# I Proper vs. coordinate acceleration

#### bob012345

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Summary
What is the difference between proper acceleration and coordinate acceleration? What role does each concept play in relativity?
[Moderator's note: Spun off from previous thread due to topic change.]

Yes, the attraction changes and in Newtonian physics that means they are applying a gravitational force to each other? So what? This thread is about GR, not Newtonian physics. Things just naturally travel along geodesics (ie are in "free fall")
Can you show me a situation where Newton says things accelerate and Einstein says they don't? Being in free fall doesn't mean things don't accelerate. I drop a ball towards the earth and it not only accelerates but it has a greater acceleration as it gets closer. If that case GR only slightly tweaks Newtonian gravity. Both say it accelerates at a greater rate as it gets closer regardless if you call it free fall along a geodesic. Right?

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#### Orodruin

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Can you show me a situation where Newton says things accelerate and Einstein says they don't? Being in free fall doesn't mean things don't accelerate. I drop a ball towards the earth and it not only accelerates but it has a greater acceleration as it gets closer. If that case GR only slightly tweaks Newtonian gravity. Both say it accelerates at a greater rate as it gets closer regardless if you call it free fall along a geodesic. Right?
You are wrong. In GR, the Earth’s surface does not specify an inertial frame. On the contrary, it is accelerating at a rate of about 9.8 m/s^2. This can be measured by an accelerometer. In fact your phone is pobably doing that right now. A freely falling object by definition has no proper acceleration.

#### bob012345

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You are wrong. In GR, the Earth’s surface does not specify an inertial frame. On the contrary, it is accelerating at a rate of about 9.8 m/s^2. This can be measured by an accelerometer. In fact your phone is pobably doing that right now. A freely falling object by definition has no proper acceleration.
Fine. Jump off a building and don't worry about because your just in free fall along a geodesic which by definition has no proper acceleration. Good luck with that.

#### phinds

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Can you show me a situation where Newton says things accelerate and Einstein says they don't?
Uh ... all of them? Newton says anything moving towards a body under the force of gravity is accelerating and GR says that nothing following a geodesic is accelerating.

#### phinds

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Fine. Jump off a building and don't worry about because your just in free fall along a geodesic which by definition has no proper acceleration. Good luck with that.
Well, now here's another misunderstanding that you clearly have. As soon as you hit the ground, you are not in freefall any more, you are not following a geodesic any more, and you ARE accelerating. And it will hurt.

#### bob012345

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Uh ... all of them? Newton says anything moving towards a body under the force of gravity is accelerating and GR says that nothing following a geodesic is accelerating.
Also fine.
Well, now here's another misunderstanding that you clearly have. As soon as you hit the ground, you are not in freefall any more, you are not following a geodesic any more, and you ARE accelerating. And it will hurt.
Any objective observer can measure the object in free fall gaining velocity and not at a constant rate either. You can call it no acceleration but that's just defining away the obvious to fit your model. You will have admit the object is closing the distance to the ground at an accelerating rate? If so, then why not just say the object is accelerating?

#### phinds

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Any objective observer can measure the object in free fall gaining velocity and not at a constant rate either.
yes
You can call it no acceleration but that's just defining away the obvious to fit your model.
Yes, I am using the standard terminology of GR. You are simply refusing to accept the formal definition of proper acceleration. I can't help you with that but be aware that no physicist is going to agree w/ your own personal definitions.

You will have admit the object is closing the distance to the ground at an accelerating rate? If so, then why not just say the object is accelerating?
see above

#### Ibix

You will have admit the object is closing the distance to the ground at an accelerating rate?
Other way around in relativity - it's the ground that's accelerating up at you. That's one of the lessons from the whole "can't tell the difference between gravity and being in a rocket" thing.
If so, then why not just say the object is accelerating?
Because accelerometers attached to the object read zero, while accelerometers attached to the ground do not.

#### bob012345

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Other way around in relativity - it's the ground that's accelerating up at you. That's one of the lessons from the whole "can't tell the difference between gravity and being in a rocket" thing.

Because accelerometers attached to the object read zero, while accelerometers attached to the ground do not.
If the ground is accelerating up at the object, I stand on the ground and my accelerometer says I'm accelerating up? That certainly is telling the difference!
You should claim that neither the object or the planet accelerate towards each other to be consistent.

#### Orodruin

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Fine. Jump off a building and don't worry about because your just in free fall along a geodesic which by definition has no proper acceleration. Good luck with that.
You are still not getting it. Obviously the same thing happens in GR as in Newtonian physics for this situation. What you have failed to realise is that it is not the acceleration during the fall that splashes you against the ground. It is the impact with some relative velocity to the ground. In GR, a body in free fall has no acceleration in the only meaningful invariant sense. However, the ground accelerates towards the body, leading to the same impact speed and the same splash. It is the same as if you were out in space during an EVA (free fall) and suddenly your ship starts accelerating towards you. The impact will still kill you if you are hit by it.

Also, I suggest you lose the attitude and realise that you are talking to people who actually know GR in this forum. Making snide remarks is not likely to make people want to help you understand things.

#### PeterDonis

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If the ground is accelerating up at the object, I stand on the ground and my accelerometer says I'm accelerating up?
Yes. Accelerometers read proper acceleration, and proper acceleration is the only invariant kind of acceleration.

The kind of acceleration you keep thinking of--the second derivative of position with respect to time--is coordinate acceleration, and coordinate acceleration is not invariant. It depends on your choice of coordinates (as its name indicates). That kind of acceleration is not really useful in relativity for understanding the physics, whereas proper acceleration is. That's why GR texts focus on proper acceleration.

You should claim that neither the object or the planet accelerate towards each other to be consistent.
Why would you claim this, since it isn't true on either interpretation (proper or coordinate) of acceleration?

#### pervect

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We call the sort of acceleration as measured by an accelerometer "proper acceleration", and when you're standing on the ground, it is not zero. The proper acceleration of the ground isn't zero, either. The proper acceleration at the center of the Earth is zero. So, the proper acceleration is not a property of the Earth as a whole , it depends on your exact location.

The word "acceleration" by itself could mean coordinate acceleration. But coordinate acceleration requires a frame of reference of some sort - usually a coordinate system - to specify. The proper acceleration doesn't require specifying a frame of reference or a coordinate system, it is something you can actually measure with an instrument. Because it reflects something one can actually measure, the proper acceleration is often considered to be more fundamental. If you're familiar with tensors or 4-vectors, it's also a 4-vector (which is a specific category of tensor).

The ambiguity can be confusing, but usually it's clear which one is meant by context.

#### bob012345

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Because accelerometers attached to the object read zero, while accelerometers attached to the ground do not.
I get that. But if your geodesic following accelerometer measures changing Doppler shifts from fixed stars, you know you are accelerating.

#### Orodruin

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But if your geodesic following accelerometer measures changing Doppler shifts from fixed stars, you know you are accelerating.
No you do not. Not for any reasonable invariant meaning of the word "accelerate". Acceleration is measured locally by an accelerometer, nothing else.

#### PeterDonis

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if your geodesic following accelerometer measures changing Doppler shifts from fixed stars, you know you are accelerating
More precisely, you know you have nonzero coordinate acceleration in a coordinate chart in which those stars are at rest. But, as already noted, coordinate acceleration is frame-dependent.

If your claim is that you prefer to look at coordinate acceleration rather than proper acceleration, that's your choice; nobody can force you to only look at proper acceleration, although we can strongly suggest that looking at proper acceleration rather than coordinate acceleration will be more helpful in understanding the physics. For example, in trying to understand gravitational wave emission, the original subject of this thread.

But if your claim is that coordinate acceleration is "true" acceleration and any other concept of acceleration is "wrong", then you are simply in error, and if you continue to make that claim you will receive a warning and you will be banned from further posting in this thread.

#### bob012345

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More precisely, you know you have nonzero coordinate acceleration in a coordinate chart in which those stars are at rest. But, as already noted, coordinate acceleration is frame-dependent.

If your claim is that you prefer to look at coordinate acceleration rather than proper acceleration, that's your choice; nobody can force you to only look at proper acceleration, although we can strongly suggest that looking at proper acceleration rather than coordinate acceleration will be more helpful in understanding the physics. For example, in trying to understand gravitational wave emission, the original subject of this thread.

But if your claim is that coordinate acceleration is "true" acceleration and any other concept of acceleration is "wrong", then you are simply in error, and if you continue to make that claim you will receive a warning and you will be banned from further posting in this thread.
That's not my claim at all. I never claimed anything like that. Please don't ban me for simply being a bit incredulous at some answers.

#### bob012345

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No you do not. Not for any reasonable invariant meaning of the word "accelerate". Acceleration is measured locally by an accelerometer, nothing else.
I don't get it. If I turn on a rocket engine I know I'm accelerating and can measure it, right? Is that proper or coordinate? If my reference frame to compare to is say the frame of the earth of the fixed stars, I should, be able to measure my acceleration wrt those frames. If I then fall through a geodesic around Jupiter, can I not figure out how my speed changed as it's happening? NASA does those maneuvers routinely.

#### PeterDonis

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Please don't ban me for simply being a bit incredulous at some answers.
I am having trouble understanding why you are "incredulous" if you don't have a problem with proper acceleration as a concept. I could see you asking for more information about proper acceleration as a concept if you're not familiar with it, but you're not doing that; you're basically arguing that coordinate acceleration is the "right" concept. Or if you're not, then I don't understand what point you are trying to make by continuing to emphasize coordinate acceleration, and even to use the word "acceleration" unqualified to refer to it (as though there were no such thing as proper acceleration).

#### PeterDonis

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If I turn on a rocket engine I know I'm accelerating and can measure it, right?
Yes, with an accelerometer.

Is that proper or coordinate?
Proper.

If my reference frame to compare to is say the frame of the earth of the fixed stars, I should, be able to measure my acceleration wrt those frames.

If you happen to be interested in your coordinate acceleration relative to distant stars, or relative to the Earth (more precisely, relative to some frame in which the Earth is at rest--there are multiple possible ones), then yes, you can find ways to measure it. The question is why you would be interested in such a thing. In most cases (including the case under discussion in this thread), measurements of coordinate acceleration tell you nothing useful for understanding the physics. That's why everyone else is focusing on proper acceleration.

#### George Jones

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I don't get it. If I turn on a rocket engine I know I'm accelerating and can measure it, right? Is that proper or coordinate? If my reference frame to compare to is say the frame of the earth of the fixed stars, I should, be able to measure my acceleration wrt those frames. If I then fall through a geodesic around Jupiter, can I not figure out how my speed changed as it's happening? NASA does those maneuvers routinely.

Consider a simple accelerometer that indicates non-zero proper acceleration. This accelerometer consists of two main parts, a hollow sphere like a basketball inside of which is a slightly smaller sphere. Initially, the centres of the spheres coincide, so that there is a small, uniform (vacuum) gap between the spheres. When proper acceleration is non-zero, the gap will be closed, contact between the spheres will be made, and an alarm that indicates non-inertial motion will sound. If the ship has zero proper accelerating, no alarm will sound.

Place the accelerometer in a rocket that has non-zero proper acceleration and the alarm sounds. Place the accelerometer near the surface of the Earth, and assume that the accelerometer is small enough that tidal forces can be neglected. When the accelerometer sits on a table, the alarm sounds, but if the accelerometer falls freely, no alarm sounds.

In both special and general relativity, the accelerometer sounds the alarm when proper acceleration is non-zero, i.e., when an observer is non-inertial.

#### Orodruin

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If I turn on a rocket engine I know I'm accelerating and can measure it, right? Is that proper or coordinate?
That would indeed be proper acceleration (assuming you are accelerating with the rocket).

If my reference frame to compare to is say the frame of the earth of the fixed stars, I should, be able to measure my acceleration wrt those frames.
As has already been pointed out to you, this is not a local and invariant measurement of acceleration but something set up to measure your coordinates. This is coordinate acceleration. That you had to insert "wrt those frames" is already a hint of this fact.

You can easily have a changing frequency shift between two free fall observers. This happens in the case of falling towards the Earth as well as in cosmology (cosmological redshift).

If I then fall through a geodesic around Jupiter, can I not figure out how my speed changed as it's happening?
You cannot compare velocities at different points uniquely in GR. Since it is often difficult to think in curved four dimensional manifolds, imagine the following instead:

You are walking on the Earth's surface. You start from the north pole and walk down to the equator. You then walk along the equator for half a lap and back up to the north pole. You changed direction twice for a total rotation of 180 degrees, yet when you came back you were walking in the exact same direction as you started walking in. The directions at the turning points simply do not correspond to the same directions as on the North pole because the surface you are walking in is not flat. Similar issues arise in curved spacetime. You cannot compare velocities at different events uniquely.

#### PeterDonis

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Acceleration is measured locally by an accelerometer, nothing else.
More precisely, proper acceleration is measured by an accelerometer, and nothing else.

There is such a concept as coordinate acceleration, and it can be measured if a suitable reference frame is set up, so I don't think we can just say "acceleration" without qualification is measured by an accelerometer. Particularly not in a thread where the distinction between proper and coordinate acceleration is now a topic of discussion.

#### Orodruin

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More precisely, proper acceleration is measured by an accelerometer, and nothing else.

There is such a concept as coordinate acceleration, and it can be measured if a suitable reference frame is set up, so I don't think we can just say "acceleration" without qualification is measured by an accelerometer. Particularly not in a thread where the distinction between proper and coordinate acceleration is now a topic of discussion.
I think in general "acceleration" would be taken to mean proper acceleration, but I guess you are right about it being relevant for this discussion.

#### FactChecker

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There seems to be some confusion between the mathematical second derivative of relative position and the physics definition of acceleration as a result of a force. They are not the same. An object in free fall toward the Earth's surface has a non-zero second derivative of position relative to Earth, but it is not accelerating due to a force (in GR).

#### Ibix

If the ground is accelerating up at the object, I stand on the ground and my accelerometer says I'm accelerating up? That certainly is telling the difference!
You should claim that neither the object or the planet accelerate towards each other to be consistent.
If you put an accelerometer at the center of the Earth it would also read zero, even if the mass of the object were sufficient to cause a noticeable gravitational effect at that distance. I think that's the "consistent" you were expecting.

But the surface of the Earth is not the center. It can't free-fall because the rest of the Earth is in the way - so accelerometers attached to it show exactly that fact.

Others have covered the distinction between proper acceleration and coordinate acceleration. I think it's worth adding that the distinction is present in Newtonian physics as well. In rotating frames of reference, for example, an object moving inertially has a $\partial r/\partial t$ that is time-varying, so it has coordinate acceleration but (by definition) no proper acceleration. Meanwhile an object at rest in this frame has zero coordinate acceleration but is undergoing centripetal proper acceleration.

"Proper vs. coordinate acceleration"

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