B How does GR explain increase/decrease in speed?

[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?

 

[SlowThinker]

I think I figured it out.

...

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.

I think you got it wrong:
1. The released kinetic energy does not depend on how long the apple was hanging before it dropped.
2. A "geodesic" is only when there are no forces acting on the object. Otherwise, it's called "worldline".

I think that I have the geometrical answer for you, but I've never heard it put this way, so I'm not sure if it's correct.
A stationary object is moving through time. As seen by a co-moving observer, it ages at 1 second per second.
In a different frame of reference, the object ages slower (or rather, it's moving faster through time). This difference is what provides the kinetic energy.
In Euclidean geometry, an object could move fast Forward and slow Left, or slow Forward and fast Left, depending on the viewer.
In Minkowski geometry, you can move slowly through time and slowly through space, or fast through time and fast through space, depending on the viewer.

I hope this is not completely wrong... again

 

[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.

I read your post again and I think you are saying the same, that the apple's stem (or, in my case, the tree branch) is providing the upwards force. But I was trying not to use force and see it in purely geometric terms as GR requires.

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.

This cannot be.
Consider the case of a feather falling, instead of the apple.
The Earth/wood would have the same KE in this case, according to your description, but the feather will not penetrate the wood; the apple will.

It is the difference in KE/momentum between wood and falling body that matters, and that difference would (must?) be an absolute value.

 

[Nugatory]

This cannot be.
Consider the case of a feather falling, instead of the apple.
The Earth/wood would have the same KE in this case, according to your description, but the feather will not penetrate the wood; the apple will.

In the case of the feather, the wood does not have the same kinetic energy (as measured using coordinates in which the feather is at rest) because it is not accelerating towards the apple nearly as quickly. This is because the feather experiences a significant (relative to its mass) upwards force from air resistance, and that force continues to accelerate the feather away from the wood, just not as quickly as the force from the support before we released it.

 

[Jakaha]

In the case of the feather, the wood does not have the same kinetic energy (as measured using coordinates in which the feather is at rest) because it is not accelerating towards the apple nearly as quickly. This is because the feather experiences a significant (relative to its mass) upwards force from air resistance, and that force continues to accelerate the feather away from the wood, just not as quickly as the force from the support before we released it.

OK, fair enough.
Doing this in a vacuum and acknowledging that the feather's smaller mass will induce a smaller acceleration in the Earth, but the point remains.
What part of pure geometry gave the wood/Earth this KE?

 

[Nugatory]

OK, fair enough, but the point remains.
What part of pure geometry gave the wood/Earth this KE?

None. The KE came from the force accelerating the surface of the Earth and the wood upwards. It's easy to see that force - just put the block of wood on a spring scale, and you'll see the spring compress so it has to be exerting an upwards force on the wood. And there's no countervailing downwards force acting on the wood to resist that force, so we know that the wood must be accelerating upwards.

Geometry comes into play when you go looking for the answer to the questions "How can the wood be accelerating upwards when the distance between it and the center of the Earth is not changing?" and "How can it be that neither the center of the Earth nor the apple are subject to any external forces so neither is accelerating and must be traveling in straight lines through spacetime... yet their spacetime trajectories intersect (or would intersect if the surface of the Earth didn't get in the apple's way as it followed its path through spacetime)?"

 

[Jakaha]

None. The KE came from the force accelerating the surface of the Earth and the wood upwards. It's easy to see that force - just put the block of wood on a spring scale, and you'll see the spring compress so it has to be exerting an upwards force on the wood. And there's no countervailing downwards force acting on the wood to resist that force, so we know that the wood must be accelerating upwards.

Geometry comes into play when you go looking for the answer to the questions "How can the wood be accelerating upwards when the distance between it and the center of the Earth is not changing?" and "How can it be that neither the center of the Earth nor the apple are subject to any external forces so neither is accelerating and must be traveling in straight lines through spacetime... yet their spacetime trajectories intersect (or would intersect if the surface of the Earth didn't get in the apple's way as it followed its path through spacetime)?"

But this 'force' is the old Newtonian gravitational force.
GR says there is no force; it's all geometry. Geometry is a static entity.
Something compelled the wood/Earth to move through this static space-time and gave it the energy to do so.
And I still don't see what part of pure geometry is producing this energy.

 

[Nugatory]

But this 'force' is the old Newtonian gravitational force.
GR says there is no force; it's all geometry.

The force of the compressed spring on the block of wood is not the Newtonian gravitational force. It's the Hooke's law force that a compressed spring exerts on the object at its end.

The key difference between the Newtonian and the GR picture is that in the Newtonian picture there is a gravitational force acting downwards and counterbalancing the upwards force from the spring pushing the block of wood up. So in the Newtonian picture we end up with gravitational force down, spring force up, the two cancel, and there is no net acceleration (as long as we use cordinates in which teh surface of the Earth is at rest). In the GR picture, there is, as you say, no gravitational force - thus there is nothing to balance the upwards froce from teh spring s the block of wood accelerates upwards.

 

[Jakaha]

The force of the compressed spring on the block of wood is not the Newtonian gravitational force. It's the Hooke's law force that a compressed spring exerts on the object at its end.

The key difference between the Newtonian and the GR picture is that in the Newtonian picture there is a gravitational force acting downwards and counterbalancing the upwards force from the spring pushing the block of wood up. So in the Newtonian picture we end up with gravitational force down, spring force up, the two cancel, and there is no net acceleration (as long as we use cordinates in which teh surface of the Earth is at rest). In the GR picture, there is, as you say, no gravitational force - thus there is nothing to balance the upwards froce from teh spring s the block of wood accelerates upwards.

Sorry, this also cannot be.

The upward force of the spring is purely a function of the wood's 'weight', but the acceleration in question (and the KE at moment of impact) depends on the feather/apple. As we noted, the wood's upward acceleration when the feather meets the wood is far smaller than when the apple meets the wood.

EDIT: Now I am totally confused. The acceleration of the feather and the apple must be the same, since it only depends on the Earth's mass. So the Earth must accelerate the same in reverse, but the reverse acceleration depends on the feather's mass v/s the apple's mass.

 

[Nugatory]

Sorry, this also cannot be.

The upward force of the spring is purely a function of the wood's 'weight'.

The wood is accelerating at the same rate whether it is moving towards a feather or an apple - the difference is that air resistance is forcing the feather away from the wood so that the accelerating wood approaches the feather less rapidly than it approaches the apple.

The mental shift that you have to make, and that you are having trouble with, is that you can describe the physics from the point of view of an ant standing on the surface of the apple the same way that you and I are standing on the surface of the earth. From the ant's point of view, there is no force acting on the apple, the apple is floating in free fall through empty space, and the surface of the Earth is accelerating towards the floating apple.

 

[Jakaha]

The wood is accelerating at the same rate whether it is moving towards a feather or an apple - the difference is that air resistance is forcing the feather away from the wood so that the accelerating wood approaches the feather less rapidly than it approaches the apple.

The mental shift that you have to make, and that you are having trouble with, is that you can describe the physics from the point of view of an ant standing on the surface of the apple the same way that you and I are standing on the surface of the earth. From the ant's point of view, there is no force acting on the apple, the apple is floating in free fall through empty space, and the surface of the Earth is accelerating towards the floating apple.

Let's remove the atmosphere and do this in a vacuum to simplify things.

As noted in my edit, I am now totally confused at a more basic level.

If the feather and the apple fall at the same rate, because the acceleration depends only on the Earth's mass, then the Earth's acceleration upwards must also be the same in both cases. But if we switch perspectives, then the Earth's acceleration must only depend on the mass of the feather or the apple, which are different, so how can the Earth's acceleration upwards be the same in both cases?

 

[Dale]

That is one of my quandaries with GR and relativity in general.

His example was not GR, it was plain old Newtonian physics in a non inertial reference frame. Energy is also frame variant in Newtonian physics.

This appears to be a challenge that was never brought to your attention in your Newtonian physics coursework, but it is not something unique to GR. This is ever bit as much a quandary for Newton as for Einstein.

 

[Jakaha]

His example was not GR, it was plain old Newtonian physics in a non inertial reference frame. Energy is also frame variant in Newtonian physics.

This appears to be a challenge that was never brought to your attention in your Newtonian physics coursework, but it is not something unique to GR. This is ever bit as much a quandary for Newton as for Einstein.

I agree this is a problem with Galilean relativity as well.
It is a problem when everything is relative and there is no absolute reference frame.

 

[Dale]

@Jakaha this thread is all over the place.

Before you go on to energy, do you understand that the geometry is not just space but rather spacetime? Do you understand how changing speed is geometrically the same as changing direction in spacetime?

In spacetime a point object is represented by a worldline. Reference frames are just coordinate "grid lines". If the point object worldline is parallel to the time grid lines then it is at rest in the reference frame. If it is not parallel then it is moving. The worldline of an accelerating point therefore bends from parallel to diagonal with the sharpness of the bend being called proper acceleration.

We can discuss the energy concepts, but first I think we need to make sure that you understand what is meant by "pure geometry".

 

[Jakaha]

@Jakaha this thread is all over the place.

Before you go on to energy, do you understand that the geometry is not just space but rather spacetime? Do you understand how changing speed is geometrically the same as changing direction in spacetime?

In spacetime a point object is represented by a worldline. Reference frames are just coordinate "grid lines". If the point object worldline is parallel to the time grid lines then it is at rest in the reference frame. If it is not parallel then it is moving. The worldline of an accelerating point therefore bends from parallel to diagonal with the sharpness of the bend being called proper acceleration.

We can discuss the energy concepts, but first I think we need to make sure that you understand what is meant by "pure geometry".

Yes, the question of speed was answered by A.T. in post #17.

In my opening post, I mentioned both speed and KE, but I only put speed in the title.
If you want I can open another thread about KE in GR.

 

[Jakaha]

Let's remove the atmosphere and do this in a vacuum to simplify things.

As noted in my edit, I am now totally confused at a more basic level.

If the feather and the apple fall at the same rate, because the acceleration depends only on the Earth's mass, then the Earth's acceleration upwards must also be the same in both cases. But if we switch perspectives, then the Earth's acceleration must only depend on the mass of the feather or the apple, which are different, so how can the Earth's acceleration upwards be the same in both cases?

OK, so I resolved this.

Both the apple/feather and the Earth are accelerating towards the common center of gravity. The pull of the Earth is the same for the feather and the apple, but the pull of the feather/apple on the Earth is different, so the Earth will accelerate upwards at a different rate for the feather v/s the apple.

So the statement that the feather and the apple fall at the same rate isn't strictly true if you consider the Earth's upward movement also.

 

[Dale]

I agree this is a problem with Galilean relativity as well.
It is a problem when everything is relative and there is no absolute reference frame.

If you want I can open another thread about KE in GR.

I think that you need to learn about KE in Newtonian physics first.

Do you understand, in Newtonian physics, that KE is frame variant? For example, if a car accelerated from 0 m/s wrt the Earth to 100 m/s then its KE increased in that frame, but there is also a frame where that exact same car at the exact same moment accelerated from -100 m/s to 0 m/s and therefore lost KE? This is not GR, this is Newtonian physics.

 

[Nugatory]

Let's remove the atmosphere and do this in a vacuum to simplify things.

If the feather and the apple fall at the same rate, because the acceleration depends only on the Earth's mass, then the Earth's acceleration upwards must also be the same in both cases. But if we switch perspectives, then the Earth's acceleration must only depend on the mass of the feather or the apple, which are different, so how can the Earth's acceleration upwards be the same in both cases?

The upwards acceleration of the Earth's surface is unrelated to the mass of the apple or the feather so is the same in both cases.

The apparent difference between the acceleration towards the feather and the apple is completely due to the air pushing the feather upwards more than the apple, so if we remove the atmosphere that difference would disappear. Remember how astronaut David Scott dropped a hammer and a feather while standing on the airless surface of the moon? The moon's surface hit them both at the same time.

 

[PeterDonis]

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

The KE from the Big Bang is irrelevant if we are talking about a ball falling to a quasar. The PE of the ball is just a function of whatever distance it happens to be from the quasar when it starts falling. That is the PE that gets converted to KE, in Newtonian terms, as the ball falls.

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

Ok, so by "frame of reference" you mean "coordinate chart". That's fine.

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.

Yes. Phrasing things in terms of the difference between the objects makes it clearer that you are talking about something that is frame-independent.

Initially, the apple is hanging motionless above the wood, so they have zero relative KE and momentum

Yes. Again, the term "relative" here makes it clear what frame-independent fact we are talking about.

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.

Yes.

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

The fact that the wood and the apple are following different paths through spacetime, that intersect with a particular relative KE/momentum. The apple's path is a geodesic--it is the straightest path the apple can follow in its local region of spacetime. The wood's path is not a geodesic; it is being accelerated towards the apple by the surface of the Earth. So, geometrically, the wood's path is curved and the apple's path is straight, and the path curvature of the wood's path ("path curvature" is just the geometric way of saying "proper acceleration") means that the two paths meet at an angle; the angle is the geometric representation of the relative KE/momentum between the two. Under the conditions of the problem, the angle is large enough to provide sufficient relative KE/momentum for the apple to punch a hole in the wood.

 

[Jakaha]

I think that you need to learn about KE in Newtonian physics first.

Do you understand, in Newtonian physics, that KE is frame variant? For example, if a car accelerated from 0 m/s wrt the Earth to 100 m/s then its KE increased in that frame, but there is also a frame where that exact same car at the exact same moment accelerated from -100 m/s to 0 m/s and therefore lost KE? This is not GR, this is Newtonian physics.

I know KE is relative. In any model of physics without an absolute reference frame, of course velocity and all derived quantities will be relative.
My question is not about the relativity of KE, but how pure geometry can explain the addition/removal of KE from an object.
It may well be that GR is silent on the subject and that's fine. I wanted to know if GR says anything.

 

[Jakaha]

The upwards acceleration of the Earth's surface is unrelated to the mass of the apple or the feather so is the same in both cases.

The apparent difference between the acceleration towards the feather and the apple is completely due to the air pushing the feather upwards more than the apple, so if we remove the atmosphere that difference would disappear. Remember how astronaut David Scott dropped a hammer and a feather while standing on the airless surface of the moon? The moon's surface hit them both at the same time.

This is getting perhaps a bit off topic so maybe we can move it to a new thread?
In any case, please see my post #50.
I would say the experiment on the moon would give different results if we have a sufficiently sensitive instrument to detect the minuscule time difference between the feather and the hammer.

My reasoning is simple: imagine the whole thing from the pov of the feather/hammer. Each of them has a completely different curvature and the moon falling into them will have a different acceleration.

 

[PeterDonis]

how pure geometry can explain the addition/removal of KE from an object.

Pure geometry explains the paths that all freely falling objects take through spacetime, and how those paths intersect. Pure geometry plus knowledge of forces applied (such as the force applied by the Earth to objects in contact with it) explains the paths that all objects, freely falling or not, take through spacetime, and how those paths intersect. Knowledge of how all the paths intersect is equivalent to knowledge of all the KE/momentum of objects relative to each other when they intersect. That's all there is to it.

 

[Dale]

I know KE is relative. In any model of physics without an absolute reference frame, of course velocity and all derived quantities will be relative.

OK, so let's then talk about KE in non inertial frames. (Still in Newtonian physics)

Suppose that you are in a non inertial frame, specifically, one that is uniformly accelerating in a straight line. In that non inertial frame, an inertial object starting at rest will accelerate, continuously gaining KE. Where did that KE come from?

 

[PeterDonis]

I would say the experiment on the moon would give different results if we have a sufficiently sensitive instrument to detect the minuscule time difference between the feather and the hammer.

In making this claim, you are claiming that GR is false in a regime where its predictions have been confirmed to many decimal places. That doesn't seem like good ground to stand on to me (pun intended).

My reasoning is simple: imagine the whole thing from the pov of the feather/hammer. Each of them has a completely different curvature and the moon falling into them will have a different acceleration.

No, they don't have "different curvature" in any sense. They are both freely falling objects, at rest relative to each other, so their paths are parallel geodesics of the same local spacetime geometry. That means the Moon's surface accelerates the same relative to both of them.

 

[Jakaha]

In making this claim, you are claiming that GR is false in a regime where its predictions have been confirmed to many decimal places. That doesn't seem like good ground to stand on to me (pun intended).

I don't think my statement contradicts GR. In fact I am saying that GR demands the feather and hammer fall at different rates. The whole thing makes more sense if you think of the moon falling into the feather or the hammer.

No, they don't have "different curvature" in any sense. They are both freely falling objects, at rest relative to each other, so their paths are parallel geodesics of the same local spacetime geometry. That means the Moon's surface accelerates the same relative to both of them.

As I wrote, it's about changing your point of view. Forget that the moon is a massive body.

Think of the feather and its curvature of space-time. Now see the moon reacting to that curvature and accelerating.

Now imagine a different scenario with the hammer and its curvature and calculate the moon's acceleration.

You will get different values because the feather and the hammer have different masses, i.e. curvature.

 

[Jakaha]

The KE from the Big Bang is irrelevant if we are talking about a ball falling to a quasar. The PE of the ball is just a function of whatever distance it happens to be from the quasar when it starts falling. That is the PE that gets converted to KE, in Newtonian terms, as the ball falls.

And where did this PE come from?
Answer: the KE that the Big Bang provided when it separated the ball and the quasar by sending them their separate ways billions of years ago.

The fact that the wood and the apple are following different paths through spacetime, that intersect with a particular relative KE/momentum. The apple's path is a geodesic--it is the straightest path the apple can follow in its local region of spacetime. The wood's path is not a geodesic; it is being accelerated towards the apple by the surface of the Earth. So, geometrically, the wood's path is curved and the apple's path is straight, and the path curvature of the wood's path ("path curvature" is just the geometric way of saying "proper acceleration") means that the two paths meet at an angle; the angle is the geometric representation of the relative KE/momentum between the two. Under the conditions of the problem, the angle is large enough to provide sufficient relative KE/momentum for the apple to punch a hole in the wood.

The wood is the same as the Earth. There is no extra acceleration. Think of a planet made of wood, if it makes the problem simpler.