Is force/mass ratio an invariant quantity in GR?

[JohnNemo]

[Mentors' note - this thread was split off from https://www.physicsforums.com/threads/invariance-of-force-and-mass.939025/]

@JohnNemo Proper acceleration is not the rate of change of velocity. It is best thought of as applied force (and as you say, gravity is not a force in GR so it doesn't count here) divided by the object's mass. So when you move your phone around and the accelerometer number changes, it's not telling you how the phone's speed changes; it's telling you how much force you are applying to move the phone (and of course this is in addition to the force of the Earth pushing you up and transmitted through you to the phone).

Are force and mass both invariant, or is it just the force divided by mass value which is invariant?

 

[PeterDonis]

Are force and mass both invariant, or is it just the force divided by mass value which is invariant?

4-force is a 4-vector. Mass is a Lorentz scalar (i.e., a number which is invariant). So 4-force divided by mass is also a 4-vector. The magnitude of a 4-vector is an invariant, but its individual components are not: they are covariant--they transform according to the appropriate coordinate transformation.

When I said proper acceleration was an invariant, strictly speaking what I should have said was that the magnitude of the proper acceleration 4-vector is an invariant, and is equal to the magnitude of the 4-force vector divided by the mass. (And the mass itself is the magnitude of the 4-momentum vector, which is why it is an invariant.)

 

[JohnNemo]

4-force is a 4-vector. Mass is a Lorentz scalar (i.e., a number which is invariant). So 4-force divided by mass is also a 4-vector. The magnitude of a 4-vector is an invariant, but its individual components are not: they are covariant--they transform according to the appropriate coordinate transformation.

When I said proper acceleration was an invariant, strictly speaking what I should have said was that the magnitude of the proper acceleration 4-vector is an invariant, and is equal to the magnitude of the 4-force vector divided by the mass. (And the mass itself is the magnitude of the 4-momentum vector, which is why it is an invariant.)

If proper acceleration and mass are both invariant, if there a particular reason why the quotient value, proper acceleration, is mentioned a lot? Does it crop up in a lot of formulas?

 

[PeterDonis]

if there a particular reason why the quotient value, proper acceleration, is mentioned a lot?

I couldn't say in general. I prefer it because it seems more intuitive to me, since it's the path curvature of the object's worldline, so it has an immediate geometric meaning. But not everyone prefers the geometric viewpoint.

 

[Nugatory]

If proper acceleration and mass are both invariant, if there a particular reason why the quotient value, proper acceleration, is mentioned a lot? Does it crop up in a lot of formulas?

Yes; it's a very useful quantity when you are calculating an object's worldline through spacetime. It has an intuitive geometric meaning that is not apparent from either the rest mass or the four-force when considered in isolation.

You will also have noticed that the ordinary boring classical Newtonian acceleration is mentioned a lot in classical physics, even though it is also just the quotient value of two other invariant (under the Galilean transforms of Newtonian mechanics) quantities. The reason is similar; an intuitive meaning not apparent in the two quantities from which it is derived.

 

[JohnNemo]

I couldn't say in general. I prefer it because it seems more intuitive to me, since it's the path curvature of the object's worldline, so it has an immediate geometric meaning. But not everyone prefers the geometric viewpoint.

When you say 'the path curvature of the object's worldline' would another way to express that be 'the extent of a object's worldline's deviation from a geodesic'?

 

[PeterDonis]

When you say 'the path curvature of the object's worldline' would another way to express that be 'the extent of a object's worldline's deviation from a geodesic'?

Yes.

 

[JohnNemo]

Yes; it's a very useful quantity when you are calculating an object's worldline through spacetime. It has an intuitive geometric meaning that is not apparent from either the rest mass or the four-force when considered in isolation.

You will also have noticed that the ordinary boring classical Newtonian acceleration is mentioned a lot in classical physics, even though it is also just the quotient value of two other invariant (under the Galilean transforms of Newtonian mechanics) quantities. The reason is similar; an intuitive meaning not apparent in the two quantities from which it is derived.

I'm all for things which are intuitive. Maths is not my strong suit!