How does GR explain increase/decrease in speed?

[Jakaha]

OK, so my basic understanding is that GR is all about geometry of space-time. It's all geometry, no other mechanism.

This explains why objects change direction due to gravity. But why does the speed increase?

How does pure geometry cause a change in speed?
Also, where does this kinetic energy come from?

I know people will talk about potential energy, but that's just a hand-waving mathematical trick to balance equations. What does geometry have to do with potential energy?

 

[SlowThinker]

OK, so my basic understanding is that GR is all about geometry of space-time. It's all geometry, no other mechanism.

This explains why objects change direction due to gravity. But why does the speed increase?

The speed of a falling object does not increase. It is you, the observer, accelerating in the other direction, as the surface of Eaarth pushes your feet.
The geometry allows you to stay at the same distance from the center of the Earth, while accelerating.

 

[Jakaha]

The speed of a falling object does not increase. It is you, the observer, accelerating in the other direction, as the surface of Eaarth pushes your feet.
The geometry allows you to stay at the same distance from the center of the Earth, while accelerating.

Forget the observer.
Are you saying that a falling ball bearing does not increase speed by 9.8 ms-2?
Sure, we can say that the Earth is increasing its speed towards the ball bearing, but the problem remains.

Why does geometry cause a change in speed and a change in kinetic energy?

 

[SlowThinker]

Forget the observer.
Are you saying that a falling ball bearing does not increase speed by 9.8 ms-2?
Sure, we can say that the Earth is increasing its speed towards the ball bearing, but the problem remains.

Why does geometry cause a change in speed and a change in kinetic energy?

Kinetic energy is not something absolute. It is always taken relative to some observer. So I can't "forget the observer".

Geometry does not change speed of objects: acceleration does that. The surface of Earth is accelerating, as viewed from an inertial frame.
As viewed from the surface of Earth, an inertial ball bearing appears to accelerate in the other direction.

 

[PeterDonis]

This explains why objects change direction due to gravity. But why does the speed increase?

"Speed" is frame-dependent; so is "direction". These are not good things to focus on if you want to understand how GR models gravity as spacetime curvature.

Why does geometry cause a change in speed and a change in kinetic energy?

Kinetic energy is also frame-dependent; as SlowThinker points out, the kinetic energy of a falling rock does not change in the inertial frame in which the rock is at rest.

I know people will talk about potential energy, but that's just a hand-waving mathematical trick to balance equations. What does geometry have to do with potential energy?

Potential energy is not exactly frame-dependent, but it is dependent on choosing a zero point, and it is only well-defined under certain conditions. But the spacetime around a massive gravitating body like the Earth is one of those conditions, so it's not correct to just discount potential energy as a "hand-waving mathematical trick"; in the conditions where it's well-defined, it is telling you something physically significant. If we choose the zero point to be at infinity, which is the usual convention, then the potential energy at any finite distance from the gravitating body is negative, and it is telling you how much energy it would take to move an object at rest at that distance out to infinity.

However, that's still not really a geometric viewpoint. Geometrically, the thing to focus on is not "gravity" in general, but specifically tidal gravity. Tidal gravity is what shows the presence of spacetime curvature (in fact, they are really the same thing). For an example of tidal gravity, consider two rocks, which at some instant are both at rest above the Earth at slightly different altitudes. Relative to the Earth, they will both start free-falling downwards, but at slightly different rates, so the separation between them will increase.

Now shift viewpoints and imagine you are falling along with one of the rocks, and keeping track of your separation to the other rock. You are in free fall, so you feel no acceleration (and you can use an accelerometer to verify that you are experiencing no acceleration)--as far as you can tell, you are at rest in an inertial frame. You can also verify that the other rock is in free fall (you can attach another accelerometer to it to verify this). But the other rock started out at rest relative to you; yet over time, it moves away from you, as viewed from the "inertial" frame in which you are at rest. That is not possible in a "true" inertial frame like the ones we have in Special Relativity; in such an inertial frame, freely falling objects can never change their speed relative to one another, so if they start out at rest relative to each other, they will always be at rest relative to each other. That isn't happening with the two rocks, so clearly SR is not exactly correct in this case.

What aspect of SR fails in this case? Geometrically, the worldlines of freely falling objects can be viewed as "grid lines" (the technical term is "geodesics") that mark out the geometry of the spacetime. In SR, this geometry is flat: grid lines that start out parallel (i.e., the objects are at rest relative to each other) always stay parallel. But in the presence of gravity, that no longer happens; grid lines that start out parallel (like the worldlines of the two rocks) don't stay parallel, because of tidal gravity. Initially parallel grid lines not staying parallel is the definition of curvature, geometrically speaking; so tidal gravity is spacetime curvature.

Finally, what about the "change in speed" of a falling rock relative to the Earth? As SlowThinker has pointed out, if we just look at one rock, we can account for its change in speed relative to the Earth by observing that the Earth's surface itself is accelerated; if you are standing on the Earth's surface, you feel acceleration (weight), so you are being pushed upward relative to the rock, which is in free fall. (Once again, you can attach accelerometers to you and the rock to verify that you are the one who is accelerated.) Where spacetime curvature comes in is when you look at rocks falling in different places, and realize that they fall at different rates (depending on height) and in different directions (depending on location on the Earth's surface--rocks in Australia fall in a different direction from rocks in Europe). Spacetime curvature is what makes all those freely falling rocks fall in different directions, and what allows different pieces of the Earth's surface to be accelerated in different directions while the Earth as a whole stays the same size and shape.

 

[Jakaha]

Kinetic energy is not something absolute. It is always taken relative to some observer. So I can't "forget the observer".

There is no third observer. It is a two body problem.
Think of a universe with only two particles P1 and P2.
Take P1's frame of reference. It sees P2 moving towards it and P2's radial speed and KE is increasing.
Where is P2 getting all this extra KE?

Geometry does not change speed of objects: acceleration does that.

Your second phrase is a tautology.
GR purports to explain gravity as a purely geometrical effect, so the question arises?
From P1's frame of reference, where is P2 getting its KE?

 

[SlowThinker]

There is no third observer. It is a two body problem.
Think of a universe with only two particles P1 and P2.
Take P1's frame of reference. It sees P2 moving towards it and P2's radial speed and KE is increasing.
Where is P2 getting all this extra KE?

No energy is created but part of P2's rest energy looks like kinetic energy.
Just like the Leaning Tower of Pisa is not getting any shorter but it looks wider.

 

[Jakaha]

the potential energy at any finite distance from the gravitating body is negative

That's what I mean. PE is a bookkeeping trick which has no physical basis in reality.
Consider the case of the ball bearing falling to Earth. Normally, we are told that the ball bearing is getting its downward KE from the PE stored into it when I lifted it up. The molecular bonds' energy in ATP -> KE in my muscle fibers -> KE in ball bearing -> PE in ball bearing -> KE in ball bearing.

But that explanation is nonsense.

If I lift the ball bearing high enough, it will fall towards the Sun instead of the Earth, and it will pick up a whole lot of KE along the way. If it falls towards a quasar 10 billion light years away, it will pick up even more. How can one claim that the ATP in my muscles gave it all that PE?

consider two rocks, which at some instant are both at rest above the Earth at slightly different altitudes. Relative to the Earth, they will both start free-falling downwards

I appreciate the example, but could you please explain the (surely simpler) case of the two-body situation.

Take P1's frame of reference. P2 starts 'falling' towards it, so P1 sees P2's radial speed and KE increase. I understand that the geodesics explain the directional change in velocity, but what is causing the change in speed and KE?

Remember we are using P1's frame of reference, so we (P1) are stationary. Only P2 is moving and increasing its speed/KE.

 

[Jakaha]

No energy is created but part of P2's rest energy looks like kinetic energy.

So you are saying that P1 sees P2's radial speed increase, but P2's total mass/energy remains constant?
That doesn't sound right. What is there to stop P2 achieving and exceeding the speed of light in that case, if its relativistic mass doesn't increase as seen by P1?

 

[PeterDonis]

Normally, we are told that the ball bearing is getting its downward KE from the PE stored into it when I lifted it up.

That's because you lifted it up through the same distance that it fell down. If you let it fall down through a hole in the Earth, all the way to the center, obviously it's going to have more KE than the PE you gave it--because you let it fall through a longer distance than you lifted it. Similar remarks apply to your other cases.

Before you even tackle relativity, it looks to me like you need to first understand basic Newtonian physics.

Take P1's frame of reference. P2 starts 'falling' towards it, so P1 sees P2's radial speed and KE increase.

Relative to him, yes.

I understand that the geodesics explain the directional change in velocity, but what is causing the change in speed and KE?

The geodesics also explain the change in P2's speed and KE relative to P1. P1 and P2 are both following geodesics of the spacetime they are in (note that this spacetime is not the same as the spacetime around a single gravitating body that I was using earlier in this thread--by hypothesis you have two gravitating bodies in this scenario). Those geodesics are such that they start out parallel (at the instant that P1 and P2 are at rest relative to each other) but don't stay parallel--they approach each other, then pass each other, recede from each other, then become parallel again for an instant (the instant when P1 and P2 have passed each other, decelerated, and are now at rest relative to each other for an instant), then start another cycle of approach, pass, recede. To each of P1 and P2, this behavior of the geodesics appears as the speed and KE of the other object changing.

Remember we are using P1's frame of reference, so we (P1) are stationary. Only P2 is moving and increasing its speed/KE.

The properties of geodesics are independent of the coordinates we choose. Which object we label as having "speed" and "KE" depend on the coordinates, but the geometry of the geodesics does not.

 

[SlowThinker]

So you are saying that P1 sees P2's radial speed increase, but P2's total mass/energy remains constant?
That doesn't sound right. What is there to stop P2 achieving and exceeding the speed of light in that case, if its relativistic mass doesn't increase as seen by P1?

When all of P2's rest energy has been converted to kinetic energy *, there is no more speed to be gained.
When the Tower falls, it won't get any longer than it already is.

* in fact, P2 will always keep a bit of rest energy, so it can always move a bit faster. As if the Tower never falls perfectly horizontal.

 

[Matterwave]

In GR, the quantity "velocity" can be more closely associated with the the 4-velocity. The 4-velocity is parallel transported along geodesics (the paths that freely falling objects follow). As such, the 4-velocity does the GR equivalent of "not changing" along free fall paths. There is no 4-acceleration. As there is no 4-acceleration, there is also no change in the 4-momentum (no change here again means parallel transported). The 4-momentum, the quantity that encapsulates Newtonian ideas of energy and momentum into a relativistic theory, does not change along free fall paths.

What the ball and the surface of the Earth are doing is that they are both following free fall paths, and these free fall paths happen to meet at some point in the future. Just like if you drew two lines which started parallel, but which eventually converged. The "curvature" effect of GR is to make these once parallel lines converge.

 

[PeterDonis]

No energy is created but part of P2's rest energy looks like kinetic energy.

No, this is not correct. First of all, you are trying to apply SR in the presence of gravity, but SR doesn't work in the presence of gravity. Second, even in an SR context, it is not true that, in a frame in which an object is moving, part of its rest energy looks like kinetic energy. An object has the same rest energy in every frame; in a frame in which it is moving, it also has kinetic energy due to that motion, in addition to its rest energy. Total energy is frame-dependent; that's all there is to it.

What is there to stop P2 achieving and exceeding the speed of light in that case, if its relativistic mass doesn't increase as seen by P1?

"Relativistic mass" is just another term for "total energy", and is frame-dependent. Hence, it is not a good tool for analysis.

As for P2 somehow exceeding the speed of light, that is impossible because of the geometry of spacetime. The speed of light limitation is not a property of P2, or any other object; it's a property of spacetime itself. A better way to describe that property in curved spacetime is that the worldlines of all objects must lie within the light cone at every event they pass through.

When all of P2's rest energy has been converted to kinetic energy

This is not correct; P2's rest energy is not being converted to kinetic energy. See above.

 

[Jakaha]

That's because you lifted it up through the same distance that it fell down. If you let it fall down through a hole in the Earth, all the way to the center, obviously it's going to have more KE than the PE you gave it--because you let it fall through a longer distance than you lifted it. Similar remarks apply to your other cases.

Let's drop PE since no one is invoking it as a deus ex machina to conjure up the needed KE in GR.

Newtonians would probably argue that the Big Bang gave the ball bearing a huge amount of PE which it now converts to KE as it falls into the quasar 10 billion light years away.

The geodesics also explain the change in P2's speed and KE relative to P1. P1 and P2 are both following geodesics of the spacetime they are in (note that this spacetime is not the same as the spacetime around a single gravitating body that I was using earlier in this thread--by hypothesis you have two gravitating bodies in this scenario). Those geodesics are such that they start out parallel (at the instant that P1 and P2 are at rest relative to each other) but don't stay parallel--they approach each other, then pass each other, recede from each other, then become parallel again for an instant (the instant when P1 and P2 have passed each other, decelerated, and are now at rest relative to each other for an instant), then start another cycle of approach, pass, recede. To each of P1 and P2, this behavior of the geodesics appears as the speed and KE of the other object changing.

I am confused.

I was told that we can consider the apple/ball bearing to be at rest and the Earth to be accelerating towards it.

In our two-body example, I (P1) am at rest and P2 is the only thing moving in my universe. Since GR only works in 3D, and I don't think anyone is invoking time dilation here, I don't see how P2 moving at a steady speed along the 3D space can be perceived as an increase or decrease in speed, no matter how convoluted the geometry.

The properties of geodesics are independent of the coordinates we choose. Which object we label as having "speed" and "KE" depend on the coordinates, but the geometry of the geodesics does not.

By coordinates, do you mean frame of reference?
Are you saying the warping of spacetime (geodesics) are an intrinsic property of an object regardless of any frame of reference?

Sorry if I misunderstand but that can't possibly be. Since KE is a relative value, then relativistic mass is a relative value and the resulting warping of space-time by that object must be a relative effect, no?

 

[Jakaha]

In GR, the quantity "velocity" can be more closely associated with the the 4-velocity. The 4-velocity is parallel transported along geodesics (the paths that freely falling objects follow). As such, the 4-velocity does the GR equivalent of "not changing" along free fall paths. There is no 4-acceleration. As there is no 4-acceleration, there is also no change in the 4-momentum (no change here again means parallel transported). The 4-momentum, the quantity that encapsulates Newtonian ideas of energy and momentum into a relativistic theory, does not change along free fall paths.

Sorry, but isn't that just a restatement of GR's mathematical model?
I am asking how to reconcile that with the physical reality where the ball does indeed increase speed and momentum as it races towards its rendezvous with the Earth.

What the ball and the surface of the Earth are doing is that they are both following free fall paths, and these free fall paths happen to meet at some point in the future. Just like if you drew two lines which started parallel, but which eventually converged. The "curvature" effect of GR is to make these once parallel lines converge.

No problem with the trajectory side of things.
My question is about the fact that the ball covers a greater and greater distance along the geodesic in a given amount of time as it gets closer to its rendezvous point.

What is there in pure geometry that compels it to do so?

 

[Orodruin]

Total energy is frame-dependent; that's all there is to it.

I think it is worth underlining that this fact is not particular of relativity. It is the case also in classical mechanics.

 

[A.T.]

How does pure geometry cause a change in speed?

Look at the geometry again. The "straight" (geodesic) world-line of a free faller deviates more and more from the time axis, towards the space axis. This corresponds to moving faster and faster through space:

 

[Jakaha]

Look at the geometry again. The "straight" (geodesic) world-line of a free faller deviates more and more from the time axis, towards the space axis. This corresponds to moving faster and faster through space:

Thank you.
Since the answer must be purely geometric, the diagram and animation help.

This also answers my question above that the curvature cannot be an intrinsic property of an object, but is a property of interaction between two particles.
The curvature will be custom-tailored for every particle that interacts with the gravitational "field", and each upcoming segment of the curvature will be instantaneously created as the particle moves within that "field". This newly created curvature must occur from local information, since there can be no communication with the other particle, which might be light years away.

This still leaves the question of where the KE comes from, and we can know that the KE is not just an apparent attribute, but there are absolute constraints on the apple's KE/momentum which every observer must validate.

For e.g. if we place a sheet of wood at ground level with a known strength that will only be punched through with at least a minimum amount of momentum/KE, and the apple punches through it, then every observer must measure the apple (or wood) to have at least that amount of momentum and KE.

 

[Dale]

OK, so my basic understanding is that GR is all about geometry of space-time. It's all geometry, no other mechanism.

This explains why objects change direction due to gravity. But why does the speed increase?

Don't forget that GR is about the geometry of spacetime, not just space. A change in speed is the same thing as a change in direction in spacetime.

 

[A.T.]

This still leaves the question of where the KE comes from,

KE comes from defining a reference frame. But that is not specific to GR.

 

[PeterDonis]

Newtonians would probably argue that the Big Bang gave the ball bearing a huge amount of PE

No, they wouldn't. They would argue that the Big Bang gave the ball bearing a huge amount of KE, since the Big Bang made the universe expand very fast. Once again, it seems to me that you need to improve your understanding of Newtonian physics.

I was told that we can consider the apple/ball bearing to be at rest and the Earth to be accelerating towards it.

Locally, yes.

In our two-body example, I (P1) am at rest and P2 is the only thing moving in my universe.

If you choose coordinates appropriately, yes. But you can't say that P1 is at rest in any absolute sense; there is no such thing as absolute rest. Rest, like motion, is always relative.

GR only works in 3D

No, it works in 4D. Spacetime is 4-dimensional.

I don't think anyone is invoking time dilation here

Correct, you don't need to invoke time dilation to explain what's going on in your example.

I don't see how P2 moving at a steady speed along the 3D space can be perceived as an increase or decrease in speed

In the 3D space relative to P1, P2 is not moving at a steady speed. It accelerates until it passes P1, then decelerates, comes to a stop, and accelerates back in the other direction, and the cycle starts again.

In 4D spacetime, both P1 and P2 are following geodesics, and those geodesics start out parallel, approach, pass each other, recede, become parallel again, then start approaching again in another cycle.

By coordinates, do you mean frame of reference?

The term "frame of reference" is often used when what is really meant is "coordinates", but there are also other meanings of "frame of reference" that are not synonymous with "coordinates". So you need to clarify what you mean by "frame of reference".

Are you saying the warping of spacetime (geodesics) are an intrinsic property of an object regardless of any frame of reference?

No, I'm saying the warping of spacetime is an intrinsic property of spacetime, regardless of any frame of reference. Spacetime is the geometric object; geodesics are just "grid lines" on that geometric object.

Since KE is a relative value, then relativistic mass is a relative value and the resulting warping of space-time by that object must be a relative effect, no?

Relativistic mass is not what causes spacetime curvature. The stress-energy tensor is. The stress-energy tensor is a frame-independent geometric object, just like spacetime curvature. The Einstein Field Equation relates the two.

My question is about the fact that the ball covers a greater and greater distance along the geodesic in a given amount of time

No, it doesn't. "Distance along the geodesic" is measured in spacetime, not space. The thing that measures it is a clock carried along with the ball. That clock "covers distance" along the ball's worldline at one second per second, regardless of how that worldline is situated in spacetime.

The curvature will be custom-tailored for every particle that interacts with the gravitational "field",

No. The curvature is caused by the stress-energy tensor, as I said above. The stress-energy tensor is a property of the object that is the source of the field, not the object that is responding to the field. In the example of a ball falling towards the Earth, the SET inside the Earth causes spacetime curvature inside the Earth; that curvature is then propagated out to the vacuum region outside the Earth, where the ball is. The ball just responds to the spacetime curvature in its vicinity; it doesn't produce any itself.

In your P1-P2 example, both P1 and P2 are sources of spacetime curvature, and they both also respond to spacetime curvature. That makes the full analysis a lot more complicated, because there are issues involving "self-interaction" that come into play. But if we assume that P1 and P2 are weak enough sources, we can ignore the self-interaction complications and treat them separately as sources of curvature and then as objects responding to curvature. As sources, they produce a self-consistent solution of the Einstein Field Equation that describes a spacetime geometry; as objects, they move on geodesics within that spacetime geometry.

there are absolute constraints on the apple's KE/momentum

Those constraints are not "absolute" constraints on KE and momentum. See below.

For e.g. if we place a sheet of wood at ground level with a known strength that will only be punched through with at least a minimum amount of momentum/KE, and the apple punches through it, then every observer must measure the apple (or wood) to have at least that amount of momentum and KE.

Incorrect. An observer at rest relative to the wood must measure the apple to have at least that amount of momentum and KE. But an observer moving along with the apple would measure it to have zero momentum and KE; instead, he would measure the wood to have some positive amount of momentum and KE (but different than the apple's relative to the wood, since the wood presumably has a different rest mass than the apple). So there is no absolute constraint on the apple's momentum and KE.

 

[Jakaha]

No, they wouldn't. They would argue that the Big Bang gave the ball bearing a huge amount of KE, since the Big Bang made the universe expand very fast. Once again, it seems to me that you need to improve your understanding of Newtonian physics.

I understand Newtonian mechanics well enough, so let me spell out the intermediate steps...

Ball falling to Earth:
KE of my muscle fibers -> KE of ball away from Earth -> PE of ball -> KE of ball towards Earth

Ball falling to quasar:
KE from Big Bang -> PE of ball -> KE of ball towards quasar

No, it works in 4D. Spacetime is 4-dimensional.
Correct, you don't need to invoke time dilation to explain what's going on in your example.

What I meant is that no one is suggesting the ball undergoes time travel. In any case, the youtube video posted by A..T. details the geometrical explanation, so I accept that.

The term "frame of reference" is often used when what is really meant is "coordinates", but there are also other meanings of "frame of reference" that are not synonymous with "coordinates". So you need to clarify what you mean by "frame of reference".

I use it in the sense of defining a (0,0,0,0) from which all events are measured.

Those constraints are not "absolute" constraints on KE and momentum. See below.

Incorrect. An observer at rest relative to the wood must measure the apple to have at least that amount of momentum and KE. But an observer moving along with the apple would measure it to have zero momentum and KE; instead, he would measure the wood to have some positive amount of momentum and KE (but different than the apple's relative to the wood, since the wood presumably has a different rest mass than the apple). So there is no absolute constraint on the apple's momentum and KE.

I thought I had explained the absolute constraints well enough, but let me try again.

The piece of wood has a certain strength which will only be breached by an impacting object if the difference in KE/momentum between the object and the wood is at least a certain amount. Initially, the apple is hanging motionless above the wood, so they have zero relative KE and momentum, regardless of which frame of reference you chose. At the point of impact, if the apple punched through the wood, then there is at least that minimum amount of KE/momentum difference between the two. This fact, too, must be explained by all observers, regardless of their frame of reference.

Where did this difference in KE/momentum come from, using a purely geometric explanation?

 

[Jakaha]

This also answers my question above that the curvature cannot be an intrinsic property of an object, but is a property of interaction between two particles.

I take back this comment.

I was having a brain freeze where I thought acceleration is a percentage change in speed, but it is an absolute change in speed (or direction), so one curvature will work for all interactions.

 

[SlowThinker]

The piece of wood has a certain strength which will only be breached by an impacting object if the difference in KE/momentum between the object and the wood is at least a certain amount. Initially, the apple is hanging motionless above the wood, so they have zero relative KE and momentum, regardless of which frame of reference you chose. At the point of impact, if the apple punched through the wood, then there is at least that minimum amount of KE/momentum difference between the two. This fact, too, must be explained by all observers, regardless of their frame of reference.

Where did this difference in KE/momentum come from, using a purely geometric explanation?

Again, all inertial observers agree that the piece of wood is accelerated (and the apple is not).

Geometry explains something different: the fact that the distance of the piece of wood to the center of the Earth stays constant, even though it's accelerating.

 

[Dale]

@Jakaha you may find this insights article helpful
https://www.physicsforums.com/insights/understanding-general-relativity-view-gravity-earth/

 

[Jakaha]

Again, all inertial observers agree that the piece of wood is accelerated (and the apple is not).

Geometry explains something different: the fact that the distance of the piece of wood to the center of the Earth stays constant, even though it's accelerating.

Sure but how does GR explain the energy provided to the apple to punch through the piece of wood?
Remember, the hole in the wood is an absolute physical reality that must be explained by all observers.

The apple and wood had zero KE/momentum relative to each other before the apple started moving.
This ties in with the question of what gives the apple its initial push to start moving in the first place.
I don't know that GR answers that question.

P.S. Sure, in this case we can say that the apple and Earth/wood were already moving and the apple just hopped on a different geodesic, but the fundamental problem remains in the P1-P2 two-body case. What causes P1 and P2 to start moving in the first place?

 

[Orodruin]

The apple and wood had zero KE/momentum relative to each other before the apple started moving.

No they did not. They were not even at the same place in spacetime and so you are trying to make a comparison which must be made locally global.

 

[Nugatory]

Sure but how does GR explain the energy provided to the apple to punch through the piece of wood?
Remember, the hole in the wood is an absolute physical reality that must be explained by all observers.

The apple and wood had zero KE/momentum relative to each other before the apple started moving.
This ties in with the question of what gives the apple its initial push to start moving in the first place.
I don't know that GR answers that question.

Nothing gave the apple an initial push towards the wood. Consider the situation at the moment that the apple is released: At that moment the apple experiences no force at all. However, the ground and the piece of wood are still subject to the same forces as before the apple was released; these forces were pushing the piece of wood up with an acceleration of 1g both before and after the apple was released. What's different after the apple is released is that the stem of the apple is no longer exerting a force on the apple to accelerate it upwards at the same speed as the wood. Without that force, the apple doesn't accelerate at the same rate as the piece of wood, so the piece of wood smashes into the apple and the collision leaves a hole in the wood.

The energy comes from the kinetic energy of the wood, which is moving rapidly in the inertial frame in which the apple is at rest.

 

[Jakaha]

No they did not. They were not even at the same place in spacetime and so you are trying to make a comparison which must be made locally global.

The apple and wood have zero relative motion so they have zero relative KE and momentum. It doesn't matter how far apart they are.

 

[Orodruin]

The apple and wood have zero relative motion so they have zero relative KE and momentum. It doesn't matter how far apart they are.

This is simply not correct. In order to define relative motion in GR you need to be at the same event in space time or have an unambiguous way of comparing velocities at different events, generally through some parallel transport.

Edit: Nugatory's post gives a good description when tidal effects can be neglected (ie, the space time is essentially flat and equivalent an accelerated motion of the wood in SR).

 

[Jakaha]

Nothing gave the apple an initial push towards the wood. Consider the situation at the moment that the apple is released: At that moment the apple experiences no force at all. However, the ground and the piece of wood are still subject to the same forces as before the apple was released; these forces were pushing the piece of wood up with an acceleration of 1g both before and after the apple was released. What's different after the apple is released is that the stem of the apple is no longer exerting a force on the apple to accelerate it upwards at the same speed as the wood. Without that force, the apple doesn't accelerate at the same rate as the piece of wood, so the piece of wood smashes into the apple and the collision leaves a hole in the wood.

The energy comes from the kinetic energy of the wood, which is moving rapidly in the inertial frame in which the apple is at rest.

I think I figured it out.

The tree branch which was upholding the apple was preventing the apple from following its 'natural' geodesic to rendezvous with the earth. This prevention was accomplished by the tree branch providing the apple an upward 'potential' momentum which kept it on the tree branch's geodesic.

When released, the apple left the tree branch's geodesic and followed its 'natural' geodesic. The 'potential' momentum which the tree branch had been provided got converted to the apple's 'real' momentum.

 

[Orodruin]

Let me ask you: where does all the kinetic energy of the Earth in your rest frame come from when you push the accelerator of your car?

 

[Jakaha]

Let me ask you: where does all the kinetic energy of the Earth in your rest frame come from when you push the accelerator of your car?

Going backwards in time,
KE car <- KE piston/explosive gas <- molecular bond energy in ethanol <- photosynthesis <- solar photons <- solar fusion <- ...

 

[Orodruin]

Going backwards in time,
KE car <- KE piston/explosive gas <- molecular bond energy in ethanol <- photosynthesis <- solar photons <- solar fusion <- ...

No, not the kinetic energy of the car - the car is at rest in its rest frame! (Which is accelerating, just line the wood rest frame hou are talking about.) I was asking you about the kinetic energy of the Earth in this frame. Given some modest amount and time of acceleration, let us say the Earth moves at 50 kph. Where did the energy come from? This is essentially the same question as the one you are asking.

 

[Jakaha]

No, not the kinetic energy of the car - the car is at rest in its rest frame! (Which is accelerating, just line the wood rest frame hou are talking about.) I was asking you about the kinetic energy of the Earth in this frame. Given some modest amount and time of acceleration, let us say the Earth moves at 50 kph. Where did the energy come from? This is essentially the same question as the one you are asking.

Yes, indeed, not just the Earth but the entire Universe acquired a backwards KE when I pushed my car's accelerator.
That is one of my quandaries with GR and relativity in general.
Now one can say, as we do in the Twin Paradox, that one entity had an injection of energy so it was the one 'really' accelerating but, without mention of energy, relativity makes no sense to me.

I am also not happy with my earlier explanation of tree branch giving the apple 'potential' momentum. The basic problem remains in the two-body problem.
In a universe with only P1 and P2, what aspect of pure geometry makes P1 and P2 move wrt each other in the first place?