Definition of distance in GTR

[wolfgang6444]

TL;DR

in nonrelativistic physics and in special theory of relativity distance is defined by the time light takes to travel from A to B. How about general theory of relativity

Up front I need to say that I have a certain understanding of STR, but never studied GTR thoroughly.
What puzzled me is the interferometric proof of gravitational waves: I always assumed that the definition of distance d=c*t for a beam of light traveling the distance also holds true in GTR. If that was the case, the effect of a gravitational wave or simply the presence of a mass affecting c and d in the same way would cancel out and the phase shift in both arms of the interferometer would not change.
So my question: How do I measure the distance to a remote mirror in GTR? Can I still point a laser at it and wait for the time it takes to see the reflection and calculate d=c*t/2 ?

Many thanks,
Wolfgang

 

[Ibix]

There's a degree of flexibility in how you define distance in a non-stationary spacetime like those containing gravitational waves. You may choose to keep the speed of light constant, in which case the gravitational wave varies the distance between the mirrors differently on the two interferometer arms. Or you may choose to keep the distance constant and allow the coordinate speed of light to vary differently on the two interferometer arms. It's a matter of coordinate choice which one you do - so generally the answer is that sometimes $d=ct$ and sometimes not.

The usual "stretch and squish" description of interferometers like LIGO uses the "changing distance" approach. It's easy to comprehend, but it makes assumptions about the state of motion and, to some extent, orientation of the interferometer that can be avoided with a more general analysis.

 

[Ibix]

Just to add, the underlying point here is that relativity is a theory of spacetime. Splitting spacetime into a series of slices you call "space at one time" can be done in a number of ways, it's a matter of choice which slicing ("foliation") you use, and distance through "space at one time" depends on that choice. Fundamentally, this is the origin of length contraction in special relativity. GR just has more complicated spacetime geometries, so there isn't always one obvious family of foliations to use and therefore there are more options for a reasonable choice and more complexity in the results of your choice.

 

[PeterDonis]

affecting c and d in the same way

But a gravitational wave doesn't affect c and d in the same way. That's why you can detect them with an interferometer. How exactly you want to describe the effect depends on how you choose coordinates: in the most common coordinates used to analyze gravitational waves, $c$ is constant and $d$ varies as the wave passes by (because the components of the spacetime metric transverse to the wave vary as the wave passes by).

 

[pervect]

I would say that the proper distance between the LIGO mirros changes with time when the gravitational wave passes. But, I'd say this is just one possible interpretation of the physics.

Probably the biggest issue is how familair you might be with the concept of a metric. Are you familiar with the SR metric $ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$? In this metric, the coordinate speed of light is globally always equal to c. In terms of invariants, we can say that ds, the Lorentz interval along any small segment of a light beam, is zero.

But a metric of this form cannot correctly model a gravitational wave space-time, because this metric describes a flat space-time, and the space-time of a gravitational wave is not flat.

Thus certain convetions of SR, such as the coordinate speed of light always being equal to a fixed number, are not followed in General relativity. Note that we do not regard the speed of light as "physically" changing, at least not in the modern interpretation. Any local observer with local clocks and rulers will measure the local speed of light to always be "c". When we try to cover large swaths of curved space-time with coordinates, we find we just aren't able to make the metric more simple.

One mathematical expression for the metric of a plane gravitational wave with the usual choices about coordinates (called the Transverse Traceless, or TT gauge) propagating in the z direction looks like this:

$$ds^2 = -c^2 dt^2 + (1+h_+(t) ) \, dx^2 + (1-h_+(t)) \, dy^2 + dz^2$$

This isn't the only possible expression for the metric of a plane gravitational wave. In fact, I'd really like to give the Fermi-normal version, because I find it easier to interpret Fermi-normal coordinates. I only have unreliable sources for the Fermi-normal metric, sadly. But I'll try to give a brief popular description of how the mathematical descriptoins can be interpreted.

In the above expression for the TT gauge, the mirrors are regarded as "not moving", but the space between them can be regarded as shrinking or expanding. This is actually one of the more common interpreations. I find it a bit spooky, I've never been fond of "expanding" or "contracting" space ideas, though they are widely used.

In the Fermi-normal interpreation, the mirros can be regarded as being set into motion by the passing gravitational wave, essentially "tidal forces" (the physical interpretation of the Riemann tensor) make the mirrors move. This is a bit less spooky, but the whole idea eventually starts to break down as the mirrors get furhter and further apart - it's only a local approximation.

To do: confirm that $R_0x0x$ and $R_0y0y$ as the only two nonzero trms in the Riemann, that they can be interpreted in the Fermi frame as "just tidal forces" that shake the test masses, and compare to
https://iopscience.iop.org/article/10.1088/2399-6528/ab9320/pdf