How can the volume of a star be determined?

[Rick16]

TL;DR Summary

How can the volume of a star be determined?

Until very recently I thought that according to general relativity masses curve spacetime as a whole, i.e. the complete abstract 4-dimensional spacetime. It came as a complete surprise to me to learn that you can take away the time-component and are left with a curved 3-dimensional space, which is a much more tangible concept than curved spacetime.

In section 5.6 of his book "Covariant Physics" Moataz H. Emam uses the spatial component of the Schwarzschild metric – i.e. the variant of the metric for the inside of a spherical mass – to calculate the volume of a star, and the result is significantly larger than the value that you get using the Euclidean formula for the volume of a sphere.

This means that GR is not just an abstract model, but that it actually describes the reality of space: each mass shapes the space that it occupies. I just wonder how we can be sure that this is correct. How can we determine the volume of a star through measurements without assuming a specific geometry of the space that the star occupies?

 

[Dale]

TL;DR Summary: How can the volume of a star be determined?

Until very recently I thought that according to general relativity masses curve spacetime as a whole, i.e. the complete abstract 4-dimensional spacetime. It came as a complete surprise to me to learn that you can take away the time-component and are left with a curved 3-dimensional space, which is a much more tangible concept than curved spacetime.

This paragraph doesn’t make a lot of sense to me. Mass (the stress energy tensor) does curve spacetime. I wouldn’t characterize spacetime geometry as any more or less abstract or tangible than spatial geometry.

You can choose to look at the geometry of a 3D slice of 4D spacetime just like you can choose to look at the geometry of a 2D slice of a 3D space. I guess maybe that is what you mean by taking away time.

 

[Ibix]

Until very recently I thought that according to general relativity masses curve spacetime as a whole, i.e. the complete abstract 4-dimensional spacetime.

They do.

It came as a complete surprise to me to learn that you can take away the time-component and are left with a curved 3-dimensional space,

It's always possible to "slice" 4d spacetime into a "stack" of 3d spatial slices. It's only really meaningful to do so in a restricted class of spacetimes called "stationary", where there is timelike Killing vector field, which provides a notion of a space that doesn't change with time. If you model a star as an eternal unchanging mass then you get such a spacetime, and it's a decent enough approximation.

This means that GR is not just an abstract model,

Well, not really.

I just wonder how we can be sure that this is correct. How can we determine the volume of a star through measurements without assuming a specific geometry of the space that the star occupies?

Eddington's measurements of gravitational lensing (and more recent more precise measurements) wouldn't match the model if you left out the curvature of the spacelike planes. In fact, I believe Einstein's predictions with and without spatial curvature were the hypotheses that Eddington tested. That's a measure outside a star, but it shows what you were asking about.

 

[Nugatory]

TL;DR Summary: How can the volume of a star be determined?

I just wonder how we can be sure that this is correct. How can we determine the volume of a star through measurements without assuming a specific geometry of the space that the star occupies?

We are making that assumption. We don't know by measurement or direct observation what the geometry is; instead we assume that it is what we have calculated from the Einstein field equations. The justification for this assumption is that the field equations have gotten the geometry right everywhere that we have been able to test their predictions, there is no evidence that suggests that the field equations should not apply everywhere, and no plausible alternative theory that suggests there might be a different geometry.

We do something similar when (for example) we consider physics deep inside Jupiter's atmosphere: we assume that the laws of physics work the same way there as here. Sure, there might be some completely different physics going in the deep inside Jupiter such that $F=ma$ doesn't work or hydrogen atoms have different rest masses... but it would be absurd to approach the problem from that point of view.

So you can consider Emam's analysis, like just about everything in modern science, to come with an implicit qualifier: "Unless our current theories are wrong..."

 

[PeterDonis]

How can we determine the volume of a star through measurements without assuming a specific geometry of the space that the star occupies?

In principle you could measure the physical diameter of the star with rulers, and the physical surface areas of successive "shells" of the star corresponding to each marking on the rulers you are using to measure the diameter, and compute the appropriate integral. That would be physically mirroring the calculation GR tells you to do to obtain the volume.

In practice, of course, nobody does this because (a) we can't get inside the star to make direct measurements, and (b) our confidence in the GR model is high enough that we accept indirect calculations as sufficient.

 

[Vanadium 50]

Is it worth pointing out that the difference between Newton and Einstein is much, much smaller than the uncertainty on where a star's surface actually is?

 

[Rick16]

We are making that assumption. We don't know by measurement or direct observation what the geometry is; instead we assume that it is what we have calculated from the Einstein field equations. The justification for this assumption is that the field equations have gotten the geometry right everywhere that we have been able to test their predictions, there is no evidence that suggests that the field equations should not apply everywhere, and no plausible alternative theory that suggests there might be a different geometry.

This is what I had suspected. The other answers also contain interesting elements, but this is basically what I wanted to know.

 

[pervect]

TL;DR Summary: How can the volume of a star be determined?

Until very recently I thought that according to general relativity masses curve spacetime as a whole,

It's unclear to me why you now apparently think it doesn't.

This means that GR is not just an abstract model, but that it actually describes the reality of space: each mass shapes the space that it occupies. I just wonder how we can be sure that this is correct. How can we determine the volume of a star through measurements without assuming a specific geometry of the space that the star occupies?

If I'm reading this right, you think space is "real" and space-time is abstract. I don't share this position. In fact, I would say space-time is "real" in the sense that it is independent of the observer. Decomposing space-time, which exists independent of the observer, into space and time, requires one to specify something more, an "observer" if you will.

What "an observer" in General Relativity is is actually a bit unclear, there are some different techniques for splitting space-time into separate entities space and time. The situation is easier in special relativity, (henceforth SR), and the issues are similar, so let's consider the SR case for now.

In SR, space can be defined as the set of events at some instant of time, which we will call "now". But different observers, depending on their velocities, have different notions of "now", a different set of events which comprise "now". Thus different observers have different notions of space, due to the relativity of simultaneity.

 

[pervect]

Something I wanted to add, as I hit return prematurely. The volume of a star, to my way of thinking, is an observer dependent quantity However, the space-time of a star is generally considered to be static (or stationary if the star rotates), which defines a particular observer, the stationary (or static) observer. Using this choice of observer, we can meaningfully talk about the volume of the star.

It is still true that to different observer, for instance space-ship moving at a relativistic velocity relative to the star, that it's volume would be different. In the case of the space-ship, "length contraction" would tend to squash the sphere, in the ultra-relativistic case it would be a disk.

 

[Rick16]

It's unclear to me why you now apparently think it doesn't.

I don’t think that it doesn’t, I still think that it does. I just thought that the effect only exists when you consider space-time in its 4-dimensional entirety. It did not occur to me that you could consider space on its own. Space may not be more real than space-time, but I can visualize space, I cannot visualize space-time. That’s why this perspective – considering space on its own – helps me to get a somewhat more intuitive idea of what is going on.

 

[PeroK]

I don’t think that it doesn’t, I still think that it does. I just thought that the effect only exists when you consider space-time in its 4-dimensional entirety. It did not occur to me that you could consider space on its own. Space may not be more real than space-time, but I can visualize space, I cannot visualize space-time. That’s why this perspective – considering space on its own – helps me to get a somewhat more intuitive idea of what is going on.

The Schwarzschild geometry is theoretically static, which means that you can talk about a time-independent spatial geometry. This geometry is spherically symmetric, described by a radial parameter $r$, but is non-Euclidean. It's almost impossible to measure the geometry around the Moon, the Earth or the Sun. Instead, you infer other measurements that depend on the overall geometry - like redshift. Newton's law of gravity could not be directly tested either (in fact, the Newtonian gravitational forces are not even measurable). Instead, it was the prediction of elliptical orbits and the discovery of Neptune that corroborated the theory - indirect measurements inferred from or implied by the theory.

When I was learning GR I thought about how you could experimentally measure the Schwarzschild radius, and it's not at all easy. How do you measure the area of a sphere a fixed distance from the centre of the Earth? To test whether it's Euclidean or not. You'd need a lot of satellites, perhaps!