In the Big Crunch Theory it says that gravity (curvature in space-time) will stop the universe's expansion and gravity will cause the universe to contract on itself. My question is if gravity is a curvature in space into which other objects fall into, how does that cause space to contract on itself?
[Physics] Does gravity actually contract space-time
general-relativitygravityspacetimeuniverse
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Imagine you live in a universe governed by extremely simple rules, like Conway's Game of Life, for example. Once you discovered those rules, you might wonder, "Why do cells come alive if they have three living neighbors? Why do they die if they have one? How does that work?" (By "how" here I am referring to "what underlying mechanism makes it work?", which is my interpretation of "how" in the original question.)
In a simulation of the Game of Life that you run on your computer, there is a good answer to this. You can examine the source code, look at the hardware of the computer, and eventually arrive at a complete description of exactly what goes on such that little squares on your computer monitor light up and go off according to the rules.
But we're imagining that these rules are just how the universe works. In that case, there may be no reason at all. Maybe it just does it, full stop.
As humans, though, I think we might find that very hard to accept. There are many cellular automata very similar to the Game of Life, but their behavior is not nearly so rich. Why did we get the one universe with the interesting laws? And how does the universe know to implement those laws without screwing up? Surely there must be some wheels and gears beneath there!
That sort of curiosity is extremely important for physicists, and it has led to a lot of new understanding. Peter Shor pointed out in the comments that wondering about how quantum mechanics works led to quantum information and computation. Famously, Einstein wondered about how electromagnetism worked, leading to understanding relativity. Frequently, a theory of physics doesn't quite feel right to us. That drives curiosity. We demand an answer, and eventually it leads to breakthroughs and new theories.
Physicists have derived great benefit from this approach of taking the pieces that don't feel right or don't feel well-enough explained and using that as a springboard to go deeper, but sometimes it also leads to complete frustration. It turns out that the universe isn't obliged to be the way we want.
If you lived in the Game of Life universe, once you figured out the rule it was following, you could keep asking forever, "Why does it have that rule? How does it implement it?" without getting anywhere. The rule itself is just a short little description. It just says that there's a grid of cells and that they light up and turn off according to a simple pattern, and that's all it says. If there was nothing deeper going on than that, oh well. We wouldn't have to give up trying to find a deeper explanation, but we aren't owed one.
My argument is that real laws of physics are the same. So in General Relativity, we posit that the Einstein equation is true. The theory of General Relativity itself makes no comment on this, just as the theory of the Game of Life makes no comment on why cells with three living neighbors come alive.
So when you ask, "How does something about the mass energy tensor alter geodesics or 4-velocity vectors? I see no explanation of gravity in GR merely a more detailed description of the motions it effects," you are right. GR doesn't say how it does it.
It could be that there's an explanation, but it doesn't seem likely to me that the fundamental problem will go away. For example, suppose someone tells you that gravitation works by sending particles called gravitons around, and gives a detailed description of the theory of gravitons. Couldn't we then ask the same question? How do the gravitons interact with spacetime? We could describe the precise mathematical rules, but fundamentally, this anthropocentric feeling of dissatisfaction would remain. Why those rules for gravitons? If they're derived from some set of appealing principles, why those principles?
Elsewhere in physics, how do wavefunctions know to obey the Schrodinger equation? What forces them to obey that equation rather than doing something else? Nothing. They just do that. It's purely a description of how the wavefunctions behave. The problem is the same, as far as I can see. (You can recast QM in some new formulation, but I don't think this averts the "problem".)
To answer your question as best I understand it, you are right that GR is just a description, nothing more. That may not always be true for GR in particular, but it seems likely to me it will always be true for something. (I can't say for sure, of course, since I don't know what the "something" will be!) It is the nature of theories of physics to be just descriptions. We don't have to accept that as a final word, and our desire to understand more deeply fuels our greatest communal quest for knowledge, but ultimately the universe will do what it will do, and can't be bullied into explaining itself just because things don't feel mechanistic enough for us.
note: This answer is completely rewritten after reading the helpful comments from Qmechanic, Peter Shor, and dmckee. Thank you for your input. This answer is essentially philosophical, so disagreement on it is inevitable, and it represents only my personal opinion.
The answer by @peterh is accurate on the factual information about the Einstein Field Equations and that it describes how the matter distribution affects spacetime. There is more that may be added that hopefully will help understand more of it.
First, just to be totally clear, gravity as described by GR (general relativity, through Einsteins Field Equations) is due to the curvature of spacetime, which is caused by any kind of matter energy. Thus, you can say matter-energy causes gravity, which is the curvature of spacetime. As the spacetime curves matter then follows the curves in spacetime that are the shortest path between 2 points, called the geodesics. Yes, one creates gravity and spacetime, which affects everything in it. It is a set of very nonlinear equations. Thus, in this geometrical description of gravity, there is no force. We still call it the effect the gravity, or the gravitational field effect
Still, it turns out that the geometric description, when the gravitational field is not too strong, can be described as a force and Newton's equations and description of gravity are a very good approximation in those cases. The gravitational force of the earth can for the most part be described that way, and all orbits computed accurately enough that way. The same is true for the sun. There are some minor effects that Newtons equations cannot describe in those cases: 1) there is a small time dilation effect, where time is just a tiny bit slower on the surface of the earth than it is where the GPS satellites keep track of time, and those are then slightly adjusted. 2) the orbits of the planets have their perihelion shift just slightly, and it's been observed. GR describes those perfectly.
Your question of why the critical density affects whether the uNiverse keep expandinged forever (an open universe), collapses back (closed universe, Big Crunch), or just barely keeps going (flat) is actually a little more complex. Remember the Einstein Field Equations. They can be written as
Einstein Tensor = k X [Energy-momentum tensor + dark energy term]
(The dark energy term was originally on the left side, as a cosmological constant term. Different words for the same thing)
When one solves this set of equations (there are 10 independent equations, the different components of the Tensors), for homogeneous isotropic spacetimes (a very limited set, called Robertson Walker solutions, really 3, with positive, negative or zero SPATIAL curvature), you get the Friedman equations. Those relate the density of matter energy to the curvature. The critical density is simply that density which makes the SPATIAL curvature zero, or flat, a so called flat universe (words again, it really is only the spatial part that has no curvature, there is an expansion which makes the spacetime curved).
So if the density is equal to the critical density, spacetime is so called flat. It has been measured and estimated to be so within about 2%. And yes about 70% of that density is the dark energy, and about 25% dark matter. Both dark matter and dark energy are mysterious, but there is evidence of their existence.
For dark matter it is well determined that they are around and inside galaxies, and help keep them as such. Galaxies rotate too fast to not have their stars fly away due to the centrifugal effect, and that has been used to determine the density of dark matter around and in galaxies. It is thought they are massive particles that are remnants of the Big Bang because they interact very weakly (no strong nuclear nor electromagnetic interactions, only the weak nuclear and gravity) with themselves and other matter. The specific particles have not yet been directly detected, so there can always be some surprises.
Dark energy is even more mysterious. Try some of the answers on this site about it or Wikipedia for a quick summary. We don't know what it is, but there is also evidence that it exists. Galaxies further away from us are expanding faster, accelerating, and the numerical observations are consistent with a constant dark energy density at about 70% of critical. When and if we find out what it really is there can also be surprises. See also https://en.m.wikipedia.org/wiki/Dark_energy
In both cases, the most accurate measurements are due to the cosmological microwave background. See https://en.m.wikipedia.org/wiki/Cosmic_microwave_background. It predicts the cosmological parameters very accurately, but still some uncertainties
So, yes, energy and matter density affects spacetime, and for the universe its expansion. If the total energy density is critical it is a flat universe.
Best Answer
You've undoubtably seen the rubber sheet analogy for spacetime curvature, and I'd guess you're thinking that things fall into the dimples on the sheet. This is certainly true and is an analogy for how gravity works between astronomical bodies like stars.
However the rubber sheet as a whole can expand and contract, and this is an analogy for how spacetime as a whole expands and contracts. For a closed universe you have to imagine the rubber sheet expanding at early times as the universe expands, reaching a maximum stretch, then shrinking again at later times as the universe contracts.
The usual caveats apply: be cautious about taking the rubber sheet analogy too literally. Googling will find you many articles describing the deficiencies of the rubber sheet analogy e.g. this one. Also note that in the contraction phase we are not talking about a finite sheet contracting to a point. The sheet is infinite at all times - the contraction to a singularity means the spacing between any two randomly chosen points on the sheet goes to zero at the Big Crunch.