# How does light slow in the presence of gravity?

## Main Question or Discussion Point

I'm interested in some physical interpretations of how light slows in free space in the presence of gravity according to general relativity. I believe the reference frame is therefore distant, essentially infinity....say, as when we observe light moving toward a black hole or passing a star.

For example, in contrast to free space, when light enters a dense optical medium, a nice picture is to view photons being absorbed and new ones being emitted as photons move from atom to atom. The absorption and subsequent emission of a new photon delays the passage of photons hence "slowing" light.

But in free space, what happens? There are no atoms, only gravitons, and I don't think gravitons can absorb and subsequently emit photons, analogous to my example above, so what are some explanations for the electromagnetic (photon or wave ) and gravitational(graviton or field) interaction? It would also be interesting to know if all frequencies slow the same amount. I'm pretty sure they do. Why is this different than a dense optical medium where different frequencies slow different amounts. And is this slowing in free space related to the Shapiro time delay.

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George Jones
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I'm interested in some physical interpretations of how light slows in free space in the presence of gravity according to general relativity.
The physically measure speed of light does not slow in the presence of gravity, only the coordinate speed does. Depending on the coordinates, the coordinate speed of light can be anything, even in special relativity.

Well, what I am asking is when a suitable reference frame is chosen to observe a change in speed, what physical phenomena is taking place between the electromagnetic and gravitational entities? There IS some physcial causality! I don't what to choose a reference frame whether there is nothing to observe.

Here is an example of wikipedia...if this isn't s change in speed for light what's happening:

Closely related to light deflection is the gravitational time delay (or Shapiro effect), the phenomenon that light signals take longer to move through a gravitational field than they would in the absence of that field. There have been numerous successful tests of this prediction.
http://en.wikipedia.org/wiki/General_relativity#Consequences_of_Einstein.27s_theory

Here is one source that generated my question:

Contrary to intuition, the speed of light (properly defined) decreases as the black hole is approached. In fact, one way to understand the bending of light by the gravitational field of a star is to regard it as resulting from the refraction of the wavefront due to the fact that the part of the wavefront that is nearer to the star moves more slowly than the part farther away from the star. The result is that the direction of advance of the wavefront is deflected toward (or around) the star.

If the photon, the 'particle' of light, is thought of as behaving like a massive object, it would indeed be accelerated to higher speeds as it falls toward a black hole. However, the photon has no mass and so behaves in a manner that is not intuitively obvious.

The reason for the qualification 'properly defined' above is that the speed of light depends upon the vantage point (frame of reference) of the observer. When we say that the speed of light is decreased, we mean from the perspective of an observer fixed relative to the black hole and at an essentially infinite distance. On the contrary, to an observer free falling into the black hole, the speed of light, measured locally, would be unaltered from the standard value of c.

Here is a quote from another post here:

The speed of light in a gravitation field:

If, however, the distance through which the light travelled in the course of measuring its speed was too great, the deviation of the reference frame from being 'flat' would become apparent. That is, gravitational effects would begin to become apparent.

So, it is absolutely true that the speed of light is _not_ constant in a gravitational field [which, by the equivalence principle, applies as well to accelerating (non-inertial) frames of reference]. If this were not so, there would be no bending of light by the gravitational field of stars. One can do a simple Huyghens reconstruction of a wave front, taking into account the different speed of advance of the wavefront at different distances from the star (variation of speed of light), to derive the deflection of the light by the star.

Indeed, this is exactly how Einstein did the calculation....

c' = c0 ( 1 + V / c^2 )*

where V is the gravitational potential relative to the point where the speed of light c0 is measured.

So, the fact that the speed of light changes in a gravitational field was expressed by Einstein himself in 1911 (though he made in this paper an error in the derivation of the bending of light, which he later, luckily for him, corrected before the experiments were made)QUOTE]

* I modified the "C2" term on the post as I believe the correct term is c^2 as shown above.

atyy
And is this slowing in free space related to the Shapiro time delay.

The Shapiro time delay is a coordinate time delay. We can choose any coordinates we wish, so there are many viable definitions of the speed of light in General Relativity, and we just have to make it clear which definition is being used. The one definition that is always available is that the speed of light is the same in every local Lorentz frame. Since there is a (different) local Lorentz frame at every point in spacetime, the local speed of light in spacetime is constant in this sense.

This is presumably why the link you provided says: "The reason for the qualification 'properly defined' above is that the speed of light depends upon the vantage point (frame of reference) of the observer. When we say that the speed of light is decreased, we mean from the perspective of an observer fixed relative to the black hole and at an essentially infinite distance. On the contrary, to an observer free falling into the black hole, the speed of light, measured locally, would be unaltered from the standard value of c."

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atyy
I don't what to choose a reference frame whether there is nothing to observe.
What you observe is independent of the reference frame. How you describe what you observe is dependent on the reference frame.

atyy
And is this slowing in free space related to the Shapiro time delay.
The usual textbook presentation of the Shapiro time delay is that it is the delay compared to what would happen if Newton's law of gravity were correct, and light were not affected by gravity. But when we do the experiment, we only have real light, which we do not yet know obeys General Relativity (since that is what we wish to test), and there is no such thing as Newtonian light against which to compare the experimental result for real light - so how can such a comparison be made?

Here is the quote from Will, Theory and experiment in gravitational physics, CUP 1993:
"Since one does not have access to a "Newtonian" signal against which to compare the round trip travel time of the observed signal, it is necessary to do a differential measurement of the variations in round trip travel times as the target passes through superior conjunction and look for the logarithmic behavior. To achieve this accurately however, one must take into account variations in round trip travel time due to the orbital motion of the target relative to the Earth ... The resulting predicted round trip travel times in terms of the unknown coefficient (1/2)(1+gamma) are then fit to the measured travel times using the method of least squares, and an estimate obtained for (1/2)(1+gamma). [This is an oversimplification of course. The reader is referred to Anderson (1974) for further discussion]"

Shapiro time delay experiment is very simple and straightforward. A radio signal is sent from Earth to Mars, reflects from the Mars surface and returns to the Earth observatory. If Sun is far from the Earth-Mars line then the measured travel time is

T = 2L/c (1)

where L is the Earth-Mars distance. If Sun is close to the Earth-Mars line the measured roundtrip time was found to be greater than (1). There could be only two logical explanations of this time delay.

(a) The Mars-Earth distance L increases when the Sun is between the two planets.

(b) The speed of light (c) decreases when it passes near the Sun.

The explanation (b) seems more reasonable to me.

Note that this slowdown effect can be calculated with a simple Hamiltonian describing the photon-Sun interaction. In the absence of such interaction the Hamiltonian of the 2-body system Sun+photon is

H_0 = Mc^2 + pc

where M is Sun's mass and p is photon's momentum. If the gravitational interaction is turned on, the Hamiltonian is

H = H_0 - 2GMp/(cr) = Mc^2 + p[c - 2GM/(cr)] (2)

where r is the Sun-photon distance and G is the gravitational constant. The term in square brackets can be interpreted as (distance-dependent) reduction of the speed of light. It is not difficult to show (using standard Hamilton's equations of motion) that Hamiltonian (2) yields exactly the measured Shapiro time delay. Note that the speed reduction does not depend on the photon's frequency (momentum). Moreover, the same Hamiltonian explains quantitatively the light deflection by the Sun's gravity.

Meopemuk,

Thank's for that reference...I saw it somewhere but could not find it to post here.

In that description it seemed to me maybe the curving (warping) of space by the sun does make the distance traveled by light longer than would be measured as "L" from earth. (If we used a light based measure, we'd be unaware of any change.) So although the path would be the geodesic seen by photons as they travel we would not be aware of such a path observing from earth. Is that a correct interpretation?

Another question: What is "coordinate speed"...I can't find that term anywhere...not Wikipedia, not the popular physics and cosmology texts I use. I'm think I understand that the speed of light is always measured as "c" locally in general relativity, that is, if one sits adjacent to a photon when ones "measures" the speed of light and distances are small relative to curvature they appear flat (Euclidean) and speed is "c".

Is it correct to say when we measure sufficiently large distances for curvature to be a factor, the speed of light is "observed" to vary from c??

atyy:

What you observe is independent of the reference frame. How you describe what you observe is dependent on the reference frame.
I don't understand that...sounds like my description is inconsistent!!.....part of my general difficulty in interpretations is that I am unsure what assumptions are attributed to various observers. For example, all earth bound observers are immersed in earth's gravitational field...but an observer in nearly free space experiences far less gravity...so local time measure, for example, is different for each. I believe each therefore measures light speed differently at the others locale, but at c locally...is that the idea?

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Wikipedia provides one possible interpretation:
Several decades after the discovery of general relativity it was realized that general relativity is incompatible with quantum mechanics. It is possible to describe gravity in the framework of quantum field theory like the other fundamental forces, such that the attractive force of gravity arises due to exchange of virtual gravitons, in the same way as the electromagnetic force arises from exchange of virtual photons.[
http://en.wikipedia.org/wiki/Gravity#Gravity_and_quantum_mechanics

granpa....if you understand any of the google search results a brief interpretation would be appreciated....the first reference says:

Velocity and time, relative to a gi
ven inertial frame in special relativity, is ordinarily described with reference to the behavior of clocks at rest in the inertial frame of interest. We refer to this as the coordinate kinematic'' or time/velocity pair.
But my question(s) pertain to general relativity, that is, in the presence of gravitational fields, while special relativity excludes gravitational effects....the difference between inertial frames (no acceleration) in SR and acceleration in GR prevents me from putting "coordinate" terms in perspective...

atyy
Another question: What is "coordinate speed"...I can't find that term anywhere...not Wikipedia, not the popular physics and cosmology texts I use.
Coordinate speed or coordinate velocity is not such a common term, but it is what people mean when they refer to a velocity that is coordinate-dependent or observer dependent.
http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html

I don't understand that...sounds like my description is inconsistent!!.....part of my general difficulty in interpretations is that I am unsure what assumptions are attributed to various observers. For example, all earth bound observers are immersed in earth's gravitational field...but an observer in nearly free space experiences far less gravity...so local time measure, for example, is different for each. I believe each therefore measures light speed differently at the others locale, but at c locally...is that the idea?
This can happen even in everyday space. Say you set out cartesian axes x,y,z, and you find there is a cat at x=1. Your friend sets out different cartesian axes which he also calls x,y,z and he finds the same cat at y=99. Is the cat really on the x axis or is it really on the y axis?

Coordinate speed or coordinate velocity is not such a common term, but it is what people mean when they refer to a velocity that is coordinate-dependent or observer dependent.
OK!!....that sounds like a generic term for either SR or GR......any coordinates for observers other than local inertial coordinates for special relativity, and other than free fall for observers for general relativity....all those frames measure local light speed as "c"...

Quoting from a post above:
The reason for the qualification 'properly defined' above is that the speed of light depends upon the vantage point (frame of reference) of the observer. When we say that the speed of light is decreased,(going towards a black hole) we mean from the perspective of an observer fixed relative to the black hole and at an essentially infinite distance. On the contrary, to an observer free falling into the black hole, the speed of light, measured locally, would be unaltered from the standard value of c.

So a single coordinate frame of reference is used in that explanation..and that is ok by me because if I understand that explanation correctly, a difference is observed in the speed of light in free space and the speed of light approaching near a black hole.

So I am now back to my original post: So what's the physcial interaction causing that change? Is there one?...Is it Shapiro time delay an explanation??

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atyy
OK!!....that sounds like any coordinates for observers other than local inertial coordinates for special relativity, and other than free fall for observers for general relativity....all those frames measure local light speed as "c"...
Yes. The key is that in special relativity, spacetime is flat, and an observer can use global Lorentz inertial coordinates that cover all of spacetime. In curved spacetime, all observers have Lorentz inertial coordinates, but only locally. To describe something nonlocal, he must extend his local coordinates to global coordinates, which will be different from the global Lorentz inertial coordinates of flat spacetime.

I'm trying to post in this thread various references for the benefit of all readers because interpretations vary, terminology varies, and some appear to give different answers regarding the speed of light.

Here are two explanations of the speed of light from:

http://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html

(page 2)

we need first to be very clear about what we mean by constancy of the speed of light, before we answer our question. We have to state what we are going to use as our standard ruler and our standard clock when we measure c. In principle, we could get a very different answer using measurements based on laboratory experiments, from the one we get using astronomical observations.
I take the above quoted reference to mean that local measures will differ from distant measures because curvature of spacetime effects measurements.

And (page 4)

In the 1920 book "Relativity: the special and general theory" (Einstein)wrote: . . .

according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity [. . .] cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position.

Since Einstein talks of velocity (a vector quantity: speed with direction) rather than speed alone, it is not clear that he meant the speed will change, but the reference to special relativity suggests that he did mean so. This interpretation is perfectly valid and makes good physical sense, but a more modern interpretation is that the speed of light is constant in general relativity.

The problem here comes from the fact that speed is a coordinate-dependent quantity, and is therefore somewhat ambiguous. To determine speed (distance moved/time taken) you must first choose some standards of distance and time, and different choices can give different answers. This is already true in special relativity: if you measure the speed of light in an accelerating reference frame, the answer will, in general, differ from c.

In special relativity, the speed of light is constant when measured in any inertial frame. In general relativity, the appropriate generalisation is that the speed of light is constant in any freely falling reference frame (in a region small enough that tidal effects can be neglected). In this passage, Einstein is not talking about a freely falling frame, but rather about a frame at rest relative to a source of gravity. In such a frame, the speed of light can differ from c, basically because of the effect of gravity (spacetime curvature) on clocks and rulers.
Note the language describing Einstein's "freely falling reference frame"...he apparently is describing local measures when at rest relative to the source of gravity...as when sitting in a lab on earth measuring local light speeds.

This appears to conform with the prior post by ATYY: Here on earth, if I measure light locally I'll measure c; but if I observe very distant light ,say approaching a black hole lightyears distant, I will measure a slower speed than c. This is analagous to the Shapiro effect, posted earlier: It is a result of measurement of great distance where curvature is significant...hence it it as coordinate dependent.

I take the term "coordinate dependent" to mean the same as the term "reference frame dependent"....

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atyy
Reading Naty1's link to Carlip and Gibbs's article http://math.ucr.edu/home/baez/physic..._of_light.html [Broken], I'm rethinking my earlier answers. I previously said that the local speed of light is the same at all points in space, because there is a local Lorentz inertial frame at every point in spacetime. I think this is not quite correct, because the local Lorentz inertial frame is only a first order approximation, which would seem to imply that the speed of light is constant only to first order.

However, in General Relativity the speed of light is exactly constant because light follows a null geodesic exactly. So a better answer might be that the speed of light is defined to be constant. Presumably this is why Carlip and Gibbs say, "in the light of well tested theories of physics, it does not even make any sense to say that it varies." Hence http://physics.nist.gov/cgi-bin/cuu/Value?c

I guess the interesting thing is that you cannot make a measurement until you have defined what space and time are. And at least for now, General Relativity is our definition of space and time.

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Here is another lightspeed tidbit from Fabric of the Cosmos, Brian Greene, footnote 3-21:

Speed of light is calculated from Maxwell's equations as

c = 1/([epilson x mu]1/2)....
where epsilon is electric permittivity and mu is the magnetic permeability....

From Halliday and Resnick:
these are measurements, experimentally determined, right here on earth....

epilson was in fact measured via capacitor measurements while magnetic permeability was apparently measured from Ampere's law experiments....

So the constant for "c" was based on measures within earth gravity. I'm not sure if some other more modern experimental data now defines c.

atyy
So the constant for "c" was based on measures within earth gravity. I'm not sure if some other more modern experimental data now defines c.
I think in the latest definition it is not possible to measure c (no error bars!): http://physics.nist.gov/cgi-bin/cuu/Value?c.

atyy posts:
I think this is not quite correct, because the local Lorentz inertial frame is only a first order approximation, which would seem to imply that the speed of light is constant only to first order.
Seems to me that local measures over short distances eliminates gravitational curvature except in extreme cases of density. So I'm inclined to think your earlier statement might well be entirely accurate.

On a separate issue, Brian Greene in Fabric of the Cosmos makes no bones about it: When Einstein said "the benchmark for general relativity is a feely falling observer who has given in to gravity and are not acted on by any other forces", that's what Einstein meant. Yet a quote alread posted, #18, claims what Einstein really meant was an observer at rest with respect to the source of gravity...

From post #6:
"Indeed, this is exactly how Einstein did the calculation....

c' = c0 ( 1 + V / c^2 )*

where V is the gravitational potential relative to the point where the speed of light c0 is measured...."

Does anyone know the origin, assumptions as frame of reference and proper interpretation behind this equation? There appear three speeds for light!! If the speed of light varies locally according to different gravitational potential (V) then it seems there is some sort of local frame of reference where speed of light varies....
All in all, a tad confusing!!!!

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Al68
So I am now back to my original post: So what's the physcial interaction causing that change? Is there one?
No. There is no physical interaction or force causing the change. Similar to the pseudo force or fictional force in Newtonian physics.

Al

Al68... I am beginning to think you are right!! but it seems to me Shapiro time delay is an experimental basis for coordinate velocity changes.

Yet, from an above cited reference: "Since Einstein talks of velocity (a vector quantity: speed with direction) rather than speed alone, it is not clear that he meant the speed will change, but the reference to special relativity suggests that he did mean so. This interpretation is perfectly valid and makes good physical sense, but a more modern interpretation is that the speed of light is constant in general relativity..."

"makes good physical sense.." really bugs me....Is this now considered an invalid explanation?

I found an explanation of the above formula in Post #22....as suspected, it appears to be a coordinate reference frame...... and applies to a uniform gravitational field, not one from a spherical body which is the second formula here:

http://www.geocities.com/physics_world/gr/c_in_gfield.htm

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doesnt that equation imply that the velocity of light can not only be less than c but can even go to zero or become negative? how is that possible if the change in the speed of light is due to space stretching and distances becoming longer near a massive object?

or have I completely misunderstood it?