# Acceleration and Curvature of space-time

## Main Question or Discussion Point

I'm confused, but when objects travel along the straight lines in curved space-time, do they undergo acceleration? We know that following geodesics is equivalent to inertial motion (one example is free-fall), but when these inertially moving objects travel in curved spacetime, they accelerate, do they (since the geometry of space-time tells matter how to move)? If yes, wouldn't that give us the generalization that gravitation, the curvature of spacetime, causes acceleration? Or gravitation is acceleration?

If all the above is true, can we say free-fall is an inertial motion and acceleration?

Answer the bold part first, please. We can move on after that.

Many thanks,

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Fredrik
Staff Emeritus
Gold Member
The answer to the bold part is no, but this is more of a definition than anything else. Note that the coordinate acceleration can be made to be anything you want it to be, just by choosing an appropriate coordinate system.

I'm confused, but when objects travel along the straight lines in curved space-time, do they undergo acceleration? We know that following geodesics is equivalent to inertial motion (one example is free-fall), but when these inertially moving objects travel in curved spacetime, they accelerate, do they (since the geometry of space-time tells matter how to move)? If yes, wouldn't that give us the generalization that gravitation, the curvature of spacetime, causes acceleration? Or gravitation is acceleration?

If all the above is true, can we say free-fall is an inertial motion and acceleration?

Answer the bold part first, please. We can move on after that.

Many thanks,
First you have to define acceleration. In relativity they tend to use proper acceleration which the acceleration measured by an accelerometer attached to the accelerating object. By this definition, if you are seated comfortably in a "stationary" chair reading this, you are accelerating upwards as can be proved by holding an accelerometer in your hand and you can feel this acceleration as the pressure exerted by your chair on your behind. When an object falls an accelerometer attached to the object shows no acceleration so it is not accelerating. Granted, its velocity relative to a non inertial observer is changing, but the falling observer can interpret this as non inertial observer accelerating past him.

In Newtonian physics an object continues in a straight line or remains in a state of rest unless a force is acting on it. In relativity, an objects follows a geodesic unless a force is acting on it. There are no forces acting on a free falling or or orbiting particle (if you ignore tidal effects) and "no forces" is defined as no reading on an accelerometer.

The path of a free falling object as observed by a non-inertial observer can appear to be curved, but by a suitable transformation (even in strongly curved spacetime) the path of the freefalling object is a straight line as observed by an inertial observer.

If you insist on defining acceleration in terms of motion, then difficulties arise. An object resting on a table appears to have no motion and so by that definition its acceleration is zero. With no acceleration the force f=ma acting on the object is zero and since the table is not moving either, then the object exerts no force on the table it is resting on. (We know that is not true because we can measure the stress in the table due to the weight of the object ;)

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So is the falling (inertial) object traveling at constant velocity or stationary as we accelerate towards it?

DrGreg
Gold Member
So is the falling (inertial) object traveling at constant velocity or stationary as we accelerate towards it?
Being "at constant velocity" or "stationary" has to be relative to something else to make sense. If you have several objects all falling near to each other, they will all move (approximately) at constant velocities relative to each other. But if you wait long enough, you'll find that they eventually start to slowly "accelerate" apart. In Newtonian physics this would be explained as the gravitational field not being uniform. In general relativity it is due to the curvature of spacetime.

I'm confused, but when objects travel along the straight lines in curved space-time, do they undergo acceleration?
A point mass would not. A mass of any volume would, the effect would be extremely small though.

A point mass would not. A mass of any volume would, the effect would be extremely small though.
What do you mean by that?

Thanks.

atyy
I'm confused, but when objects travel along the straight lines in curved space-time, do they undergo acceleration?
The free fall of a planet around the sun is "geodesic" "straight line" "non-accelerated" "inertial" motion in the framework of general relativity in which spacetime is curved - this is almost equivalent to curved accelerated motion in the framework of Newtonian gravity in which space is flat, and time is separate from space.

I say "almost equivalent" because where the frameworks differ in predictions, general relativity has been observed to be more accurate.

No-one has mentioned the fact that the acceleration between worldlines depends on the double covariant derivative of separation of the world lines. When worked out this shows that the magnitude of the acceleration depends the Riemann tensor, which is the best measure of the curvature. So acceleration is closely related to ( caused by ?) curvature.

DrGreg
Gold Member
A point mass would not. A mass of any volume would, the effect would be extremely small though.
What do you mean by that?
The centre of mass of a free-falling object does not experience proper acceleration. But, if the object is semi-rigid and the gravitational field is not uniform, other parts of the object will try to move relative to the centre of mass (and would move if they were free to move inertially). The retaining "tidal force" will cause some proper acceleration of those parts. The tidal forces are usually tiny except for large objects (e.g. planets) or close to a black hole.

I see many different answers. So, which is it?

EDIT: Oh, the "acceleration" in my bold part is coordinate acceleration. Objects traveling along geodesics in curved space-time don't experience proper acceleration.

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DrGreg
Gold Member
I see many different answers. So, which is it?

EDIT: Oh, the "acceleration" in my bold part is coordinate acceleration. Objects traveling along geodesics in curved space-time don't experience proper acceleration.
If by "acceleration" you mean "coordinate acceleration" then the answer is "yes or no".

Sorry if that seems unhelpful, but it depends which coordinates you are using. In some coordinates the answer is yes. In other coordinates the answer is no (as Fredrik said in post #2).

Coordinate acceleration is just $d^2\textbf{x}/dt^2$ in whatever your choice for x and t.

Which coordinates will give me the answer yes?

Thanks much.