Space Warps + Black-holes

I realized I had a misunderstanding about how space is warped in gravity. I am trying to gain intuition from the Schwarzschild solution for a blackhole. This can be used to see how space reacts far away from a kind of ideal massive object. I am using wikipedia as references:

https://en.wikipedia.org/wiki/Schwarzschild_metric
https://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution

The initial solution presented however is not isotropic in the coordinates, which makes it hard to understand how space is really behaving from a measurable point of view. If these equations don't make any sense to you I will try to explain so don't freak out. I will copy here for reference, but I will explain why this one is deceiving:

\[ds^2 = \left (1-\frac{2Gm}{c^2 r} \right)^{-1} dr^2 + r^2 \left ( d\theta^2 + sin^2(\theta)d\phi^2 \right) - \left (1-\frac{2Gm}{c^2 r} \right )c^2 dt^2 \]

The solution (or metric) in the second page for isotropic coordinates I found more revealing and I will copy here.

\[ds^2 = \left (1+\frac{Gm}{2c^2 r_1} \right)^4 \left (dr_1^2 + r_1^2 \left ( d\theta^2 + sin^2(\theta)d\phi^2 \right)  \right) - \frac{\left (1-\frac{Gm}{2c^2 r_1} \right )^2}{\left (1+\frac{Gm}{2c^2 r_1} \right )^2}c^2 dt^2 \]


The \(dt\) represents a tick of a reference clock, and \(dr\), \(rd\theta\), and \(r sin(\theta) d\phi\) represent reference lengths. The factors multiplying them represent a scaling factor of what the actual clock ticked, or length is, at that radius relative to the reference.

The factor on the time coordinate gives the time dilation.

Time dilation as a function of radius: \(\frac{Gm}{2c^2}=1\)

The factors are a function of radius, \(r\), but the radius does not mean the same thing that it means in flat space. The solution is defined from imagining space being built up from a series of concentric spherical shells. The coordinate \(r\) tells which shell you are sitting on by equating the surface area of the shell to \(4 \pi r^2 \). It does not, however, necessarily tell you how far from the center you are. And actually it doesn't even tell you what the actual surface area is, since you also have to look at the scaling factor for those dimensions.

In flat space, each shell has to be bigger than the one inside of it, and smaller than the one outside, by a fixed amount. This is the limit where \(m = 0\), and so the scaling factor is a constant. As m increases, the scaling factor becomes a function of radius.

Length expansion as function of radius: \(\frac{Gm}{2c^2}=1\) 

There is a limit to this solution at \(r = \frac{Gm}{2c^2}\). The metric at that radius is

\[ds^2 = 4^2 \left (dr_1^2 + r_1^2 \left ( d\theta^2 + sin^2(\theta)d\phi^2 \right) \right)\]

This is called the event horizon because the time component vanishes, which means nothing can ever cross this boundary from the point of view of someone outside. Now, from what I have read, other coordinate systems allow the solution to progress past the event horizon. It's not important right now whether this is physically real since I only care about events far away from this limit. I think what may be more important is to see that nothing too crazy is happening to the spatial coordinates here.

However, in the first solution it looks like the length factor in the radial coordinate blows up as the event horizon is approached. This is because of the choice of radial coordinate. The problem is that the surface area of each shell as measured by a distant observer starts to approach a constant value as the event horizon is approached. That is, each concentric shell has about the same surface area as the one just outside, and the one just inside. This means space is not flat.

The problem with this solution is that in order to get to a spherical shell of a smaller surface area, one must drop a much further distance toward the event horizon. And at the event horizon itself the areas become constant, which makes it look like the radial coordinate blows up. However, the distance to the event horizon is actually a finite distance through space.

Radial factor accounting for length expansion near horizon

The second solution scales the surface areas as one gets close to the event horizon which is what one would actually experience. We can see that the radial length factor is actually only 4x that of a reference length far away. But also every dimension is 4x bigger there, not just the radial lengths.

For things that are not blackholes, what this means is that essentially there is slightly more space inside and around a planet than one would expect from far away, in addition to time running slightly slower.

Gravitational Field Energy

I meant to talk about how I am thinking about the gravitational field energy. My first thought experiment is to imagine a single photon of sufficient energy (and frequency) is converted to matter. That matter falls into a gravitational well, adding to the mass that is already there. As it fell it also gained kinetic energy.

The total amount of energy gained by matter falling into a gravity well is the energy of assembly. But since it gained energy by assembling into one big mass, the energy of assembly for gravity should be negative.

But I realized that time-dilation and gravitational acceleration are proportional. Now imagine that instead of the matter colliding with the planet surface, it is converted back into a single photon and reflected back into space. The new photon starts with a frequency higher than the original photon, corresponding to the gain in kinetic energy. But as it travels back out of the potential well it is red-shifted back to the frequency of the original photon: total conservation of energy.

This means that the kinetic energy gained through gravity only has a relevant meaning within the gravitational well. From outside observers, there is no change of energy at all. Gains of kinetic energy through gravitational collapse are exactly matched by a time-dilation factor, which for observers away from the well, there is no net change of frequency (or energy) at all!

A second thought experiment: imagine two massive bodies well separated with total mass M. The two bodies then collapse gravitationally. In the gravity well energy is gained increasing causing the energy of the single new body to be higher than the original bodies. Since the new body is 2M in mass, it also experiences a higher time dilation. However, from outside observers the total energy of the system is still only 2mc^2. The extra thermal energy is folded into the total energy. If the thermal energy is radiated away (out of the gravity well), observers will actually see a total mass less than 2M, even though observers inside the well still see 2M worth of mass.

This now brings me to black holes. If a massive body simply collapses without any interruptions, and ignoring radiation taking away thermal energy, the total mass of a body will remain the same for outside observers even though the internal thermal energy increases. As the radius of the body approaches the Schwarzschild radius (although the definition of that radius seems a bit wonky), the time dilation goes to zero, and the internal energy approaches infinity. But because of the near infinite time-dilation, outside observers still see the same total mass, energy, and temperature no matter how much the body collapses under gravity.

So what does this mean for field energy? In a way, the energy is taken from the time dimension. Object gains energy falling, but then causes time-dilation which makes it appear as total energy of the matter is unchanged. If the total energy is unchanged, then there is no need to invoke the idea of a field energy.

What is really wonky is the perspective of the rest of the universe from inside a gravity well.

General Relativity and Quantum Mechanics

Over the past few days I have made some realizations about how to think about gravity, and what it is exactly. General relativity describes the effect of gravity as objects simply following their natural trajectories of shortest path, but that space and time itself are warped such that the shortest path is actually a curved line, and so we see acceleration due to gravity.

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]