What does the bending of "The Fabric of Space Time" really look like?

Question

My question is in relation to the graphical depiction of the distortion of the “Fabric of Space-time” by a massive object in space. Often showing the lines as a single layer that intersect at 90° bending (concave) underneath an object in space must be false since that fabric is continuous in all directions. What it means to me is that if the fabric was shown above the object it should create a dimple not a concave disturbance. Can you please explain what is actually thought to be a occurring but not shown?

Here is an excerpt from the top-voted answer for that question: “The rubber sheet analogy, and all pictures that go with it have mislead many people (including me) and so now I must request that you completely and utterly banish that thought from your head. No sheet.” — G. Smith, Dec 11 '20 at 21:05
There is no fabric in General Relativity. There is only the 4-dimensional geometry of spacetime. Pictures like this are just trying to convey that that geometry is non-Euclidean/non-Minkowskian. — G. Smith, Dec 11 '20 at 21:15
@G.Smith, Re, "4-dimensional geometry..." That doesn't sound right. I haven't studied GR, but I know that the spacetime manifold is four dimensional. When a manifold is curved, don't we need at least one more dimension to define a space in which that curvature can be described? (e.g., Surface of the Earth is a 2-D manifold, but we need 3 dimensions to describe its spherical shape.) — Solomon Slow, Dec 11 '20 at 21:31
@SolomonSlow No. Riemann showed that you don’t have to embed a curved manifold in a higher-dimensional flat space in order to describe its curvature. (Actually, Gauss may have understood this earlier. I don’t remember.) Riemannian curvature is the intrinsic curvature of an $n$-dimensional manifold, not its extrinsic curvature. You don’t need 3D to understand the curvature of 2D spheres, and you don’t need 5D to understand the curvature of 4D spacetime. — G. Smith, Dec 11 '20 at 21:40

pela · Answer 1 · 2020-12-12T13:48:05.627

You cannot see space, and hence there is no way to show how bent space itself looks.

In fact, the term "bent space" is perhaps misleading: It is a way for us to describe what happens to space near massive objects, namely that its metric deviates from Euclidian$^\dagger$ geometry.

What you can see is light. Light travels from the object emitting it, in straight lines away from it. If a photon comes near a massive object, that photon will still travel in a straight line through space, but since the metric is no longer Euclidian, that straight line no longer looks like a straight line to a distant observer. But a (sufficiently) local observer would still see the photon travel in a straight line.

The rubber sheet

The rubber sheet analogy is a way to visualize how geometry changes from Euclidian to non-Euclidian, but it's important to remember that it's an analogy, and that it has its limitations.

For instance, the sheet is depressed into a third dimension, whereas in reality, space is not bent into a fourth dimension (at least, it doesn't have to be). And in pictures like the one you post, you see straight "coordinate system" lines following the depression, but continuing in the same direction as they entered.

So, What does the bending of "The Fabric of Space Time" really look like?

While I don't know the best way to visualize bent space, a quite good way would be to consider an array of distant light sources behind a massive object. If there were no massive object between you and the distant sources, you would see them where they actually are. But when their light travels past the massive object, it reaches you from a slightly different angle. That is, you see the background objects slightly different directions than their actual directions. If the shiny objects are extended, you may even see them being distorted, as the light from different parts of the shiny objects travel through regions that are bent slightly different.

Whoa, I just described gravitational lensing! The Universe has performed this exact visualization for you. Behold, the impressive view of distant (bluish) galaxies being gravitationally lensed and distorted by a foreground cluster of (orangish) galaxies:

^{The galaxy cluster Abell 370. Credit: NASA, ESA, and J. Lotz and the HFF Team (STScI).}

Many of the background galaxies are even multiply lensed, meaning that they appear at different positions in the sky. Abell 370 lies at a distance of 4.9 billion lightyears (Glyr). The background galaxies lie at typical distances of 5–10 Glyr, but the most distant are some 25 Glyr away (Bezecourt et al. 1999).

$^\dagger$_{Euclidian geometry is the geometry you're used to, i.e. parallel lines stay parallel, the angles of a triangle sum up 180º, etc.}

score 0 · Answer 2 · answered Dec 11 '20 at 21:41

Here is a slightly more easily-imagined analogy for visualizing what it means for 3-dimensionsal space to be "curved".

Think of the "fabric of space-time" not as a rubber sheet but instead as foam rubber that fills up space in 3-D. We embed a massive object into that rubbery volume and determine that in the vicinity of the massive object, the squishy space is bunched together slightly, and the closer you get to the object or the more massive it is, the more bunched together the space becomes.

Now we shoot a bullet on a grazing trajectory near the massive object, through the rubbery space surrounding it. The bullet moves more slowly through the bunched-up space than through space that is not bunched-up, and this causes the path of the bullet to get bent inwards as it passes close to the massive object.

That bullet behaves like a photon of light would when passing close to a massive object embedded in "curved spacetime".

What this leaves out is that the temporal component of the spacetime metric ($g_{tt}$), not the spatial components, is responsible for Newtonian gravity. (Its deviation from 1 or -1 is proportional to the Newtonian gravitational potential in the weak-field limit.) Envisioning squished space doesn’t really convey the proper physics, although I suppose it is better than envisioning a rubber sheet. — G. Smith, Dec 11 '20 at 21:50
@G.Smith, I left all that out on purpose, since the OP did not seem like he was ready for that, but please post your own answer. -NN — niels nielsen, Dec 12 '20 at 06:51

What does the bending of "The Fabric of Space Time" really look like?

2 Answers2

The rubber sheet

So, What does the bending of "The Fabric of Space Time" really look like?