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## Faster than a speeding photon? Precursors test whether light can be faster than light

Posted on October 9, 2011 by

Over the past two weeks, the biggest physics news has been the apparent observation of neutrinos (nearly undetectable subatomic particles) moving faster than the vacuum speed of light.  At first glance, this would seem to violate Einstein’s special theory of relativity, which fixes the vacuum speed of light at $c = 3\times 10^8$ meters per second, and as a consequence makes it in principle impossible to travel faster than that speed.  The theoretical implications are in fact a bit more subtle, but before we worry too much about those implications the experimental results will need to be checked carefully and independently verified.

While we wait, it is worth noting that in June 0f 2011 a group of researchers performed an experiment to see if light itself could move faster than light!  In particular, the scientists used a little known optical phenomenon known as an optical precursor to see if individual photons might travel faster than $c$ while propagating in a material.  In the end, the experiment suggests that these single photons did not in fact violate Einstein’s speed limit, though the results still got a significant amount of press.

The response of many physicists to the news was a collective, “Well, duh!”  The prevailing attitude seems to have been: “What’s so interesting about proving something we already knew?”  In this post I’d like to explore that question a little bit, and explain how some uncertainty remains about the behavior of light in materials.  Along the way, we’ll introduce the fascinating phenomenon of precursors, and see how they can be used both to probe the nature of matter as well as the nature of light.

To start, we need to understand a bit about how light and matter interact with one another, and how this relates to the speed of light in matter.  This is a surprisingly tricky subject, which I’ve covered several times previously on this blog; we’ll review the important aspects, with a few additional details.

Let’s talk about how we measure the speed of an object first.  If we’re looking at the motion of a rigid object, like a speeding car or a thrown baseball, the speed $v$ can be determined simply by measuring how much time $\Delta t$ it takes for an object to travel a distance $\Delta x$.  The speed is simply the distance divided by the time:

$\mbox{speed} = \mbox{distance}/\mbox{time} = \Delta x/\Delta t$.

There’s a small subtlety to this definition: cars and baseballs are extended objects!  To accurately measure an object’s speed, we have to be consistent in how we define its position.   For a car driving down the road, for instance, we should do all measurements of its position from a fixed position, such as the front bumper, to measure the speed.

But what do we do when the object doesn’t have a fixed position on it?  For example, what is the best way to measure the speed of a hurled bucketful of water?

Water is, of course, not a rigid body, and the volume of water changes shape and spreads out as it travels.  It is strange to realize, after some thought, that a blob of water like this doesn’t really have a well-defined speed.  We arrive at the same problem with a pulse of light traveling in matter: in general, a light pulse has no “fixed point” upon it, and it can change shape as it travels.  There isn’t a single definition for the speed of light in matter that is useful in all cases.

There are three different descriptions of the speed* of light used in physics, each with its own significance and usefulness.  We can introduce each of these by looking at similar definitions for our bucketful of water.

We imagine that our volume of water may be broken up into a collection of well-defined droplets.  The most straightforward and thorough thing to do is to measure the speed of each individual droplet: the velocity of each droplet will be referred to as its phase velocity.  If there is only one or a handful or drops, this set of numbers will be a good description, but for a large number of droplets this collection of numbers will not necessarily tell us anything about the motion of the entire bulk of the water.

A better option is to define the speed of the body of water by its center of mass.  This definition, referred to as the group velocity, works quite well in many cases, but can also be misleading.  If we’re interested, for instance, in the time it takes for water to first reach its target (say a fire), there may be droplets well ahead of the center of mass and the use of the group velocity can overestimate the time.

We may also choose to define the speed of the body of water as the speed of the fastest drop in the collection; this definition is called the signal velocity.  One problem with the signal velocity is the opposite of that of the group velocity: if we need to know when the bulk of a body of water reaches a fire, the signal velocity will underestimate the time.

Physicists use roughly analogous definitions to characterize the speed of light in matter.  Instead of “droplets”, a pulse of light can be decomposed into a collection of waves of different frequencies (colors).  It can be shown that a narrow pulse of light necessarily consists of a very broad spectrum (range of colors), while a very broad, slowly varying pulse has a narrow spectrum.  Also, and significant for our later discussion, a pulse with a very sharp edge necessarily has a very broad spectrum, as well.  This idea is illustrated crudely below:

Atoms respond differently to light of different frequencies, in a phenomenon known as dispersion.  The result of this is that some frequencies of light are absorbed more than others, and in general every frequency of light travels at a different speed.  The term phase velocity is used to refer to the different speeds of each frequency of light.  For a pulse narrow in frequency (like the upper one above), this can sometimes serve as an approximation to the overall speed of light.  However, in many circumstances this phase velocity can be faster than the vacuum speed of light $c$, and therefore does not accurately represent the speed of the pulse.

Because of difficulties with the phase velocity, the group velocity is the standard method of defining the velocity of a pulse in a medium.  It may be considered, in essence, the speed of the “center of mass” of the pulse.  However, if the shape of the pulse is highly distorted as it propagates, the group velocity can lose all useful meaning.  Also, it has been shown that the group velocity can be greater than the vacuum speed of light, a result that generated quite a bit of controversy at first.

The absolute speed of light in a medium is the signal velocity, which is loosely defined as the speed at which information can be conveyed in a medium, or the fastest possible speed that the front of a light signal can travel.  This is generally thought to be no faster than the vacuum speed of light $c$, in agreement with Einstein’s special theory of relativity.

But is $c$ the fastest speed of light in matter?  It seems likely, but a few gaps in our knowledge leave it a partially open question.  The quantum-mechanical nature of light-matter interactions leaves open the possibility that some “quantum weirdness” allows Einstein’s speed limit to be broken in a subtle way.

A more concrete concern involves our understanding of how light propagates in matter.  It is well-known that the “causality” of a light signal — and the absolute speed of light — is built into the frequency-by-frequency response of light to matter.  If one knows the speed of light, and the absorption properties of light, for every frequency, one can determine whether or not $c$ is the top speed.

But we don’t know these properties for every frequency!  In particular, we don’t know exactly what happens to light in matter for arbitrary high frequencies (frequency approaches infinity) or for arbitrary low frequencies (frequency approaches zero).  There is always the possibility that some small violation of relativity is “hiding” in these extreme frequency ranges.

This is where the idea of optical precursors becomes useful.  Let’s consider the temporal and spectral properties of another pulse, with a square envelope:

This pulse has a large central peak in frequency, but also has long frequency “tails” that stretch out, to some degree, to arbitrarily high frequencies (to infinity) and arbitrarily low frequencies (to zero).  These tails arise due to the instantaneous start and end of the pulse in time: the rapid rise and fall that give the “square pulse” its name.

At the beginning of the 20th century, physicists Arnold Sommerfeld and Léon Brillouin independently performed theoretical work to investigate what happens to such sharp-edged pulses as they propagate a long distance through matter.  In 1914, they each published results** showing that the main body of the pulse (traveling at the group velocity) is preceded by faster-moving waves, now known as “precursors”.  The precursors are general broken into two types: the Sommerfeld precursor, which actually travels at the vacuum speed of light $c$, and the Brillouin precursor, which travels at the speed of light $c/n(0)$, where $n(0)$ is the refractive index of the medium at zero frequency.  A crude illustration of the arrangement of precursors is shown below:

The most remarkable thing about the precursor fields is that they are very weakly absorbed by matter, much less so than the main body of the pulse.  If we send our original square pulse through a very thick piece of absorbing material, the main body will be almost completely absorbed while the precursors will be absorbed only weakly.  This fact has led to some researchers suggesting that precursors could be used to “see” through normally opaque objects, like clouds.

It took quite a few years for Sommerfeld and Brillouin’s theoretical work to be confirmed.  The first observation of precursors was performed in 1969, using microwaves***.  Since then, they have been observed for a variety of wavelengths, and have been observed even in “mundane” materials such as water****.

What is the origin of these precursors?  I wasn’t able to find a simple explanation for them in the literature, perhaps not surprising in terms of their mathematical complexity.  I would like to introduce a (hopefully accurate) description of them, in what I will refer to as the “pendulum slapping” model!

We have all been taught, at some point in our education, the planetary model of the atom.  In this model, an atom consists of a positively-charged nucleus surrounded by one or more orbiting negatively-charged electrons, much like the planets in the solar system orbit the Sun:

With our modern understanding of quantum mechanics, we know that this model isn’t a terribly good one: electrons act much more like “clouds” of negative charge centered on the nucleus, and don’t “orbit” in a well-defined sense.  However, the planetary model does get one thing right: electrons have a characteristic frequency associated with their motion, much like the planets each orbit the Sun with a characteristic frequency (the Earth orbits the Sun with a frequency of 1/365 days).  In a very loose sense, we can picture atoms (and molecules) as collections of oscillating electron pendulums, each vibrating with a characteristic frequency:

What happens when we apply an oscillating electric field (i.e. a light wave) to our pendulum?  The electric field applies a force to the electron, driving its oscillation.

The effectiveness of the electric field in moving the pendulum depends on how close its frequency is to the pendulum frequency.  The pendulum will have the largest swing when the electric field matches its own frequency; this is like a child pumping his legs on a swing to increase his motion.  Any other frequency of the electric field will be less effective in moving the pendulum, but pretty much every frequency will force some oscillation in it.  The energy that is imparted to the pendulum is lost by the electric field, and consequently the light wave; this is the origin of the absorption of light by matter.

There are two extreme limits of frequency in which no energy is transferred to the pendulum: the limit of infinite frequency and the limit of zero frequency.  We can visualize what happens in these cases by imaging that we are “slapping” a pendulum with our hands on either side (or you can try it yourself, if you have a pendulum). When we slap a pendulum at a frequency much higher than its characteristic frequency, a push on the left is almost immediately canceled by a push on the right — the pendulum doesn’t move, and we’ve transferred no energy to it!  In the limit of very low frequency, the pendulum moves, but doesn’t oscillate: we “lift” it to the right with our left hand, then slowly lower it back to its rest position and then “lift” it to the left with our right hand.  At no point does the pendulum oscillate freely, and therefore we have transferred no energy to it.

These limits, in which no energy is transferred to the pendulum, are the origins of the precursors.  We have seen that a square wave has components of arbitrarily high frequency and arbitrarily low frequency, and these components are only very weakly absorbed by the medium.  The high frequency components of the wave result in the Sommerfeld precursor, and the low frequency components of the wave result in the Brillouin precursor.

For tests of the absolute speed of light, the Sommerfeld precursor is of particular interest, because (a) it tests the whether a violation of relativity is “hiding” at high frequencies, and (b) the speed of a precursor is theoretically supposed to be equal to $c$, making “faster than light” violations relatively easy to spot.

All of this brings us back at last to the June paper in Physical Review Letters on the “Optical precursor of a single photon”, by a research group in Hong Kong.  A photon is a single quantum particle of light; all of the discussion of precursors up to this point have concerned pulses consisting of many, many photons.  Two questions arise when considering precursors and single photons:

(1) Do precursors even exist for a single photon?  One would naturally be inclined to say “yes”, but it may be that, on a quantum level, precursors inherently involve the interaction of many photons at once.

(2) Can single photons travel faster than $c$?  One would be inclined to say “no” in this case, but again the behavior of single photon precursors (if they exist) might be subtly different than a group of many photons.

The Hong Kong researchers investigated these possibilities by producing coupled pairs of photons using the following experimental configuration (adapted and simplified from the article):

The fundamental components of the system are a pair of magneto-optical traps (MOT) containing Rubidium atoms.  Magneto-optical traps use light and magnetic fields to localize a group of atoms and cool them to a low temperature, forming a low-density medium.  The first MOT is excited by a pair of lasers — a pump beam and a coupling beam — and through a complicated light-matter interaction pairs of photons are produced.  The physics of this production is too complicated to discuss here, but the result is a higher frequency “Stokes photon” and a lower-frequency “anti-Stokes photon”.  These photons are produced at the same time and are therefore correlated in time.

The Stokes photon is picked up by detector 1 and sends a signal to a function generator, which triggers an electro-optic modulator that the anti-Stokes photon passes through.  This modulator “chops off” the front end of the anti-Stokes photon signal, producing a sharp-edged pulse necessary for precursor generation*****.  This sharp-edged pulse propagates into the second MOT, in which its waveform is expected to take a precursor shape.

It is essential for this experiment that a single anti-Stokes photon be measured at a time; after the second MOT, the light signal is split by a beam-splitter and sent to a pair of detectors, detectors 2 and 3.  Because a single photon can only arrive at a single detector, the simultaneous tripping of both detectors implies the presence of more than one photon and the event is thrown out.  The time of arrival of the single anti-Stokes photons can be tallied, producing a profile of the average behavior of the photons.

The coupling laser illuminating the second MOT could modify the optical properties of the Rubidium atoms into one of two configurations.  In the first configuration, the absorption properties of the medium can be completely suppressed, in a technique called electromagnetic induced transparency (EIT).  With EIT, both the main body of the pulse and the precursor signal make it through the MOT and can be measured by the detectors.   Furthermore, the main signal is slowed considerably in speed and is separated in time from the optical precursor.  In the second configuration, the group velocity of the main signal is faster than $c$, i.e. the group velocity is “superluminal”.  However, this main signal is highly absorbed.

What were the results of the experiments?  The researchers measured the speed of the main body of the pulse and the precursor for both MOT configurations.  The precursor was clearly visible in both cases, and its speed was found to be exactly $c$, regardless of the MOT behavior.  Therefore no true “faster than light” behavior was observed, even when the group velocity was greater than the vacuum speed of light.

An important aspect of this result is a partial answer to a debate in quantum information theory — how fast does a single photon transmit information?  As we’ve noted, there are multiple definitions of the speed of light in matter, and it has remained a bit of a mini-mystery (at least to some) which of these described the speed at which information is transmitted.  The observation that a photon does have a precursor signal suggests that single photons can travel at the vacuum speed of light in matter, at least under the right conditions.

The result isn’t as earth-shattering as the possible discovery of superluminal neutrinos, but it does highlight a number of unusual aspects of optics, and further strengthens our understanding of both relativity and quantum mechanics!

************************

* In physics, “velocity” is used to refer to the vectorial motion of an object: not only how fast it is going, but in what direction.  ”Speed” is used to refer to the magnitude of velocity, or simply how fast the object is going.  We use the terms interchangeably here in the sense of “speed”.

** A. Sommerfeld, Ann. Phys. (Leipzig) 349 (1914), 177.  L. Brillouin, Ann. Phys. (Leipzig) 349 (1914), 203.

*** P. Pleshko and I. Palócz, “Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain,” Phys. Rev. Lett. 22 (1969), 1201.

**** S-H. Choi and U. Österberg, “Observation of optical precursors in water,” Phys. Rev. Lett. 92 (2004), 193903.

***** Actually, the anti-Stokes photon will already have a precursor signal due to its propagation in MOT 1!  This precursor is also chopped off by the EOM.

*************************

Zhang, S., Chen, J., Liu, C., Loy, M., Wong, G., & Du, S. (2011). Optical Precursor of a Single Photon Physical Review Letters, 106 (24) DOI: 10.1103/PhysRevLett.106.243602

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### 13 Responses to Faster than a speeding photon? Precursors test whether light can be faster than light

1. Pingback: Faster than a speeding photon? Precursors test whether light can be faster than light | benjamin junior

2. Pingback: RB Editor’s Selections: The Expanding Universe, Triggered Earthquakes, Statistical Badness, and More Neutrinos

3. CHRIS says:

Typo in last paragraph. “are” should be “our”.

• skullsinthestars says:

Ah, thanks! Fixed.

4. Ray Kilburn says:

Previous measurements of light velocity traveling through matter is akin to the photoelectric effect for which Einstein received his Noble prize. The current description is incomplete as electrons are incorrectly described, as in this blog, as a :”pendulum” or a spring is no more accurate than the orbital model. When we correlate the wave description of the electron’s orbit with its particle description, we realize a different view of the model of the atom. Rutherford’s model of the atom, and all subsequent variations, propose an orbital model of sorts with material electrons flying around the proton nucleus. Subatomic physics and quantum theory reduced this model to electron clouds that supported the quantum uncertainty of finding the electron in any particular position around the proton. Quantum Field Theory proposes that perhaps, the electron is not orbiting the proton at all, in this sense, but that the proton is actually inside of the electron! Each subsequent electron “shell” encompasses the previous shell. When light encounters an atom, corresponding wavelengths of the photonic wave are absorbed by the electron bubble of the corresponding frequency. While the time delay between absorption and emission explains normal ionic activity, apparent increase light velocity is directly proportional to the uncertainty of the speed of light to begin with. Keep in mind that Einstein’s velocity of light is ONLY accurate in a vacuum but not accurate within a gravitation field which effect light’s true path thru space as confirmed by relativity. Therefore light could be said to NEVER travel in a straight line by any terrestrial measurement than can be made and therefore its accurate velocity can never be determinate. In absence of a quantum description of gravity, it could be said that no gravity exists within an atom as we see no gravitational effect on particles with the atom and the gravitational effect is clearly only present outside of an atomic system. Therefore, one could postulate that our measure of light velocity within matter cannot be accurately determinate. Unlike a classical particle having mass, the Photon does not emit a rotating spherically symmetrical wave system with a continuous degree of freedom as do material particles. Nor is it normally polarized in a specific directional time orientation as observed in Electrons and Protons. Instead, it appears to interact with particles and antiparticles equally in both time domains. For this reason the Photon is said to be its own antiparticle. Following this reasoning further, one could postulate that one or more of the following is true: 1) the photon exists simultaneously in both time domains, 2) it rotates back and forth between time domains, or 3) is unaffected by time domains. When a Photon is compared to a particle with mass in motion, the Photon cannot be said to have a momentum in a single vector direction, however it appears to be propagated as a directional wave packet. However the Photon appears to propagate its momentum along a three-directional vector corresponding to physical construction of the space-time lattice structure with a definitive wave structure. Along each directional vector there is an angular momentum in the direction of motion called a heliocity. But just like particles having mass, this heliocity is restricted normally to integer values. The exception to this observation is in the case of a polarized photon (neutrino/antineutrino) which has a heliocity value of half-integer values. Because virtual particles imply both the inversion of space and time, both positive and negative energy states could be said to exist. It is only a mathematical trick that requires massless particles to be anti-unitary and anti-linear because the Standard Model of classical physics does not allow negative energy states, that is, energy less than that of the so-called vacuum. Because of this inexplicable paradox, classical physics does not allow the reversal of its time components either. Both cases to be incorrect but too lengthy to describe here.

5. Сергей Морозов says:

May I translate it in my blog (moogoo.ru)?

6. Pingback: Links for Tuesday « Galileo's Pendulum

7. Pingback: The Value of Experimenting With What We “Know” | Citizen Scientists League

8. Primoz Gabrijelcic says:

I don’t understand how the electro-optic modulator can react quick enough to clip the photon, which is traveling at c. That would imply that the signal that triggers the modulator travels with velocity > c. Or is the anti-Stokes photon traveling through some materil with low “speed of light”?

• skullsinthestars says:

One of the things I left out of my simple picture of the experimental setup is the delay line the anti-Stokes photon passes through! This photon is coupled into a long optical fiber, and with a long enough fiber the photon can be delayed as long as one likes compared to the modulator trigger — within reason!

9. Ray Kilburn says:

The result of any observation of a “superluminal neutrinos” may result in the realization that neutrinos may not really be an independent particle at all as currently considered by the Standard Model and may explain the difficulty in detecting them. Consider how the neutrino was “invented” to explain differences in mass between a neutron and its resulting beta decay model which supposedly results in an electron, proton and a neutrino that apparently flies away at light speed in violation of relativity and doesn’t interact with any matter. What is generally ignored is that the additional mass measured in the neutron may be a result of the retrograde spin of the neutron (which could be considered a micro-model of a hydrogen atom (proton+electron)) and therefore may be relativistic mass and not a real mass at which explains why the neutrino is always considered massless. During beta decay, when the proton and electron are released from the neutron, the remaining relativistic mass spins off as a photon that is moving as a helicoid instead of a plane-wave photon due to the angular momentum imparted upon it from the spin of the originating neutron. Since a helicoid looks like a spring, the photon’s angular momentum tunnels the so-called “neutrino” through all mass without stopping. The is the same principal used by electrons to tunnel thru matter as well. I think that upon further examination we will find that the neutrino is really photon traveling as a helicoid and therefore unpolarized and incapable of interaction with matter. Plane-wave photons which are polarized along two planes run into everything. Real super-luminal communication of information is probably the result that the photon as a discreet entity really doesn’t travel anywhere. Consider that the photon is really at both its origin and destination at the same time, more akin to a string connecting point A to B. The phontonic wave we observe is the jiggling of the string, nothing is really moving anywhere except the wave along the string, the string doesn’t move anywhere. From an outside observer we see the wave moving along the string and measure its velocity at c. From the perspective of the photon, it is already there since zero time dilation occurs and the photon appears (from its perspective) to arrive at the same moment it departed it source. Another point to consider in the measurement of light speed in a vacuum is that true vacuums exist nowhere in reality. Even when we peer into the deepest darkest part of space, we still measure background microwave radiation in 3micron wavelength. Therefore a photon at some energy level exists at all points in space and no point in space could be said not to contain either traditional matter or a photon at some energy level. So no true vacuum could be said to exist nor can be created and therefore light true velocity will always be indeterminate in all true circumstances. Our current measurement of light’s velocity is an approximation. Theories surrounding the big bang suggests that the speed of light varies as the universes expands….the string is getting longer?

10. zephirawt says:

My stance is, OPERA results are real and they should be published ASAP. IMO we can model the space-time (brane) with density gradient at the phase interface of two elastic fluids. After then a two kinds of solitons will appear: A) the one, which corresponds to photons and it spreads with slightly lower speed, than the transverse surface waves (which are serving as an analogy of light waves) B) the faster one, which corresponds the neutrinos and it will spread with slightly higher speed, than the surface ripples. The first kind of solitons results from coupling of surface ripples with longitudinal bulk waves of more dense phase, the second one from coupling of surface ripples with longitudinal waves of less dense phase. From this perspective the neutrinos would behave like the superpartners of gamma ray photons, i.e. like the lightweight photinos.

11. Anirudh Kumar Satsangi says:

Dear Mr. Jug Suraiya

Congratulations for your very thought provoking article ‘Einstein Won’t Mind’ (The Speaking Tree, Oct. 9, 2011, pg. 7). Einstein is great, but Newton is all time great. I would like to quote your few excellent lines from this article before I come to main theme of this article: “Faith and religious beliefs are destinations reached; science and skepticism are journeys without end”. But it’s to be modified. In Veda it is written, ‘Neti, Neti’, ‘Not this, Not this’. So in Vedic religion journeys never end. I consider Vedas the most honest scriptures which do not limit the scope of further exploration of truth. Now I come to the speed of light. This has long been proved since the time of discovery of black holes that the speed of light is not the fastest. Black holes do not allow even light to escape. It means the escape velocity at the black holes is much higher than the speed of light. Black holes are the infinitely dense ball of gravitation force. All creational forces of the universe have originated from the gravitational force field and will end up in it. The speed of light is no doubt fastest in our solar system. The source of light is Sun in our solar system. But how this light is originated? We should study the various stages involved in the formation of a star. Our Sun is also a star.

The starting material for the formation of a star is mainly hydrogen gas and helium gas. If the hydrogen cloud contains a very large number of atoms, each atom feels the gravitational pull of all the atoms in the hydrogen cloud. (Here is NO LIGHT)

The gas cloud becomes a permanent entity, held together by the mutual attraction of all the atoms present in it. The cloud then begins to contract under its own gravity setting off the process which will convert this huge condensed gas cloud into a star. Such a tight contracting cluster of atoms held in the grip of its own gravity, is called a protostar. The protostar is not yet a star and does NOT emit LIGHT. The temperature of this star is as low as -173 degree C.

The force of gravity acting on different atoms in the protostar draws every atom towards centre. As a result, the protostar shrinks in size and its density increases. As the atoms in the protostar fall towards the centre, they pick up speed. Because of the high speed and greater density of atoms, the atoms in the gas cloud collide with one another more frequently, thereby raising its temperature from -173 degree C to about 10 ^7 degree C. At these extremely high temperatures the proton (hydrogen nuclei) at the centre of the protostar collide together and undergo a nuclear fusion to form helium nuclei. In this reaction a tremendous amount of energy is released. This further raises the temperature and pressure. The release of nuclear energy marks the birth of the star. The protostar now beings to GLOW and becomes a STAR. Here at this stage LIGHT is ORIGINATED. Thus light is NOT ETERNAL. It has a beginning and an end. So LIGHT cannot be claimed as Cosmic Constant. However, Gravitation Force is eternal.

It is evident from the above description that light is latent before the birth of star. Light originates and become kinetic only after the action of gravitation force. So the speed of light can never exceed the speed of gravitation force. It cannot be ruled out that the speed of gravitation force is infinitely greater than the speed of light at black holes.

1. Gravitation Force is the Ultimate Creator
2. In Scientific Terminology Source of Gravitational Wave is God

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