# I Proper vs. coordinate acceleration

#### A.T.

What is the difference between proper acceleration and coordinate acceleration?
Proper acceleration: What an accelerometer measures (frame independent)
Coordinate acceleration: Second derivative of position (frame dependent)

#### Nugatory

Mentor
Coordinate acceleration: Second derivative of position (frame dependent)
And when we say “the second derivative of position” we’re really saying “the second derivative of the spatial coordinates”. Phrased that way, it is more clear that coordinate acceleration is something that is created by our choice of coordinates.

#### Dale

Mentor
If I turn on a rocket engine I know I'm accelerating and can measure it, right? Is that proper or coordinate?
That is proper acceleration since it can be measured with an accelerometer. All frames agree on it. In the rocket’s coordinate system there is no coordinate acceleration but in an inertial frame (eg the fixed stars frame) there is also coordinate acceleration.

If I then fall through a geodesic around Jupiter, can I not figure out how my speed changed as it's happening?
Yes, of course. By keeping track of your position in the fixed-stars frame you can determine your speed (magnitude of first derivative of position) and your coordinate acceleration (second derivative of position) in that frame.

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
No, it isn’t a difference. If you have an accelerometer in your hand and on he floor both will read 1 g whether you are on the ground or on a 1 g rocket in space. So was that the inconsistency or did you see some other inconsistency?

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#### bob012345

Gold Member
As promised, I took out my books last night and reviewed the concepts. Specifically, I looked at MTW, the book Gravitation that serves as it's own physical pun because it's so massive :).

Ok, I NOW see where people are coming from. Mainly, the idea that free falling along a geodesic means that an object is not accelerating but is in a local, invariant inertial frame. Specifically, if I remember the number correctly, the image of figure 1.7 shows the concept nicely. The conditions stated was that there was no rotation and no acceleration, I presumed that meant no actual forces of some kind such as a rocket engine acting or collisions with objects or radiation pressure.

So, the International Space Station approximates such a frame to the extent that one can ignore small tidal effects over an extended object or experiment and the resulting rotation of once per orbit due to the tidal lock (the station always faces the earth as it orbits) which amounts to about 4 degrees per minute.

#### bob012345

Gold Member
That would indeed be proper acceleration (assuming you are accelerating with the rocket).

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).

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.
You are walking in the same direction (south) but not along the same path unless you went halfway around at the equator. Nice that there are infinite ways to go south at the pole. This is the source of many fun riddles. But regarding unique velocities, as previously discussed by others, I assume what you want is some basis to compare against if your purpose isn't doing a physics experiment but just trying to navigate. Can there actually be multi-valued paths in GR?

#### Nugatory

Mentor
Can there actually be multi-valued paths in GR?
Yes... or to be precise, the result of parallel-transporting a vector in curved spacetime depends on the path along which it is transported. This is a general property of curved manifolds, not unique to GR.

Because comparing two vectors requires parallel-transporting one of them to the other, and there is no unique way of doing this, it is only possible to compare velocities at spatially separated points if they are close enough together that we can treat the spacetime in between as flat.

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#### DrGreg

Gold Member
And when we say “the second derivative of position” we’re really saying “the second derivative of the spatial coordinates”. Phrased that way, it is more clear that coordinate acceleration is something that is created by our choice of coordinates.
Or, to make it even more clear, “the second derivative of the spatial coordinates with respect to the temporal coordinate”.

#### Nugatory

Mentor
You are walking in the same direction (south) but not along the same path unless you went halfway around at the equator. Nice that there are infinite ways to go south at the pole. This is the source of many fun riddles.
@Orodruin’s example has nothing to do with there being an infinite number of ways to go south at the pole. It works just as well if you start in any direction from any point on the surface of the earth. (The only reason for specifying that you start at the pole is that it's a bit easier to visualize because you're following latitude and longitude lines).

#### vanhees71

Gold Member
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.
Indeed, that's correct. Jump off the building, and you won't feel much as long as you are in free fall. The trouble only comes, when you hit the ground, but that's not a contradiction to the GR picture of free fall since at the moment you hit the ground you are no longer in free fall but subject to other interactions (mostly electromagnetic) ;-)).

To experience the correctness of the GR picture of free fall rather go to the IRS (rumor has it NASA offers the possibility some time in the future if you are willing to pay the price ;-)) than jumping off your window.

#### bob012345

Gold Member
@Orodruin’s example has nothing to do with there being an infinite number of ways to go south at the pole. It works just as well if you start in any direction from any point on the surface of the earth. (The only reason for specifying that you start at the pole is that it's a bit easier to visualize because you're following latitude and longitude lines).
Thanks. What you said is true and I understood the pole wasn't necessary. But it does make for better riddles too.

#### bob012345

Gold Member
Indeed, that's correct. Jump off the building, and you won't feel much as long as you are in free fall. The trouble only comes, when you hit the ground, but that's not a contradiction to the GR picture of free fall since at the moment you hit the ground you are no longer in free fall but subject to other interactions (mostly electromagnetic) ;-)).

To experience the correctness of the GR picture of free fall rather go to the IRS (rumor has it NASA offers the possibility some time in the future if you are willing to pay the price ;-)) than jumping off your window.
Thanks, I understand the terminology better now thanks to MTW.

#### A.T.

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.
There are literally pop songs explaining that:

#### Ibix

There are literally pop songs explaining that
Unfortunately, only peer-reviewed songs are acceptable sources for PF.

"Proper vs. coordinate acceleration"

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