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It is often said that matter curves space (or rather spacetime) in general relativity.

But why should matter curve space one way and not the other way? So it seems like a metaphor, I guess.

I read somewhere that space (spacetime) in general relativity works like an indexing. The relation between two nearby measured points (events) can be realized in many different ways, by an infinite number of spacetime curvatures between them, if I understood correctly.

At the same time, a manifold must locally be like Minkowski spacetime.

So what is spacetime really like according to general relativity?

I find it hard to combine these thoughts to a clear idea.

Guess I'm just asking if we could get some improved metaphors - that is, in natural language some better descriptions - for layman on this subject.

Atlantis Vel
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You're confusing a lot of points:

1.Curvature is in GR not "external" but rather internal. We do not need an "embedding" into 5D space to make sense of GR. So the question on in which way it curves is ill defined.

  1. Einsteins field equations (although that's more math) gives a unique metric (the fundamental object in GR from which one can compute the curvature) for a given energy/matter source. (The energy-momentum tensor to be exact)

  2. Locally, it looks flat, i.e. Minkowski, not eulcidean.

Confuse-ray30
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  • Thx a lot :) Perhaps it should be obvious, be I do not understand why curvature being internal means space(time) does not curve one way or the other? – Atlantis Vel Jan 26 '24 at 13:55
  • You need to clarify what you mean by "one way or the other". I thought you meant whether it curves "outwards" or "inwards", which is only reasonable for external curvatures. The curvature tensor $R^\mu_{\nu\rho\sigma}$ can be calculated from the metric tensor. And this is uniquely given by the Einstein eqs. https://en.wikipedia.org/wiki/Riemann_curvature_tensor – Confuse-ray30 Jan 26 '24 at 14:18
  • What I am, naively, wondering, is what spacetime is like, how it could, perhaps, be better described, also in daily language - when this is uniquely given (as it must be for the theory to make sense). The somewhat standard metaphor of a ball or whatever curving some surface, does not seem very meaningful to me. – Atlantis Vel Jan 26 '24 at 14:57
  • I don't think daily language will ever be meanigful to describing a physical situation accurately that isn't newtonian mechanics. If you asked me what an electron is in daily language, I'd jokingly say a blue small ball. If you asked me with the same constraint what gravity is, then I would probably also state the ball metaphor. But to truly give a picture, maybe you could go into 1+1 dimension and actually visualize it geometrically. I don't think it gives you any deeper insight though. – Confuse-ray30 Jan 26 '24 at 17:50
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But why should matter curve space one way and not the other way?

The direction that spacetime is curved is determined by the sign of the energy density. If there were some exotic form of matter with a negative energy density then it would curve spacetime in the opposite direction than for ordinary matter with positive energy density.

At the same time, a manifold must locally be like Euclidean.

So what is spacetime really like according to general relativity?

In GR spacetime is modeled as a pseudo Riemannian manifold, with a metric that is determined by the matter content via the Einstein field equations.

The signature of the metric is (-+++), where - indicates measurements with clocks and + indicates measurements with rulers. At any location you would use one clock and three orthogonal rulers.

The path of an object in spacetime is called its worldline. An object with an accelerometer that reads 0 travels on a geodesic worldline, which is a “straight line”. This is the relativistic version of Newton’s first law.

Spacetime curvature represents tidal gravity. In the absence of gravity, if two nearby objects start off going at the same velocity and with 0 accelerometer readings, then the distance between them will not change. Geometrically this is the fact that in flat spacetime parallel straight lines always remain equidistant. In the presence of tidal gravity, such objects will not always remain equidistant. Geometrically this is straight lines that are parallel at one point but not parallel at other points. That is only possible in curved geometry.

So spacetime curvature is physically measurable. Curvature is tidal gravity. The “direction” of curvature around a large spherical object leads to stretching in the radial direction and compression in the tangential directions. If the object were composed of exotic matter then the opposite “direction” of curvature would imply compression in the radial direction and tension in the tangential directions.

Dale
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I think much of the problem is saying that spacetime "curves". While mathematically true, it brings to mind the "fabric" of spacetime as a sheet that can be bent in a higher dimensional space. It doesn't help that it is commonly explained by using the surface of the Earth as an example. Yes you can see the intrinsic curvature. But you can also see the surface is a sheet that is bent in the 3rd dimension.

It might be better to say that spacetime is distorted. Time runs slower on the surface of a neutron star than in orbit. The distance to the surface of a neutron star is farther than you would think from measuring the circumference from orbit.

The way to measure this distortion is to walk around a small square loop. Or equivalently, walk half of it in two different directions. For example, take a square whose sides are 1 km in the radial direction and 3.3 microseconds in the time direction. To walk one half, wait then drop. For the other half, drop then wait. Since time runs slower nearer the surface, you will arrive at two different events separated in time.

This separation shows the space is curved, but isn't the measurement. That requires parallel transport of a vector around the loop. .

mmesser314
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