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Equation of accelerated motion in GR
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[QUOTE="Decibit, post: 5454758, member: 592046"] Hello forum members, 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. [h2]Homework Statement [/h2] 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: [URL]https://www.physicsforums.com/threads/accelerated-motion.118435/[/URL] 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: [B]how can I integrate it?[/B] [h2]Homework Equations[/h2] [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. [h2]The Attempt at a Solution[/h2] 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,[B] what is the proper way to integrate[/B] [itex]\frac{d^2x^\mu}{d\tau^2} = a^\mu[/itex] (with a possibility to extend it to non-flat space-time)? Thanks! [/QUOTE]
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Equation of accelerated motion in GR
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