# Rindler Motion in Special Relativity, Part 2: Rindler Coordinates - Comments

• Insights
Staff Emeritus
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The definition of proper acceleration is ##\dot{u}_a =\nabla_b u_a u^b## and for the metric ##diag(-g_{00}, 1,1,1)## the acceleration of ##u^a=1/\sqrt{g_{00}}\ \partial_t## is just what you got in your post, ##-\partial_x g_{00}/(2\,g_{00})## ( in the x direction ).

I don't know if that helps or hinders.
Yes, so in the particular case in which only ##g_{00}## is different from ##\pm 1##, it works like a potential. Okay. Good to know when it works.

• Mentz114
Numerical Simulation with Riedler Coordinates:

The Equations of Motions (geodetic Equations) are:
$$\frac{d^2 x^\alpha}{d t^2}=-\Gamma^\alpha_{\beta\gamma}\frac{d x^\beta}{d t}\frac{d x^\gamma}{d t}\quad \quad (1)$$
With c=1 we get for the Riedler Coordinates :
$$\begin{bmatrix} t_I\\ x_I \\ y_I \\ z_I \\ \end{bmatrix}= \begin{bmatrix} \left(x+\frac{1}{a}\right)\sinh(a\,\tau)\\ \left(x+\frac{1}{a}\right)\cosh(a\,\tau)\\ y\\ z \end{bmatrix}$$
$$\Rightarrow$$

$$\frac{d^2 s^\alpha}{d \tau^2}=- \left( {\frac {d}{dt}}\tau \left( t \right) \right) ^{2}{x(t)}^{2}{a}^ {2}-2\, \left( {\frac {d}{dt}}\tau \left( t \right) \right) ^{2}x(t)a- \left( {\frac {d}{dt}}\tau \left( t \right) \right) ^{2}+ \left( { \frac {d}{dt}}x \left( t \right) \right) ^{2}\quad \quad (3)$$
and the Metric g is:
$$g=\begin{bmatrix} -\left(x\,a+1\right)^2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}$$
So the EQM with equation (1) and the metric g are :

Equation (4)
$${\frac {d^{2}}{d{t}^{2}}}\tau \left( t \right) +2\,{\frac {a \left( { \frac {d}{dt}}x \left( t \right) \right) {\frac {d}{dt}}\tau \left( t \right) }{x \left( t \right) a+1}}=0\\ {\frac {d^{2}}{d{t}^{2}}}x \left( t \right) + \left( {\frac {d}{dt}} \tau \left( t \right) \right) ^{2}a \left( x \left( t \right) a+1 \right) =0\\ \frac{d^2}{d t^2}y(t)=0\\ \frac{d^2}{d t^2}z(t)=0\\$$
Numerical Simulation

To do the simulation we need the initial conditions for the differential equations (4). These must satisfied also the equation (2).
Data :

$$a=10\,, x(0)=2\,,\frac{d x}{d t}\big|_{t=0}=0.1 \,, \frac{d y}{d t}\big|_{t=0}=0.0 \,, \frac{d z}{d t}\big|_{t=0}=0.0$$
we can calculate with equation (3) and
$$\frac{d^2 s}{d \tau^2}=0\quad \Rightarrow\quad \frac{d \tau}{d t}\big|_{t=0}=\pm 0.004761904762$$

Result:

can i put hier JPEG Format?

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Yes, you're right.

I guess there are three different types of acceleration, with components given by:
1. ##\frac{d^2 x^\mu}{dt^2}##
2. ##\frac{d^2 x^\mu}{d\tau^2}##
3. ##\frac{d^2 x^\mu}{d\tau^2} + \Gamma^\mu_{\nu \lambda} \frac{dx^\nu}{d\tau} \frac{dx^\lambda}{d\tau}##
I guess #3 is "proper acceleration", but I'm not sure whether "coordinate acceleration" means 1 or 2.
For the 2nd one,
$$\frac{d^2x^\mu}{d\tau^2}=\frac{dU^\mu}{d\tau}$$
Generally ##dU^\mu## does not give the relative velocity, so we need something can be directly subtracted, i.e. ##\theta##
For the uniformly accelerating object, the proper acceleration is defined as
$$\alpha=\frac{d\theta}{d\tau}$$