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I decided to post in the homework section because my question seems very basic to me. Still I'm getting stuck with it and would appreciate any help.

## Homework Statement

I am teaching myself foundations of GR with the goal of simulating numerically some motion in flat and curved space-time. So I start with the geodesic equation

[tex]\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = 0[/tex]

According to the forum thread: https://www.physicsforums.com/threads/accelerated-motion.118435/ one can use for constant acceleration the equation like this:

[tex]\frac{d^2x^\mu}{d\tau^2} + \Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\tau}\frac{dx^\beta}{d\tau} = a^\mu[/tex]

where [itex]a^\mu[/itex] is a constant vector with the timelike being zero.

Let us first consider flat Minkowski space so [itex]\Gamma^\mu_{\alpha\beta}=0[/itex] and the equation of motion takes the form:

[tex]\frac{d^2x^\mu}{d\tau^2} = a^\mu[/tex]

At this point it almost looks like the Newton's equation. But the problem remains:

**how can I integrate it?**

## Homework Equations

[tex]\eta=diag(-1,1,1,1)[/tex]

[tex]d\tau=-\eta_{\alpha\beta}dx^\alpha dx^\beta[/tex]

Greek indices run from 0 to 3, latin indices are from 1 to 3.

## The Attempt at a Solution

If I use ##\tau## as an independent variable the result doesn't make any sense

[tex]\begin{array}{l}

x^i=C^i_0+C^i_1 \tau+a^i\frac{\tau^2}{2}\\

x^0=C^0_0+C^0_1 \tau\end{array}[/tex]

Where ## C^\mu_0## and ##C^\mu_1## are some constants. I suppose that we can fix these constants at ## C^\mu_0 = 0## , ##C^i_1=0## and ##C^0_1=1## if we start at MCRF. So the equations are further simplified to

[tex]\begin{array}{l}

x^i=a^i\frac{\tau^2}{2}\\

x^0=\tau\end{array}[/tex]

Quick consistency check with ##a^i=(1,0,0)##.

[tex]\begin{array}{l}

x^1=\frac{\tau^2}{2}\\

x^0=\tau\end{array}[/tex]

At ##\tau = 4## the particle has traveled 8 units of lengths and 4 units of time. As FTL travel is not possible I'm sure that I'm making a mistake. So,

**what is the proper way to integrate**[itex]\frac{d^2x^\mu}{d\tau^2} = a^\mu[/itex] (with a possibility to extend it to non-flat space-time)?

Thanks!