Am I accelerating while sitting on my chair in general relativity?

[ArthurB]

Hello,
I am wondering,
if I am sitting on my chair, here at home, planet Earth, am I accelerating (in GR)?
Does the tangent four-vector $u$ to my worldline obey the equation
$$\nabla_{u} u = a$$
or instead
$$\nabla_{u} u = 0$$
??
and if the first one is correct, is the four-vector $a$ constant?

 

[Dale]

Hello,
I am wondering,
if I am sitting on my chair, here at home, planet Earth, am I accelerating (in GR)?
Does the tangent four-vector $u$ to my worldline obey the equation
$$\nabla_{u} u = a$$

Yes, with a not equal to 0.

and if the first one is correct, is the four-vector $a$ constant?

a is the proper acceleration which is the acceleration measured by an accelerometer. Is the acceleration measured by an accelerometer constant?

 

[ArthurB]

a is the proper acceleration which is the acceleration measured by an accelerometer. Is the acceleration measured by an accelerometer constant?

I have no idea. Is a scale an accelerometer? If yes, then when I stand on a scale I see constantly the same weight.
And the other question now is: if $a$ is different than zero, what is its cause? It must be some non-gravitational interaction, right? Is it the reaction of the floor under my feet?

 

[Dale]

I have no idea. Is a scale an accelerometer? If yes, then when I stand on a scale I see constantly the same weight.

With a known mass, a bathroom scale can be used as a crude accelerometer, yes. If you have an iPad or an iPhone there is an app you can download which allows you to see the output of the built in accelerometer and magnetometer.

And the other question now is: if $a$ is different than zero, what is its cause? It must be some non-gravitational interaction, right? Is it the reaction of the floor under my feet?

Exactly, yes.

 

[DrStupid]

if I am sitting on my chair, here at home, planet Earth, am I accelerating (in GR)?

It depends on the frame of reference. In an inertial reference frame (according to GR) you are accelerating. In the rest frame of Earth you are not accelerating but this frame of reference is.

 

[D H]

I have no idea. Is a scale an accelerometer? If yes, then when I stand on a scale I see constantly the same weight.

Yep. A scale (a spring scale) is essentially an accelerometer. It only captures acceleration in one direction, so it is a directed one dimensional accelerometer. But an accelerometer nonetheless.

And the other question now is: if $a$ is different than zero, what is its cause? It must be some non-gravitational interaction, right? Is it the reaction of the floor under my feet?

Exactly. Accelerometers measure all real forces except gravitation (Newtonian explanation), or all real forces, period (GR explanation). Gravitation is a fictitious force in GR.

The floor is pushing up on you; it is the normal force. That normal force is real in both Newtonian physics and GR; this is what a stationary accelerometer measures. If you put an accelerometer on the floor it would register an acceleration of about 1 g upward.

 

[ArthurB]

Ok, so my acceleration is constant. So I can say that the frame of reference made by the walls of house is a constantly accelerated frame. Then can I say I am in Rindler coordinates?

 

[Dale]

Ok, so my acceleration is constant. So I can say that the frame of reference made by the walls of house is a constantly accelerated frame. Then can I say I am in Rindler coordinates?

No, Rindler coordinates are for flat spacetime (no tidal effects). The spacetime near the Earth is essentially a Schwarzschild spacetime, which is not flat (there are tidal effects).