user1139
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- Homework Statement
- Please refer below.
- Relevant Equations
- Please refer below.
An exact gravitational plane wave solution to Einstein's field equation has the line metric
$$\mathrm{d}s^2=-2\mathrm{d}u\mathrm{d}v+a^2(u)\mathrm{d}^2x+b^2(u)\mathrm{d}^2y.$$
I have calculated the non-vanishing Christoffel symbols and Ricci curvature components and used the vacuum Einstein equation to obtain
$$\frac{1}{a}\frac{\mathrm{d}^2a}{\mathrm{d}u^2}+\frac{1}{b}\frac{\mathrm{d}^2b}{\mathrm{d}u^2}=0,$$
where ##a=a(u)## and ##b=b(u)##.
How do I show using the above differential equation that an exact solution can be found, in which both ##a## and ##b## are determined in terms of an arbitrary function ##f(u)##?
$$\mathrm{d}s^2=-2\mathrm{d}u\mathrm{d}v+a^2(u)\mathrm{d}^2x+b^2(u)\mathrm{d}^2y.$$
I have calculated the non-vanishing Christoffel symbols and Ricci curvature components and used the vacuum Einstein equation to obtain
$$\frac{1}{a}\frac{\mathrm{d}^2a}{\mathrm{d}u^2}+\frac{1}{b}\frac{\mathrm{d}^2b}{\mathrm{d}u^2}=0,$$
where ##a=a(u)## and ##b=b(u)##.
How do I show using the above differential equation that an exact solution can be found, in which both ##a## and ##b## are determined in terms of an arbitrary function ##f(u)##?