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    Gravitational waves: Computing full metric

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    Write down the full metric (flat metric plus the perturbation) which represents a sinusoidal wave of frequency 200Hz traveling in the z direction with the cross polarization. The maximum strain of 10^-21 occurs when t=0.

    © BrainMass Inc. brainmass.com December 24, 2021, 11:41 pm ad1c9bdddf
    https://brainmass.com/physics/gravity/gravitational-waves-computing-full-metric-592903

    SOLUTION This solution is FREE courtesy of BrainMass!

    The full metric in the most generic case of gravitational perturbation is:
    (see attached file)
    Where the traceless tensor (see attached file) is known as the strain tensor. (see attached file) is the lat background metric and (see attached file) is the perturbation. In the transverse gauge:
    (see attached file)
    and in weak fieid approximation where we keep time derivatives in Einstein's equations but
    put (See attached file) and working in transverse traceless gauge in which
    (see attached file)

    Thus the full metric (1) reduces to:
    (see attached file)

    We can write the strain tensor as:
    (see attached file)

    Where (See attached file) is a purely spatial, symmetric traceless constant tensor. Here (See attached file) is the wave vector which in general is:
    (see attached file)

    With (See attached file) being the frequency. Since the Wave is moving in z direction and the frequency
    is 200 Hz We have in this case:
    (see attached file)

    and since the Wave vector of the gravitational Wave is a null vector (traveling at the light
    speed) We have:
    (see attached file)

    Thus the Wave vector is:
    (see attached file)

    Equation (6) now becomes:
    (see attached file)

    Since We are in transverse gauge, in general:
    (see attached file)
    and thus:
    (see attached file)

    In the present case (see attached file) and we have:
    (see attached file)

    So we are only left with (see attached file). Since (see attached file) is symmetric and since and since (see attached file) is traceless we have:
    (see attached file)

    So the only independent components of (see attached file) are (see attached file) and (see attached file) and (see attached file) can be written as:
    (see attached file)

    Now since the wave is one with a cross polarization, the only non vanishing component of
    (see attached file) should be (see attached file) and thus (see attached file) and we have:
    (see attached file)

    So up to now the strain has only one component which using the above and (see attached file) can be
    written as:
    (see attached file)

    Using this, and the condition that the maximum of strain happens at (see attached file) and is (see attached file), we
    have for the maximum:
    (see attached file)

    and thus:
    (see attached file)

    Using this, (19) and (5) we can finally write the full metric (reduced to traceless transverse
    gauge) as:
    (see attached file)

    and if we only take the wave to be sinusoidal, we can write:
    (see attached file)

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    © BrainMass Inc. brainmass.com December 24, 2021, 11:41 pm ad1c9bdddf>
    https://brainmass.com/physics/gravity/gravitational-waves-computing-full-metric-592903

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