitting of the
energy-momentum tensor:
\begin{equation}T_{\alpha\beta}=\{T_{(\alpha\beta)}- \frac{1}{4}\,
  T^{\gamma}{}_\gamma\;g_{\alpha\beta}\}
  +T_{[\alpha\beta]}+\frac{1}{4}\,T^{\gamma}{}_\gamma
  \;g_{\alpha\beta}\;.\end{equation}

Coming back to \eqref{KI-fieldeq}, after some algebra, we find the
following decomposition:
\begin{subequations}\label{eq:wave}
\begin{align}
  \square \eta_\alpha - \frac{1}{2} \vartheta_\alpha \wedge
  \left(e_\gamma \rfloor \square\eta^\gamma \right)- \frac{1}{4}
  e_\alpha \rfloor \left( \vartheta^\gamma \wedge \square
    \eta_\gamma\right) & =\>
  \ell^2\,\overset{\frown}{\Sigma\kern-1.1em\nearrow}_\alpha \;,
  \label{eq:ki-symmetric}\\ \frac{1}{2} \vartheta_\alpha \wedge
  \vartheta_\beta \wedge \left( e_\gamma \rfloor \square \eta^\gamma
  \right) & = \ell^2\,\vartheta_{[\alpha} \wedge \Sigma_{\beta]} \;,
  \label{eq:ki-antisymm}\\ \vartheta^\gamma \wedge \square \eta_\gamma
  +4\lambda(x)\,\eta &= \ell^2\,\vartheta^\gamma \wedge \Sigma_\gamma
  \;. \label{eq:ki-trace}
\end{align}
\end{subequations}

We can combine the antisymmetric and the symmetric-tracefree part of
the wave equation in order to get its tracefree part.  The fine
splitting \eqref{eq:wave} simplifies to the tracefree and the trace
part of the field equation (\ref{eq:ki-correct}):
\begin{subequations}
\begin{align} \square \eta_\alpha - \frac{1}{4} e_\alpha \rfloor
  \left( \vartheta^\gamma \wedge \square \eta_\gamma \right)
  &= \ell^2\, \Sigma\kern-1.1em\nearrow_\alpha
  \;,\label{eq:trace-decomp} \\  \vartheta^\gamma \wedge \square
  \eta_\gamma +4\lambda(x)\,\eta
&= \ell^2\,\vartheta^\gamma \wedge \Sigma_\gamma \;.
\label{eq:trace}\end{align}
\end{subequations}

Sometimes it may be useful to rewrite \eqref{eq:trace-decomp} by using
\begin{equation} \begin{split} e_\alpha \rfloor \left(
      \vartheta^\gamma \wedge \square \eta_\gamma \right) & = -
    e_\alpha \rfloor \ensuremath{{}^\star} \left\{
      \ensuremath{{}^\star} \left( \vartheta^\gamma \wedge \square
        \eta_\gamma \right)\right\} = - \ensuremath{{}^\star} \left(
      \ensuremath{{}^\star} \left( \vartheta^\gamma \wedge \square
        \eta_\gamma \right) \wedge \vartheta_\alpha \right) \\ & = -
    \ensuremath{{}^\star} \left( \vartheta^\gamma \wedge \square
      \eta_\gamma \right) \wedge \ensuremath{{}^\star}
    \vartheta_\alpha = - \ensuremath{{}^\star} \left(\vartheta^\gamma
      \wedge \square \eta_\gamma \right) \eta_\alpha \;,
\end{split} \end{equation}
thereby finding
\begin{equation}\left[  \square + \frac{1}{4} \ensuremath{{}^\star}
    \left( \vartheta^\beta \wedge \square \eta_\beta \right) \right]
  \eta_\alpha = \ell^2\,\Sigma\kern-1.1em\nearrow_\alpha\;,
  \label{eq:decomp-second}
\end{equation}or, alternatively:
\begin{equation}\left[  \square - \frac{1}{4}
    \left(e_\beta\rfloor\square\vartheta^\beta \right) \right]
  \eta_\alpha = \ell^2\,\Sigma\kern-1.1em\nearrow_\alpha\;.
  \label{eq:decomp-second'}
\end{equation}

\subsection{Yilmaz-Rosen and Schwarzschild\\ solution compared}
\label{sec:YR-sol}
The Yilmaz-Rosen metric and the corresponding orthonormal coframe were
displayed in \eqref{YILMAZ} to \eqref{COFRAME}. In order to compare
the Yilmaz-Rosen metric with the Schwarzschild metric, we transform
the former one from the isotropic coordinates used in \eqref{YILMAZ}
into Schwarzschild coordinates and Taylor expand it:
\begin{subequations}
\begin{align}
g^{\text{YR}}_{00} & = 1-\frac{2m}{r} \qquad \quad \,+ \frac{25}{6}
\frac{m^3}{r^3} + \mathcal{O}
\left( \tfrac{m^4}{r^4}\right) \;,\\
g^{\text{YR}}_{11} & = 1+ \frac{2m}{r} + \frac{5m^2}{r^2} +
\frac{9m^3}{r^3}\; +\, \mathcal{O}\left( \tfrac{m^4}{r^4} \right) \;.
\end{align}
\end{subequations}
For the Schwarzschild metric in Schwarzschild coordinates we find:
\begin{subequations}
\begin{align}
  g^{\text{SS}}_{00} & = 1-\frac{2m}{r} \qquad \text{(exact)}\;,\\
  g^{\text{SS}}_{11} & = 1+ \frac{2m}{r} + \frac{4m^2}{r^2} +
  \frac{8m^3}{r^3}\, +\, \mathcal{O}\left( \tfrac{m^4}{r^4} \right) \;.
\end{align}
\end{subequations}

Their $g_{00}$ components are equal up to second order. The radial
components $g_{11}$ begin to differ slightly in the second order.
Therefore the Yilmaz-Rosen solution is consistent with the classical
tests of general relativity and can, in particular, describe the
post-Newtonian perihelion advance correctly, see Synge \cite[page 296,
footnote 1]{synge71}. It requires further investigations to decide
whether the two solutions can be observationally distinguished by
strong gravity effects in close binary pulsar systems.

\subsection{Yilmaz-Rosen metric motivated}\label{YR-mot}
A viable theory of gravitation should be consistent with the
\emph{local} equivalence principle. Let us consider an electromagnetic
wave of frequency \(\omega\) in the gravitational field of a point
mass $m$. The frequency shift due to the propagation from a point with
radial coordinate $r$ to one with $r+\Delta r$ reads ($c=G=1$):
\begin{equation}
\frac{\Delta \omega}{\omega} = \Delta U = -\frac{m\Delta r}{r^2} \;.
\label{eq:equiv-local} \end{equation}

According to Mashhoon \cite{mash97}, Yilmaz effectively proposed to
extend the local equivalence principle to a \emph{global} one.  With
this idea in mind, we can tentatively integrate \eqref{eq:equiv-local}:
\begin{equation} \int_{r_0}^{r_1} \frac{d\omega}{\omega} =
  \int_{r_0}^{r_1} dU \;.\end{equation} We solve the integrals and
find
\begin{equation}\frac{\omega(r_1)}{\omega(r_0)} = \exp[U(r_1)-U(r_0)]\;.
\end{equation} Using the Newtonian potential explicitly,
\(U=-\frac{m}{r}\), and taking the limit \(r_1\rightarrow \infty\),
yields
\begin{equation} \frac{\omega(\infty)}{\omega(r)}= \exp\left(\frac{m}{r}
  \right)\;, \quad \text{and thus} \qquad \frac{\Delta
    t(\infty)}{\Delta t(r)} = \exp \left( - \frac{m}{r} \right) \;.
\end{equation} Since \(\Delta t(\infty)\) is not influenced by
gravity, one can directly read off \begin{equation}
  \vartheta^{\hat{t}} = e^{-\frac{m}{r}} \,dt \quad \text{or} \qquad
  {g}_{00} = e^{-\frac{2m}{r}}\;.\end{equation}

Following the pattern of the components of the Schwarzschild metric,
we now define \(g_{11}\) as the inverse of \(g_{00}\):
\begin{equation}\label{YR-Schwarz} \widetilde{g} =
  e^{-\frac{2m}{r}}dt^2-e^{\frac{2m}{r}}dr^2-r^2\left(d\theta^2+
    \sin^2\theta\,d\varphi^2\right)\;.  \end{equation} However, the
Taylor approximation of this metric (which is \emph{not} the
Yilmaz-Rosen metric of \eqref{YILMAZ}) reads:
\begin{subequations}
\begin{align}
\widetilde{g}_{00} & = 1-\frac{2m}{r} + \frac{2m^2}{r^2} - \frac{4}{3}
\frac{m^3}{r^3} + \mathcal{O}
\left( \tfrac{m^4}{r^4}\right) \;,\\
\widetilde{g}_{11} & = 1+ \frac{2m}{r} + \frac{2m^2}{r^2} +
\frac{4}{3}\frac{m^3}{r^3} + \mathcal{O}\left( \tfrac{m^4}{r^4}
\right) \;.
\end{align}
\end{subequations}
The $\widetilde{g}_{00}$ component differs from $g^{\text{SS}}_{00}$
already to second order. And the deviation of $\widetilde{g}_{11}$
from $g^{\text{SS}}_{11}$ is doubled in comparison to that of the
Yilmaz-Rosen $g^{\text{YR}}_{11}$.  Therefore, in order to approximate
the experimentally well verified Schwarzschild metric in an optimal
way, we choose the forefactors of \eqref{YR-Schwarz} as metric
components in \emph{isotropic} coordinates, which eventually leads to
the Yilmaz-Rosen metric:
\begin{equation} g^{\text{YR}}=e^{-\frac{2m}{r}}\,dt^2-e^{\frac{2m}{r}}
  \,\left(dx^2 +dy^2 +dz^2\right)\;.\end{equation}

\subsection{Yilmaz-Rosen solution and the vacuum field equation}
The Yilmaz-Rosen metric \eqref{YILMAZ}, keeping in mind
\eqref{ORTHONORMAL} and \eqref{COFRAME}, fulfills the tracefree field
equation \eqref{eq:trace-decomp} [or, alternatively,
\eqref{eq:decomp-second} or \eqref{eq:decomp-second'}] with vanishing
source. For a proof compare the corresponding computer algebra program
in Sec.\ \ref{sec:ca}.
As we saw, the Yilmaz-Rosen solution is consistent with the classical
tests of general relativity. However, its integration constant $m$
cannot be directly identified with the source of a spherical body.

For that reason, we take recourse to the trace equation
\eqref{eq:trace}. In the vacuum case we find
\begin{equation}\lambda(x)=\frac{1}{4}{}\,^\star\left(
    \vartheta^\gamma\wedge\square\eta_\gamma\right)\,.
\end{equation} The right hand side of this equation can be easily
calculated with our computer algebra program, see also \cite{kani97}.
Therefore the Yilmaz-Rosen metric solves the vacuum field equation
\eqref{KI-fieldeq} provided the `cosmological' function is prescribed
as follows:
\begin{equation}\lambda(x)=-\left( \frac{m}{r^2} e^{-\frac{m}{r}}
  \right) ^2\,.
\end{equation}

Since such an ad hoc structure looks too implausible to us, we can
change horses at this moment: We reject the `cosmological' function of
Kaniel-Itin and put it to zero. Then we can mimic this function
$\lambda(x)$ by means of the energy-momentum trace
$-\ell^2\,T^\gamma{}_\gamma/4$, cf.\ \eqref{eq:trace} and
\eqref{eq:energy-tensor}:
\begin{equation} T^\gamma{}_\gamma =
    \left( \frac{2m}{\ell r^2} e^{-\frac{m}{r}}\right) ^2 \;.
\label{eq:matter-distrib}\end{equation}
This energy-momentum trace is plotted in Fig.\ \ref{fig:matter}.
Therefore, the Yilmaz-Rosen solution fulfills the field equations
\begin{equation} \square \eta_\alpha - \frac{1}{4} e_\alpha
\rfloor \left( \vartheta^\gamma \wedge \square \eta_\gamma \right)
=0\qquad\text{and}\qquad \ensuremath{{}^\star} \left( \vartheta^\gamma
\wedge \square \eta_\gamma \right) = -\left( \frac{2m}{\ell r^2}
e^{-\frac{m}{r}}\right) ^2 \;.\end{equation}

Taking an ideal fluid for the description of matter, then, for
vanishing pressure, \(p=0\), we have \(T^\gamma{}_\gamma=\rho\).
Therefore we can understand the above computation as a matter
distribution \eqref{eq:matter-distrib} which can be viewed as
(probably unphysical) star model. The matter of such a star reaches to
infinity, but it decreases exponentially. We find the
maximum of the distribution at \(r=\kern.1em%
\raise.5ex\hbox{\scriptsize \ensuremath{m}}%
\kern-.1em/\kern-.15em%
\lower.25ex\hbox{\scriptsize \ensuremath{2}}\), compare with Fig.\
\ref{fig:matter}. Most of the star mass is concentrated inside the
Schwarzschild radius \(r_s=2m\). The volume integral over
\(T^\gamma{}_\gamma\) yields the total mass \(m\) of the
star:
\begin{multline}
  \int T^\gamma{}_\gamma\, dV=4\pi \int\limits_0^{\infty}
  T^\gamma{}_\gamma\, r^2\, dr = \int\limits_0^\infty \frac{16\pi
    m^2}{\ell^2 r^2}\,e^{-\frac{2m}{r}} \, dr\;
  \\\stackrel{x:=-\frac{2m}{r},\; dr=\frac{2m}{x^2}\,dx}{=}\;
  \frac{8\pi m}{\ell^2} \int\limits_{-\infty}^0 e^x \, dx = \frac{8\pi
    m}{\ell^2}=M\;,
\end{multline} where $M$ is the mass in conventional units.

As a result, we can interpret the constant \(m\) of the Yilmaz-Rosen
metric as the mass of a star, but this mass
is distributed in a probably unphysical way.

\section{Conclusion}

We posed four questions about the Kaniel-Itin model in Sec.\ 1. We
will try to answer them in turn:
\begin{itemize}
\item[(i)] We can put the energy-momentum 3-form of matter on the
  right hand side of the KI-field equation \eqref{WAVE}, see
  \eqref{KI-fieldeq}:
\begin{equation}\label{KI-fieldeq'}
  \left[\square +\lambda(x)\right]\eta_\alpha=\ell^2\,
  \frac{\delta
    L_{\text{mat}}}{\delta\vartheta^\alpha}=:\ell^2\,\Sigma_\alpha\,.
\end{equation}
\item[(ii)] The Yilmaz-Rosen metric can be accommodated to
  \eqref{KI-fieldeq'} in the following sense: We decompose
  \eqref{KI-fieldeq'} in vacuum into
\begin{subequations}
\begin{align} -\square \eta_\alpha + \frac{1}{4} e_\alpha \rfloor
  \left( \vartheta^\beta \wedge \square \eta_\beta \right) &=0
  \;,\label{eq:trace-decomp'} \\ \frac{1}{4}\,
  ^\star\left(\vartheta^\beta \wedge \square \eta_\beta\right) &=
  \lambda(x)\;.
\label{eq:trace'}\end{align}
\end{subequations}
These equations are fulfilled by the Yilmaz-Rosen metric, provided we
prescribe the `cosmological' function $\lambda(x)$ in the following
way:
\begin{equation}\label{lambdaofx}
  \lambda(x)= \frac{1}{4}\, ^\star\left(\vartheta^\beta \wedge \square
    \eta_\beta\right) =-\left( \frac{m}{r^2} e^{-\frac{m}{r}}\right)^2
  \;.\end{equation} If one used a wave equation as field equation,
i.e., if in \eqref{KI-fieldeq'} one put $\lambda(x)=0$, then one could
find the Yilmaz-Rosen solution for the matter distribution
$T^\gamma{}_\gamma=\left( \frac{2m}{\ell r^2} e^{-\frac{m}{r}}\right)^2$.
\item[(iii)] There doesn't exist a consistent variational
  principle for arriving at \eqref{KI-fieldeq'}. Maybe one is able to
  find one for the tracefree vacuum equation \eqref{eq:trace-decomp'}.
\item[(iv)] The constraints of Kaniel-Itin on the variations amount
  to getting rid of the independence of the variations of the metric,
  provided the variations of the coframe are prescribed.
\end{itemize}
Is the model of Kaniel \& Itin viable? Well, it is presently in
intensive care~\dots And it is a beautiful model anyways.

\paragraph{Acknowledgments:} We are grateful to Shmuel Kaniel and
Yakov Itin for interesting discussions and most helpful remarks. We
thank Bahram Mashhoon, Eckehard Mielke, and Yuri Obukhov for useful
comments and hints.

redpajama_set_name: RedPajamaArXiv