Does an object moving in a geodesic accelerate?

[Lunct]

So in GR, objects orbiting the sun, for example, move in a geodesic - a straight line something curved. Without GR (using Newtonian Gravity), I can easily say because planets orbiting the sun are doing so in a ellipse, they are accelerating. However, would they still be accelerating when you add in GR and they move in a straight line on curved spacetime?
Let's assume for sake of simplicity that these planets are moving at a constant speed, although I believe in reality they are not.

 

[martinbn]

A geodesic is a curve whose acceleration is zero.

 

[Lunct]

A geodesic is a curve whose acceleration is zero.

so they don't accelerate?
Geodesics really aren't my strong point.

 

[PeterDonis]

so they don't accelerate?

You have to carefully distinguish two different meanings of the term "acceleration".

The first meaning, which is the one GR uses, is "proper acceleration"--what you measure with an accelerometer. Objects moving on geodesics of spacetime, in GR, have zero proper acceleration; that's what @martinbn is saying, in more precise terminology. That is true whether spacetime is flat or curved; and these geodesics are the closest things to "straight lines" that exist in spacetime.

The second meaning, which you seem to be implicitly using, is "coordinate acceleration"--the second derivative of position with respect to time. It is called coordinate acceleration because any time you use it, you are, whether you realize it or not, choosing some particular system of coordinates. (Note that you do not have to choose any coordinates to define or measure proper acceleration; that is an invariant property of a curve, independent of coordinates.) When you say planets orbiting the Sun are accelerating, you are implicitly adopting coordinates in which the Sun is at rest. But you could also adopt, for example, coordinates in which the Earth is at rest, and in these coordinates the Earth is not accelerating (in the coordinate sense), while the Sun is. (Note that these coordinates are not just theoretical; astronomers use them all the time, when they locate objects, including the Sun, by their distance from Earth and their angular position on the sky.) But regardless of your choice of coordinates, the Earth has zero proper acceleration and is moving on a geodesic of spacetime. (So is the Sun, for that matter.)

 

[Lunct]

You have to carefully distinguish two different meanings of the term "acceleration".

The first meaning, which is the one GR uses, is "proper acceleration"--what you measure with an accelerometer. Objects moving on geodesics of spacetime, in GR, have zero proper acceleration; that's what @martinbn is saying, in more precise terminology. That is true whether spacetime is flat or curved; and these geodesics are the closest things to "straight lines" that exist in spacetime.

The second meaning, which you seem to be implicitly using, is "coordinate acceleration"--the second derivative of position with respect to time. It is called coordinate acceleration because any time you use it, you are, whether you realize it or not, choosing some particular system of coordinates. (Note that you do not have to choose any coordinates to define or measure proper acceleration; that is an invariant property of a curve, independent of coordinates.) When you say planets orbiting the Sun are accelerating, you are implicitly adopting coordinates in which the Sun is at rest. But you could also adopt, for example, coordinates in which the Earth is at rest, and in these coordinates the Earth is not accelerating (in the coordinate sense), while the Sun is. (Note that these coordinates are not just theoretical; astronomers use them all the time, when they locate objects, including the Sun, by their distance from Earth and their angular position on the sky.) But regardless of your choice of coordinates, the Earth has zero proper acceleration and is moving on a geodesic of spacetime. (So is the Sun, for that matter.)

does that mean that coordinate acceleration changes when you use Newton Gravity or GR.

 

[jbriggs444]

does that mean that coordinate acceleration changes when you use Newton Gravity or GR.

Coordinate acceleration will depend on what coordinates you use. Nothing stops you from using Newtonian gravity with freely falling coordinates. Well, nothing, except the fact that a flat coordinate system will only transform gravity away locally and you'll run off the edge eventually.

Nothing stops you from using GR with non-inertial coordinates.

 

[PeterDonis]

does that mean that coordinate acceleration changes when you use Newton Gravity or GR

Coordinate acceleration doesn't depend on what theory you use; it only depends on what coordinates you choose.