Light & Gravitation: How Do We Know & Prove It?

[paweld]

How do we know that light travels on null geodesic in arbitrary curved spacetime.
Could anyone give me reason for this assumption (many GR textbooks assume
this without any justification).

In flat space time the above fact may be proven by means of the Green function.
It tells us that the potential at some point is determined by the sources on past light cone
of this point. Does we know the formula for the Green function in curved spacetime?

Thanks for answer.

 

[atyy]

I think it is an approximation (MTW, chapter 22).

http://arxiv.org/abs/0806.0464
"So in this case, light moves at the speed of light, as well as all lower velocities!"

 

[bcrowell]

I don't think it can be proved. Einstein simply decided to make GR a geometrical theory, in which particles move along geodesics. I think you have it backwards about what is more fundamental than what. The fact that GR is a geometrical theory of gravity is more fundamental than anything about Green functions.

 

[jambaugh]

How do we know that light travels on null geodesic in arbitrary curved spacetime.
Could anyone give me reason for this assumption (many GR textbooks assume
this without any justification).

In flat space time the above fact may be proven by means of the Green function.
It tells us that the potential at some point is determined by the sources on past light cone
of this point. Does we know the formula for the Green function in curved spacetime?

Thanks for answer.

One way to view it is to first ask: How do we know the geometry of space-time to determine what is a null geodesic? Given we do this through empirical observation of e.g. how light behaves, the definition of a null geodesic is the path light takes under purely gravitational influences.

Like the speed of light this is not so much an observation as a definition.

 

[atyy]

http://relativity.livingreviews.org/Articles/lrr-2004-6/

"While in flat spacetime the retarded Green’s function has support only on the future light cone of x', in curved spacetime its support extends inside the light cone as well ...

While in flat spacetime the advanced Green’s function has support only on the past light cone of x', in curved spacetime its support extends inside the light cone ..."

 

[paweld]

One way to view it is to first ask: How do we know the geometry of space-time to determine what is a null geodesic? Given we do this through empirical observation of e.g. how light behaves, the definition of a null geodesic is the path light takes under purely gravitational influences.

Like the speed of light this is not so much an observation as a definition.

I understand that we have to assume certain things in order to create the theory but
once we have the theory we can always check if the theory is in agreement with
our assuptions.

It turns out that the statement that light travels on null geodesic (which was the assumption) is
only approximately true because according to: "[URL
http://relativity.livingreviews.org/Articles/lrr-2004-6/
"in curved spacetime, electromagnetic waves propagate not just at the speed of light, but
at all speeds smaller than or equal to the speed of light; the delay is caused by an
interaction between the radiation and the spacetime curvature."

 

[bcrowell]

I think jambaugh is right, and atyy and paweld are overinterpreting the Living Reviews paper. BTW, here is a link to the section that actually contains this statement: http://relativity.livingreviews.org/open?pubNo=lrr-2004-6&page=articlesu4.html

When we say that particles travel on geodesics in GR, there are some technical restrictions on that statement. These restrictions include (1) that the particle have a small mass-energy (i.e., it can be considered a "test particle") and (2) that the particle be localized (i.e., it can be considered a particle).

If you don't apply restriction #1, then the decay of the orbits of the neutron stars in the Hulse-Taylor binary would falsify GR.

If you don't apply restriction #2, then it's not clear what is even meant by the trajectory of the particle, so there is no way to define whether the trajectory is or is not a geodesic. If you ignore this restriction, then you can convince yourself that the gradual recession of the moon from the Earth falsifies GR.

When an electromagnetic wave is propagating in curved spacetime, as in the LR article, it fails restriction #2. The effects they're talking about depend on the fact that the wavefront has finite spatial extent, and the effects vanish in the limit where the wavefront's spatial extent goes to zero.

If you want to overinterpret the significance of the discussion in the LR article, you can come to all kinds of ridiculous conclusions. For example: "The potential depends on the particle’s state of motion for all times [earlier than the given time]." If I wanted to ignore the context and act as though this kind of thing was fundamental rather than derived, then I could conclude that particles in GR retain an inertial memory of their own motion infinitely far into the past. This would be contrary to the entire notion of inertia ever since Galileo; Galilean physics, unlike Aristotelian physics, doesn't allow particles to retain a memory of their previous state of motion beyond an infinitesimal time in the past. Such an interpretation would also invalidate the foundations of GR as a classical field theory, in which you're supposed to be able to find unique solutions to boundary-value problems.

 

[atyy]

When we say that particles travel on geodesics in GR, there are some technical restrictions on that statement. These restrictions include (1) that the particle have a small mass-energy (i.e., it can be considered a "test particle") and (2) that the particle be localized (i.e., it can be considered a particle).

That's the same as saying that the test particle limit is an approximation (which incidentally, the Poisson article is about justifying this approximation from full GR without test particles, as opposed to GR+test particles).

 

[bcrowell]

That's the same as saying that the test particle limit is an approximation (which incidentally, the Poisson article is about justifying this approximation from full GR without test particles, as opposed to GR+test particles).

Right, I think we all agree on what's an approximation. The issue is what's more fundamental.

IMO the approximate statement (test particles follow geodesics) is fundamental, and the exact statement (wavefronts can propagate at <c) is derived.

 

[atyy]

If you don't apply restriction #1, then the decay of the orbits of the neutron stars in the Hulse-Taylor binary would falsify GR.

BTW, do the binaries really not travel on geodesics of the full spacetime (ie. background + gravitational waves)?

I know that elements of the star don't travel on geodesics, but that would be true even without the decaying orbits, just by virtue of being acted on non-gravitationally by other elements of the star.

 

[atyy]

Right, I think we all agree on what's an approximation. The issue is what's more fundamental.

IMO the approximate statement (test particles follow geodesics) is fundamental, and the exact statement (wavefronts can propagate at <c) is derived.

Sure that's fine. I had in mind EH action + minimally coupled matter action.

So from your point of view, you would take the result proven in http://arxiv.org/abs/gr-qc/0309074 as axiomatic?

It doesn't really matter anyway, as long as it's all self consistent. Axioms are a matter of taste, except for generalization, since GR is presumably wrong at some level.

 

[bcrowell]

Thanks for the pointer to the Ehlers paper, atyy -- that's cool.

So from your point of view, you would take the result proven in http://arxiv.org/abs/gr-qc/0309074 as axiomatic?

I would respond to that question by being weasely :-) Physics isn't math, and physical theories aren't axiomatic systems. The physical principles behind GR (the equivalence principle, ...) have resisted precise mathematical formulation, and maybe they always will. The theorem in the Ehlers paper isn't a general theorem in GR; it's a theorem about certain types of physical systems -- those that obey the dominant energy condition and are nonsingular. (The latter isn't explicitly stated in the paper, but I think it's implicit, since they assume the metric is well defined in a certain neighborhood.) We have strong reasons to believe that there are physical systems that violate the DEC ( http://arxiv.org/abs/gr-qc/0205066v1 ), and that there are also physical systems that are singular. What I would say is that if we came across a real-world physical system (a singular one, or one that violated the DEC) for which there was no result conceptually analogous to the theorem in the Ehlers paper, then that would falsify GR's geometrical picture of gravity.

BTW, do the binaries really not travel on geodesics of the full spacetime (ie. background + gravitational waves)?

I think the first two paragraphs on p. 2 of the Ehlers paper do a good job of explaining why this question doesn't have a well-defined answer unless the terms are more clearly defined.

If you want a specific example where bodies of finite mass clearly don't travel along geodesics, then replace one of the neutron stars in the Hulse-Taylor binary with a black hole. In this situation we can't possibly talk about whether the black hole follows a geodesic of the full metric, since in that spacetime the singularity isn't a point in the manifold; all we can hope to say is whether the black hole follows a geodesic of the background metric. It is then clear that it can't follow a geodesic of the background metric, since the rate of radiation is proportional to the square of the mass of the black hole.

 

[atyy]

Thanks for the pointer to the Ehlers paper, atyy -- that's cool.


I would respond to that question by being weasely :-) Physics isn't math, and physical theories aren't axiomatic systems. The physical principles behind GR (the equivalence principle, ...) have resisted precise mathematical formulation, and maybe they always will. The theorem in the Ehlers paper isn't a general theorem in GR; it's a theorem about certain types of physical systems -- those that obey the dominant energy condition and are nonsingular. (The latter isn't explicitly stated in the paper, but I think it's implicit, since they assume the metric is well defined in a certain neighborhood.) We have strong reasons to believe that there are physical systems that violate the DEC ( http://arxiv.org/abs/gr-qc/0205066v1 ), and that there are also physical systems that are singular. What I would say is that if we came across a real-world physical system (a singular one, or one that violated the DEC) for which there was no result conceptually analogous to the theorem in the Ehlers paper, then that would falsify GR's geometrical picture of gravity.


I think the first two paragraphs on p. 2 of the Ehlers paper do a good job of explaining why this question doesn't have a well-defined answer unless the terms are more clearly defined.

If you want a specific example where bodies of finite mass clearly don't travel along geodesics, then replace one of the neutron stars in the Hulse-Taylor binary with a black hole. In this situation we can't possibly talk about whether the black hole follows a geodesic of the full metric, since in that spacetime the singularity isn't a point in the manifold; all we can hope to say is whether the black hole follows a geodesic of the background metric. It is then clear that it can't follow a geodesic of the background metric, since the rate of radiation is proportional to the square of the mass of the black hole.

Yes, the black hole clearly cannot follow a geodesic of the background metric. And the singularity is clearly not part of the manifold. But it seems that under some circumstances a gravitationally radiating point particle can be seen as moving on a geodesic of background+perturbation - I had in mind sections 5.3 of Poisson's review where he gives this interpretation, then derives the same equations more rigrously in 5.4 using a black hole, where he then doesn't give this interpretation, presumably for the reasons you brought up. I don't know whether the approximations in his section 5.3 are relevant for the binary pulsar.

 

[yuiop]

I think it is an approximation (MTW, chapter 22).

http://arxiv.org/abs/0806.0464
"So in this case, light moves at the speed of light, as well as all lower velocities!"

In that paper I noticed this statement:

• By taking a different path through spacetime than
the field, which let's it encounter even the parts of
the field that move at the speed of light.

I could not let it pass without asking if it possible for something that is traveling at less than the speed of light to catch up with something that is traveling at the speed of light, by taking a shortcut through spacetime? Secondly, is that possible in normal curved spactime if the light does not reflect off anything and without the subluminal or luminal particle passing through any event horizons?

 

[jambaugh]

In that paper I noticed this statement:


I could not let it pass without asking if it possible for something that is traveling at less than the speed of light to catch up with something that is traveling at the speed of light, by taking a shortcut through spacetime? Secondly, is that possible in normal curved spactime if the light does not reflect off anything and without the subluminal or luminal particle passing through any event horizons?

Without invoking wormholes and such, yes you can but its more like the light taking the long way round.

Imagine you're on a ship buzzing by a very massive neutron star, nearly the mass to form a black hole. You flash a laser at the horizon of the star and some of the light can be bent all the way around back at you so you see the flash. Not really much different from a mirror but technically the light is taking the (locally to its path) shortest route back to you but you beat it to the second location (or tie it there which means with a tweak of conditions you can beat it there.)

 

[yuiop]

Without invoking wormholes and such, yes you can but its more like the light taking the long way round.

Imagine you're on a ship buzzing by a very massive neutron star, nearly the mass to form a black hole. You flash a laser at the horizon of the star and some of the light can be bent all the way around back at you so you see the flash. Not really much different from a mirror but technically the light is taking the (locally to its path) shortest route back to you but you beat it to the second location (or tie it there which means with a tweak of conditions you can beat it there.)

Ah, OK. Thanks for that clarification.