Acceleration in Rindler coordinates

[Ali.Sadeghi]

[Mentors note: this thread was split off from an older discussion of Rindler coordinates]

Can somebody help me understand why acceleration along the hyperbola is constant?
To be more precise: assume (X,T) is the inertial coordinates and (x,t) the corresponding Rindler transformations.

x=Xcosh(aTc )
t=Xc sinh(aTc )

A uniformly accelerated frame moves along the curves with constant x.

What do we men by fixed acceleration here? is it second derivative of x relative to t? if x is constant then its first and second derivatives are of course zero. The second derivative of X relative to T cannot be a constant either, because that would imply a parabolic and not hyperbolic path with no limit on speed of light.

The only possibility that I can see is second derivative of X relative to t. is that indeed what we mean by constant acceleration along the hyperbola? And if so how can one prove that it is constant (equal to a)?

 

[PAllen]

Can somebody help me understand why acceleration along the hyperbola is constant?
To be more precise: assume (X,T) is the inertial coordinates and (x,t) the corresponding Rindler transformations.

x=Xcosh(aTc )
t=Xc sinh(aTc )

A uniformly accelerated frame moves along the curves with constant x.

What do we men by fixed acceleration here? is it second derivative of x relative to t? if x is constant then its first and second derivatives are of course zero. The second derivative of X relative to T cannot be a constant either, because that would imply a parabolic and not hyperbolic path with no limit on speed of light.

The only possibility that I can see is second derivative of X relative to t. is that indeed what we mean by constant acceleration along the hyperbola? And if so how can one prove that it is constant (equal to a)?

By acceleration is meant what is measured by an accelerometer. In SR, this is predicted (as a magnitude) to be the norm of the 4-acceleration, which is the derivative by proper time of the of the 4-velocity. As such, it (proper acceleration) is an invariant scalar that would be computed to be the same in any coordinates used to describe the world line (consistent with the idea that change coordinates used to describe the world do not change readings on a given measuring device).

 

[Nugatory]

Consider an inertial frame in which the spaceship is at momentarily at rest. This would be the inertial frame of a dropped object on the spaceship, because the spaceship and the object are moving at the same speed at the moment that the object is released (and we'll take that moment to be time zero in that frame).

In this inertial frame, the the velocity of the ship at time zero is zero, but the first derivative of that velocity is not. That first derivative, evaluated at t=0, is the Rindler acceleration $a$. It is easy enough to show that it is a constant, meaning that you'll come up with the same value no matter what the speed of the ship is at the moment that the object is dropped.

All of this is equivalent to what PAllen said above, of course.