# Gravity - an accelerating frame paradox

ZirkMan
If the equivalence principle is true then it means that the Earth's gravity field is a constantly accelerating frame of reference. In any accelerating frame of reference the direction of acceleration is always opposite to the direction of attraction.

That means that for all observers on the Earth's surface the Earth's surface (and with it the whole Earth) is constantly accelerating towards them. If this is true for all observers on its spherical surface how is it possible that the Earth doesn't explode (due to constant acceleration away from its center to all sides)? Is there a model of spacetime that explains it?
Or is there a mechanism of acceleration that doesn't require acceleration in space?

## Answers and Replies

Science Advisor
Because in curved space-time acceleration is not equal to the rate of change of velocity. So while the surface is constantly accelerating outwards at all points, none of these points actually move outwards. Hence, no explosion.

To demonstrate this, consider a person standing on a rotating platform. Wherever he's standing, his experience is consistent with the point he is standing at accelerating inwards towards the center of the platform. In fact, that point is accelerating towards the center of the platform. However, the platform is not imploding, and in this coordinate system, no point of the platform is moving.

Staff Emeritus
Science Advisor
If the equivalence principle is true then it means that the Earth's gravity field is a constantly accelerating frame of reference. In any accelerating frame of reference the direction of acceleration is always opposite to the direction of attraction. That means that for all observers on the Earth's surface the Earth's surface (and with it the whole Earth) is constantly accelerating towards them.

You are misreading the equivalence principle to say that the Earth's surface is physically expanding outward. The equivalence principle redefines the concept of an inertial frame of reference; what it is saying is that an object at rest with respect to the surface of the Earth is accelerating with respect to an inertial frame. This, BTW, is exactly what an accelerometer says: An accelerometer placed at rest on the surface of the Earth will indicate that it is accelerating upwards at 9.8 meters/second2.

ZirkMan
To demonstrate this, consider a person standing on a rotating platform. Wherever he's standing, his experience is consistent with the point he is standing at accelerating inwards towards the center of the platform. In fact, that point is accelerating towards the center of the platform. However, the platform is not imploding, and in this coordinate system, no point of the platform is moving.
If the rotation of the platform was not accelerating (or deaccelerating) and I was standing on the platform and there was no other frame of reference to compare I would have no way to prove that my platform is rotating or not and there would be no force that would drag me to the center of the platform, wouldn't be?

And yet I'm standing on the surface of the Earth and observe it accelerating towards me (from my reference frame). And I can still see the paradox here. How can there be something moving towards me (from my reference frame without comparison to other frames), yet not moving at all?

ZirkMan
... what it is saying is that an object at rest with respect to the surface of the Earth is accelerating with respect to an inertial frame.
What is this inertial frame? Some kind of absolute space? Sorry, I do not understand this.

Mentor
2021 Award
If the rotation of the platform was not accelerating (or deaccelerating) and I was standing on the platform and there was no other frame of reference to compare I would have no way to prove that my platform is rotating or not and there would be no force that would drag me to the center of the platform, wouldn't be?
That is incorrect. A simple gyroscope or a laser ring interferometer could prove it without reference to any external object.

How can there be something moving towards me (from my reference frame without comparison to other frames), yet not moving at all?
It sounds like you are confusing proper acceleration and coordinate velocity. When you are standing on the earth then you and the ground are both accelerating upwards at g, hence your relative velocity is constant (0). When you are free-falling towards the earth then the ground is accelerating at g and you are not, hence your relative velocity is changing, and whether or not a given object is moving depends on the coordinate system chosen, but it is all self-consistent with no true paradoxes.

Science Advisor
If the rotation of the platform was not accelerating (or deaccelerating) and I was standing on the platform and there was no other frame of reference to compare I would have no way to prove that my platform is rotating or not and there would be no force that would drag me to the center of the platform, wouldn't be?
There IS a force dragging you out towards the outer edge. The centrifugal force. That's exactly the point.

Centrifugal force is a fictitious force. Same as gravity in GR. It only arises from the fact that you chose a frame of reference in which a static object is actually accelerating. On a rotating platform, each point experiences centripetal acceleration. If you pick a coordinate system relative to ground, you can see that each point on the platform is moving and accelerating towards the center. If you pick a coordinate system fixed to the platform, each point is at rest, but they are still all accelerating towards the center. Hence the centrifugal force pushing you out towards the outer edge.

ZirkMan
There IS a force dragging you out towards the outer edge. The centrifugal force. That's exactly the point.

Centrifugal force is a fictitious force. Same as gravity in GR. It only arises from the fact that you chose a frame of reference in which a static object is actually accelerating. On a rotating platform, each point experiences centripetal acceleration. If you pick a coordinate system relative to ground, you can see that each point on the platform is moving and accelerating towards the center. If you pick a coordinate system fixed to the platform, each point is at rest, but they are still all accelerating towards the center. Hence the centrifugal force pushing you out towards the outer edge.
This is actually a very good explanation, thanks. But isn't this against the Galilean relativity which states that you cannot detect a uniform motion with help of mechanical experiments? Still, I see a way out of this because although the platform rotates uniformly the speed of relative movement will be different for points in different distances from the center of rotation and this propably gives rise to the centrifugal force. But where is the source of difference in relative kinetic energy for points (objects) in the gravitational frame?

Science Advisor
Rotation is not a uniform motion. It's always an accelerated motion. So it is detectable under Galilean Relativity. Under Mach's Principle, you can't tell the difference between the rotation of the platform and rotation of the universe around the platform. But unfortunately, General Relativity does not cover that, and we have no way at present to correct for it.

Mentz114
If the equivalence principle is true then it means that the Earth's gravity field is a constantly accelerating frame of reference. In any accelerating frame of reference the direction of acceleration is always opposite to the direction of attraction.
Wrong. Where do you get these weird ideas ?

That means that for all observers on the Earth's surface the Earth's surface (and with it the whole Earth) is constantly accelerating towards them.
What ? They are all remaining stationary wrt to each other. ( We are talking about Terra here, aren't we ?).

If this is true for all observers on its spherical surface how is it possible that the Earth doesn't explode (due to constant acceleration away from its center to all sides)?
It's not true. Why would the earth explode if the gravitational field is trying to compress it into a sphere ?

Acceleration means a change in velocity.
Anything that is stationary at a fixed distance from the Earth's centre is being held by a force, which is to say it is experiencing non-geodesic motion in 4D and is not accelerating.

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ZirkMan
Rotation is not a uniform motion. It's always an accelerated motion. So it is detectable under Galilean Relativity. Under Mach's Principle, you can't tell the difference between the rotation of the platform and rotation of the universe around the platform. But unfortunately, General Relativity does not cover that, and we have no way at present to correct for it.
Agreed.

But please, let's come back to the source of the fictitious force of gravity. My current understanding is that all fictitious forces arise from relative imbalance in kinetic energy(KE) in any (de)accelerating frame of reference (as is the case of the centrifugal force from your example).

Now with all know fictitious forces (except gravity?) the source of the force can be attributed to the imbalance in KE between observer's frame and an non-uniform motion frame in which he is dragged in. His own inertia enables him to feel the fictitious force (the transfer of the KE). But what is the source of difference in KE when I'm in the accelerated frame of gravity? I cannot detect anything accelerating (in the physical sense). Yet if gravity arises from the same mechanism as other fictitious forces some source of EK imbalance must be present.

Science Advisor
What ? They are all remaining stationary wrt to each other. ( We are talking about Terra here, aren't we ?).
And yet, the surface is accelerating.

Acceleration means a change in velocity.
That's where you are making an oversight. Just because the vector describing velocity does not change, it does not mean there is no change in velocity. Again, look at the rotating frame of reference. An object that has zero velocity in a rotating frame IS accelerating.

ZirkMan said:
But please, let's come back to the source of the fictitious force of gravity. My current understanding is that all fictitious forces arise from relative imbalance in kinetic energy(KE) in any (de)accelerating frame of reference (as is the case of the centrifugal force from your example).
What does kinetic energy have to do with anything?

The source of fictitious force is a "bad choice" of coordinate system. I can pick a free-falling coordinate system, and gravitational force disappears. The difference is that in flat space-time, I can pick one coordinate system that's inertial everywhere. In curved space-time, it's a local choice. If I pick a free-falling coordinate system here, on the other side of the Earth, it's an accelerated frame. That's why in GR you have to know how to deal with accelerated frames, and working with accelerated frames of reference in flat space-time is a good practice, because you can check your answers much easier.

Staff Emeritus
Science Advisor
Gold Member
That means that for all observers on the Earth's surface the Earth's surface (and with it the whole Earth) is constantly accelerating towards them. If this is true for all observers on its spherical surface how is it possible that the Earth doesn't explode (due to constant acceleration away from its center to all sides)?

In GR, frames of reference are local. You're assuming that there is a single frame of reference that encompasses the whole earth, but that isn't true.

lugita15
Under Mach's Principle, you can't tell the difference between the rotation of the platform and rotation of the universe around the platform. But unfortunately, General Relativity does not cover that, and we have no way at present to correct for it.
Are you saying that there's no way modify GR to make it compatible with Mach's principle? What about Brans-Dicke theory?

Mentor
2021 Award
Now with all know fictitious forces (except gravity?) the source of the force can be attributed to the imbalance in KE between observer's frame and an non-uniform motion frame in which he is dragged in.
Huh? Do you have any reference for this? I have never heard of such a thing and don't even know what you mean by "imbalance of KE" between frames.

Science Advisor
In GR, frames of reference are local. You're assuming that there is a single frame of reference that encompasses the whole earth, but that isn't true.
No. In GR inertial frames are local, and yes, in general, you can't always have a global coordinate system. But you can have a coordinate system that describes Earth and the immediate surroundings just fine. Start with polar coordinates under Schwarzschild metric, then adjust the metric as necessary to get something less idealized.
lugita15 said:
Are you saying that there's no way modify GR to make it compatible with Mach's principle? What about Brans-Dicke theory?
There might be. But how would you test it if it's the correct way to do so? We don't even have an experiment confirming Mach's Principle.

ZirkMan
Acceleration means a change in velocity.
Acceleration means a relative change of EK over mass (m). Otherwise you cannot explain inertia.
Anything that is stationary at a fixed distance from the Earth's centre is being held by a force, which is to say it is experiencing non-geodesic motion in 4D and is not accelerating.
Gravity is not a force. Every observer experiencing gravity is dragged in an accelerating frame of reference which is a big difference. Non-geodesic motion is just a consequence of existence of a force opposing the direction of acceleration of the accelerating frame. Such as the repulsive force of electrons in your body and the ground acting as a buffer zone through which (from your frame of reference) the energy from the accelerating ground is transfered to you. Of course you cannot detect any gravity when you are not in contact with the accelerating frame (such as freefall). Then there is nothing to prevent you following a geodesic.

ZirkMan
In GR, frames of reference are local. You're assuming that there is a single frame of reference that encompasses the whole earth, but that isn't true.
Apparently it is so since the Earth hasn't exploded due to its global acceleration yet
But I try to find a link that would link this local frames to the global view. Maybe the solution is the same as in SR - there is no global frame and frames of all observers are only relative to one another.

nismaratwork
I was under the impression that inertia, like mass, remains elusive in its explanation. I wouldn't look to Newtonian mechanics to answer that question however.

Mentor
2021 Award
Acceleration means a relative change of EK over mass (m).
This is false. Consider uniform circular motion in an inertial reference frame, there is acceleration without any change in kinetic energy.

ZirkMan, you have provided nothing to substantiate any of the many erroneous claims you have made in this thread. You need to stop speculating and start learning some mainstream physics.

ZirkMan
Huh? Do you have any reference for this? I have never heard of such a thing and don't even know what you mean by "imbalance of KE" between frames.
Take for example a person jumping on an accelerating train (from the person's perspective). From his perspective the train has higher KE then he has. When he jumps on it as soon as he touches it the difference of of their KE starts to level off. Even if the train was not accelerating only moving uniformly the transfer of difference in their KE would not be immediate (due to inertia) and the person would probably fell down because for a moment his feet would move quicker than the rest of his body. He would feel a kick of "fictitious force".

When the train is accelerating the transfer of KE doesn't stop because the train constantly increases its relative KE. Therefore the person will feel the constant fictitious force of acceleration.

Is such an explanation ok for you? I hope it is clear and simple enough that I do not need any formal reference to back it up because it is just an ordinary relation of basic and well-known physics concepts (at least from my reference frame :shy:). And you are free to prove it wrong.

ZirkMan
This is false. Consider uniform circular motion in an inertial reference frame, there is acceleration without any change in kinetic energy.
So you disagree with this:
Rotation is not a uniform motion. It's always an accelerated motion. So it is detectable under Galilean Relativity.

ZirkMan, you have provided nothing to substantiate any of the many erroneous claims you have made in this thread. You need to stop speculating and start learning some mainstream physics.
Yes, you have a full right of accusing me of speculation in this second part of this thread, because indeed I do not agree with some of the answers and think some other approach could worked better. But I try to do my best to explain why and teach some mainstream physics in the process from you guys. I want to see where I am wrong and the best way is to get some constructive feedback after which I can admit I have gained some new insight.

Science Advisor
Acceleration means a relative change of EK over mass (m). Otherwise you cannot explain inertia.
No. It does not. Kinetic energy is something completely different in GR.

$$E_{kinetic} = cp^t - c\sqrt{p^{\mu}p^{\nu}g_{\mu\nu}}$$

Note that this quantity is frame and metric dependent.

nismaratwork
So you disagree with this:

Yes, you have a full right of accusing me of speculation in this second part of this thread, because indeed I do not agree with some of the answers and think some other approach could worked better. But I try to do my best to explain why and teach some mainstream physics in the process from you guys. I want to see where I am wrong and the best way is to get some constructive feedback after which I can admit I have gained some new insight.

Your tone and approach is very reasonable, but the content of what you're saying isn't. I believe that's the source of the disconnect between your questions, and the frustration in the attempt to answer. It feels very much like a reasonable individual trying reasonably to tell you that the moon is made of green cheese.

If you make your mistakes in the form of trying to learn, rather than teach... you'll be far more in the spirit (and letter) of this forum's law. AFAIK. After all, it's hard to tell a crackpot with an agenda from someone trying to employ a confrontational method to test their ideas.

I will say this: I read, but don't participate in a lot of these threads. I can't think of a time that DaleSpam steered someone wrong... I'd listen to him.

Mentor
2021 Award
When he jumps on it as soon as he touches it the difference of of their KE starts to level off. Even if the train was not accelerating only moving uniformly the transfer of difference in their KE would not be immediate (due to inertia) and the person would probably fell down because for a moment his feet would move quicker than the rest of his body. He would feel a kick of "fictitious force".
All of what you describe here is due to the real contact force between the person and the train and not due to fictitious forces. This certainly does not support nor even explain your claim that fictitious forces are "attributed to the imbalance in KE".

I hope it is clear and simple enough that I do not need any formal reference to back it up because it is just an ordinary relation of basic and well-known physics concepts
No, it is not clear, and it has no discernable relationship to mainstream physics. Please provide a reference, or stop posting this line of speculation.

Staff Emeritus
Science Advisor
Gold Member
No. In GR inertial frames are local, and yes, in general, you can't always have a global coordinate system. But you can have a coordinate system that describes Earth and the immediate surroundings just fine. Start with polar coordinates under Schwarzschild metric, then adjust the metric as necessary to get something less idealized.
A coordinate system isn't the same as a frame of reference, and that's exactly what you need to understand in order to understand the resolution of the paradox. If you have a frame of reference, you can define whether objects A and B are at rest with respect to one another, even if A and B are far apart. GR does not, for example, allow us to say whether our galaxy is at rest with respect to a distant galaxy.

There might be. But how would you test it if it's the correct way to do so? We don't even have an experiment confirming Mach's Principle.
This is incorrect. Mach's principle can be tested by experiments that test predictions made by a less Machian theory such as GR and a more Machian theory such as Brans-Dicke gravity. The results come out to be less Machian, as in GR. A good popular-level discussion of this is available in Was Einstein Right? by Will.

bcrowell said:
In GR, frames of reference are local. You're assuming that there is a single frame of reference that encompasses the whole earth, but that isn't true.
Apparently it is so since the Earth hasn't exploded due to its global acceleration yet
Your statement doesn't connect logically to the material you quoted from my post.

Maybe the solution is the same as in SR - there is no global frame
SR does have global frames of reference.

Mentor
2021 Award
ZirkMan,

First, the equivalence principle only applies over regions of spacetime where the curvature is negligible. So, while it applies over a small region of the earth's surface, it does not apply over the entire surface.

Second, geometrically an inertial object's worldline is a geodesic (a straight line). Conversely an object undergoing proper acceleration has a non-geodesic worldline (a curved line).

Third, an inertial coordinate system is an orthonormal coordinate system whose coordinate lines are all straight (geodesic), conversely a non-inertial coordinate system has one or more sets of curved coordinates.

Fourth, In gravity two inertial particles may accelerate relative to each other, geometrically this is only possible in a curved spacetime.

Fifth, in a curved spacetime it is not possible to introduce a global set of coordinates which are everywhere inertial, but it is possible to do so locally.

Given the above, do you understand your concern any better?

ZirkMan
All of what you describe here is due to the real contact force between the person and the train and not due to fictitious forces. This certainly does not support nor even explain your claim that fictitious forces are "attributed to the imbalance in KE".
Is it possible to distinguish between the contact force and the fictitious force in this case? If yes, then I was really wrong.

No, it is not clear, and it has no discernable relationship to mainstream physics. Please provide a reference, or stop posting this line of speculation.
OK, no more on KE and fictitious forces until I have a set of axioms that can be put into formal language and defended in other way then just freely described concepts. Thanks for your time and valuable feedback anyway.

Mentor
2021 Award
Is it possible to distinguish between the contact force and the fictitious force in this case? If yes, then I was really wrong.
Yes, there are lots of ways to distinguish them. The contact force is a real force and is associated with an "equal and opposite" reaction force on the train, the fictitious force is not. The contact forces cause stresses, the fictitious force does not. Accelerations due to the contact forces can be measured by an accelerometer, the fictitious forces cannot. Etc.

Science Advisor
A coordinate system isn't the same as a frame of reference, and that's exactly what you need to understand in order to understand the resolution of the paradox. If you have a frame of reference, you can define whether objects A and B are at rest with respect to one another, even if A and B are far apart. GR does not, for example, allow us to say whether our galaxy is at rest with respect to a distant galaxy.
Coordinate system is the frame of reference. It's the mapping from an open set in manifold onto R4. There is no other definition for frame of reference or coordinate system.

This is incorrect. Mach's principle can be tested by experiments that test predictions made by a less Machian theory such as GR and a more Machian theory such as Brans-Dicke gravity. The results come out to be less Machian, as in GR. A good popular-level discussion of this is available in Was Einstein Right? by Will.
That does not test Mach's principle. That tests specific theory of gravity. There is no practical test of Mach's Principle.

ZirkMan
Yes, there are lots of ways to distinguish them. The contact force is a real force and is associated with an "equal and opposite" reaction force on the train, the fictitious force is not. The contact forces cause stresses, the fictitious force does not.
So a fictitious force stops being fictitious as soon as a contact is being made? So when I'm sitting on the ground on Earth there is no fictitious force pushing me down only a real contact force (which source is what)?

Accelerations due to the contact forces can be measured by an accelerometer, the fictitious forces cannot. Etc.
Maybe the answer to the question above will answer this seemingly opposite answers.
An accelerometer placed at rest on the surface of the Earth will indicate that it is accelerating upwards at 9.8 meters/second2.

Science Advisor
So a fictitious force stops being fictitious as soon as a contact is being made? So when I'm sitting on the ground on Earth there is no fictitious force pushing me down only a real contact force (which source is what)?
No. The contact force is pushing you up. It's what opposing the fictitious force and prevents you from falling towards center of the Earth.

ZirkMan
No. The contact force is pushing you up. It's what opposing the fictitious force and prevents you from falling towards center of the Earth.
If the contact force is a reaction to the fictitious force then its hard to say which one causes stresses. How can you say (DaleSpam) that it's not the fictitious force thats causing the stress?

nismaratwork
If the contact force is a reaction to the fictitious force then its hard to say which one causes stresses. How can you say (DaleSpam) that it's not the fictitious force thats causing the stress?

Doesn't this go back to the RLG or an accelerometer on the ground?

Science Advisor
If the contact force is a reaction to the fictitious force then its hard to say which one causes stresses. How can you say (DaleSpam) that it's not the fictitious force thats causing the stress?
Because when you are free-falling, the fictitious force is acting on you, but there is no stress.