However, this does not seem to immediately give us any intuition about what this represents in real life. There are several erroneous visualizations about this warped space-time that have caused me great confusion when trying to understand this concept.
The most simple false visual is the stretched piece of rubber with something heavy sitting on it. The rubber bends down. Since it represents space-time we see what warping of space-time might look like. And also if we place other objects on the rubber sheet they even appear to be pulled toward each other.
A more complicated but more physically satisfying false visual is that of space itself 'falling' in a gravitational field. Then from a relativistic point of view it makes sense that anything on the falling space would fall as well, while also following their own shortest path within that patch of space.
The reason both of these visualizations are incorrect is because neither has any physical meaning. Nothing about the two scenarios could be tested. The rubber sheet only works because gravity is already around to cause the rubber to warp and balls fall to each other. It doesn't add any explanatory power as to why the balls should move at all.
While falling space does seem to add explanatory power, or an intuition pump at least, it doesn't predict anything that can actually be measured. There is no way to detect movement of space. Mathematically it would also introduce an arbitrary preferred frame of reference; that space has a particular configuration, and that it can change and accelerate etc just like matter. None of that is a part of GR.
So, what exactly CAN we measure? There are classically only two things we have. The stick to measure length, and the clock to measure time. This is how Einstein would have visualized things. We just fill up space with little sticks and clocks, which take the place of a coordinate system.
The simplest case is the elevator thought experiment. The equivalence principle holds that an accelerating frame is indistinguishable from a gravitational field. This means that if we imagine being closed in an elevator, we can not tell if we are on Earth under the influence of gravity, or in deep space and being accelerated by the cable being pulling in some direction. Inside the elevator the two situations are indistinguishable for the purposes of GR.
This also means that any warping of space-time that is measurable must be exactly the same inside the elevator in the two cases as well. It doesn't matter what is causing space-time to appear warped; it looks the same either way.
Now, we have to make a control. If the elevator is in deep space, and not being accelerated, then we can study how space-time looks inside. Then put it under acceleration and see how space-time looks. In both cases all of the sticks appear to be pretty much unchanged. We still measure the height of the elevator cabin to be the same, as well as the width and the depth. The only thing that has changed is that the clocks at the roof of the cabin tick slightly faster than the clocks at the floor when it's being accelerated.
This means that the effect of gravity is due solely, and completely, to the fact that time moves at a faster rate higher in the cabin, and slower lower in the cabin. Ok, you might ask, but how does this explain why things fall? How does this add explanatory power?
The fact that time moves slower as you go deeper in a gravity well is not just a side-effect of gravity, but the primary cause of acceleration. I do not want it to see like space is not warped as well, because it is. However, if you are to drop a ball from a stand-still on the surface of Earth, there is no effect from the warping of space that can cause it to start moving. Warped space can only affect things that are already moving through space. Warping of space can change a trajectory, like that of light, but it can't cause acceleration without the object already possessing a velocity.
That is, warped space causes velocity dependent forces on objects. Warped time is what causes acceleration from nothing. Mathematically in GR, we fall because shortest path goes through slower time, kind of like how light is bent in a lens because it goes slower in the material. But it is hard for us to actually visualize this path in a space-time diagram.
For me the epiphany came when I coupled this with quantum mechanics. The probability of finding a particle at a particular position is inferred from the evolution of what is called its wave function. From a certain point of view, the wave function evolves in time over space just like a wave does. We normally don't use this interpretation for predictions because we can't actually measure this evolution; we can only place detectors. But, ignoring that for a second, the evolution of the wave function is essentially a local function of time, which can make the probability become higher or lower in different locations over time.
In flat space-time, all the clocks everywhere tick at the same rate, and so the wave function evolves as we expect without a gravitational field. But, with time progressing faster in one direction, the part of the wave function in that direction also evolves faster. The effect is that the relative phases along that direction begin to change. This phase-shift then causes the wave function to move downward; gradients in phase are equivalent to momentum.
This can all be easily seen from simulating the Schrodinger wave equation by simply adding a scaling factor to the time evolution which depends on position.
The following simulation starts with 1D particle in box. Gravity is implemented by causing time to progress faster towards the right, which represents a gravitational acceleration to the left.
http://kcdodd.github.io/qmgrav/
When gravity is turned on, it beings to accelerate left. Because the box limits its motion, it bounces back up due to a quantum 'tension' against the two walls. I can 'push' the particle by turning gravity on when the particle is more at the right side, and off when it is more at at the left side, increasing the total energy of the particle. Otherwise the energy is conserved. The actual time-rate difference between the two sides is hard to notice, even though it maxes at double speed at the right edge.
So, in quantum mechanics a gradient in the rate of time produces something like a force. Since we can all be described by wave functions, the reason we accelerate downward at the surface of Earth is because the rate of time is slower at our feet than at our heads.
Since the factor is on the time (not the mass, or potential, etc), every wave will experience the same acceleration, because every time factor would be the same. This is why all things fall at the same rate regardless of the mass: the mass isn't what is causing the acceleration at all. A light wave traveling upward will become red shifted also due to time running faster higher up, which relates to a change in energy. In fact, the frequency of all waves are lower at higher points, and higher at the lower points in the gravity well, which relates directly to changes in energy.
Frequency and energy are equivalent in quantum mechanics. This is what gives rise to the 'gravitational potential' energy being converted to 'kinetic' energy. The potential was created by the fact that time is running slower deeper in the well. When objects fall, they increase kinetic energy, which has a certain frequency, and the further it falls the higher that frequency is, directly proportional to the blue shifting caused by time-rate changes. I will want to revisit the issue of what the gravitational field energy represents, which by the way must be negative to account for the increase in energy of things falling into the gravity well.
For me this is a fairly complete picture of why things fall in warped space-time. But it doesn't answer why matter warps space to begin with. The space elevator thought experiment explains why warped space causes acceleration, but remember that doesn't depend on a gravitating mass; that is any accelerating frame.
The effect of matter on space-time is described by the Einstein field equations. But, like most equations, they don't provide much of an intuition pump. It basically says curvature of space-time is constrained, and energy and stress can alter those constraints.
This is very difficult because even with no matter or energy, it does not mean space-time is flat! It is not that rigid. This is where the idea of a rubber sheet might be helpful; in visualizing how space-time reacts in the absence of any matter. It is completely determined by the boundary conditions. If the boundary is flat (the hoop holding the rubber), and there is no matter anywhere of course, then it will be flat. If the hoop were deformed, the rubber would not be flat anymore, but would find a kind of smooth transition between the edges of the boundary.
This is like how the universe is. We usually assume a boundary out at infinity that is flat, which makes space-time flat when there is no matter in the universe. But that does not mean space-time is flat everywhere there is no matter. And it doesn't even mean this is how the universe is shaped, when it probably is not.
When matter is introduced, it interrupts the nature of space-time. Suppose the introduction of a ball specified that time progressed at half the rate at the surface of the ball, than at the boundary at infinity. Well, the space in between the surface of the ball and infinity will warp to transition between the two values. Close to the ball the time rate is x0.5, and further away it is x0.75, and further it is x0.99, etc. and at infinity it is back to x1. (the surface of the ball can also specify 'rates' for the 3 spacial dimensions as well which would cause space warps in addition to time warps).
This gradient in time around the ball then causes things to accelerate toward the ball. Suppose there are two balls, they will accelerate toward each other. A problem with this model is that the rates at the surface of each ball is fixed to x0.5 time rate, when really it should be like x0.25 since there is twice the mass now. Each ball is looking at the boundary and saying x0.5 that, while ignoring where the other ball is. That is because I'm treating them as boundaries, instead of sources.
As a source it doesn't specify a fixed rate of time, but specifies how much the rate of time is decreased relatively, which then propagate out according to the field equations. Simply put, for some reason the presence of matter and energy causes time to slow down.
The effects of matter on space are a little more complicated. [edit: I need to deal with this separately because I missed some things]
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