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I have read these questions:

Does a photon in vacuum have a rest frame?

Based on dmckee's answer, the answer is no to a photon's rest frame.

In the modern view each particle has one and only one mass defined by the square of it's energy--momentum four vector (which being a Lorentz invariant you can calculate in any inertial frame): $$ m^2 \equiv p^2 = (E, \vec{p})^2 = E^2 - \vec{p}^2 $$ For a photon this value is zero. In any frame, and that allows people to reasonably say that the photon has zero mass without needing to define a rest frame for it.

Now I understand that the graviton is hypothetical.

But it is meant to be massless like the photon. It is a gauge boson like the photon.

So we can write the same for a graviton: $$ m^2 \equiv p^2 = (E, \vec{p})^2 = E^2 - \vec{p}^2 $$

And this should be 0 in any inertial reference frame for the graviton too.

Now as per SR (second postulate):

  1. the speed of light in vacuum is the same for all observers, regardless of the light source

Now for GWs (like EM waves), it should be true that:

  1. the speed of GWs in vacuum is the same for all observers regardless of the motion of the GW's source

In this case, gravity could be thought of as the same thing as light, and we could basically build SR onto GWs (instead of light). We could build the postulates of SR onto GWs, derive the laws of SR from the behavior of GWs (instead of light).

Question:

  1. Does a graviton (I understand it is theoretical, but what is it meant to be) in vacuum have a rest frame?

  2. Do GWs obey the laws of SR, so is the speed of GWs in vacuum the same for all observers regardless of the motion of the GW's source?

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Special relativity is part of general relativity and gravitational waves should travel with the velocity of light.

Gravitons are hypothetical particles of a quantized general relativity. There exists a graviton that is massless and is like the photon, but also Kaluza Klein gravitons which have a mass and thus have variable velocities depending on the specific problem. These are research projects as can be seen by searching for "kaluza klein graviton"

Does a graviton (I understand it is theoretical, but what is it meant to be) in vacuum have a rest frame?

Some gravitons in some models may have a rest frame, but not the lowest order one, which is photon like massless and moving with velocity c.

Do GWs obey the laws of SR, so is the speed of GWs in vacuum the same for all observers regardless of the motion of the GW's source?

Classical gravitational waves also follow special relativity, in flat spaces . See here.

See also my answer here for kaluza klein models

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  1. There is no rest frame for the hypothetical graviton for the reason you specified in your post. It has 0 mass, and anything with 0 mass has no rest frame.
  2. SR explicitly applies for Minkowski (flat) space time. A gravitational wave spacetime is not Minkowski and so the results from SR do not strictly have to hold. However, since a gravitational wave is a weak field approximation, its spacetime is approximately Minkowski - and so at least to first approximation, gravitational waves travel at the speed of light for all observers regardless of the motion of the source. We assume, when deriving gravitational waves, that we are sufficiently far from the source to make this approximation. If one is very close to the source, it becomes unclear how to even really define "gravitational wave" since the spacetime wouldn't have a nice "flat metric + GW perturbation" kind of structure to it there.
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