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Imagine a + polarized gravitational wave propagating in the z direction. A particle in the field of this wave has the following lagrangian:

[tex] L = -c^2(\frac{dt}{d\tau})^2 + (1 + hcos(wt))(\frac{dx}{d\tau})^2 + (1 - hcos(wt))(\frac{dy}{d\tau})^2 + (\frac{dz}{d\tau})^2[/tex]

where tau is the proper time in the particle's frame, x,y,z & t are the spacetime coordinates used to describe the particle's trajectory, h is a number far smaller than 1 and w is the wave's angular frequency.

the problem starts off by asking me to show that there are 3 constants of the motion. The Euler-Lagrange equation yields the constants A, B and C:

[tex]A = \frac{dz}{d\tau}[/tex],

[tex]B = (1 + hcos(wt))\frac{dx}{d\tau}[/tex],

and

[tex]C = (1 - hcos(wt))\frac{dy}{d\tau}[/tex].

The problem then gets me to prove that [tex] \frac{dy}{dx} = M(1 + 2hcos(wt))[/tex], which it turns out is done with the combination of the [tex] \frac{dy}{d\tau} [/tex] and [tex] \frac{dy}{d\tau} [/tex] expressions above, and the use of 1st order power series expansions. (It turns out that the constant M = C/B.)

Now i've reached the last part where I'm not sure how to proceed - this is surely the whole point of doing all the previous calculations and is the bit I am actually supposed to learn something from.

I am asked "Is a straight line trajectory in the x direction a geodesic here? Is a straight line trajectory along the line x=y a geodesic?" If the answer to either of these is no, I am also asked to sketch a geodesic as h tends to zero.

What I'm thinking is, I'm looking at the dy/dx expression and seeing that it's non-zero. Is that telling us that you cannot move in the x-direction without moving in the y-direction, so that you cannot move in a straight line trajectory in the x-direction?

Also, the question is saying that the x-direction is perpendicular to the wave propagation direction (the z-direction). I'm not sure what bearing this has, because if space is distorted, surely that could affect all directions, right? I also have no idea what I would sketch either.

I really appreciate any answers, getting to this stage has taken me ages of mindless mathematics but I don't feel like I've learned any physics yet.

Can anyone help out?

thanks.

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# Homework Help: General relativity - gravitational wave/geodesic problem?

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