Proving property of lorentz transforms

Question

I've been trying to learn relativity from Weinberg's Gravitation and Cosmology without very good knowledge of the mathematical background. I am developing it alongside, but there is one particular point where I am stuck and could use help.

While introducing Lorentz invariance and Lorentz transforms, he starts with defining a coordinate transform which we shall call Lorentz transforms as $$x^{'\alpha} = \Lambda^{\alpha}_{\ \beta} x^{\beta} + a^{\alpha}$$

where $\Lambda^{\alpha}_{\ \beta}$ is subject to the following condition: $$\Lambda^{\alpha}_{\ \gamma} \Lambda^{\beta}_{\ \delta} \eta_{\alpha \beta} = \eta_{\gamma\delta}$$

He assumes it to be true for a while, proves the invariance of proper time. He now assumes arbitrary co-ordinate transforms and invariance of proper time to get the equation,

$$0 = \frac{\partial ^2x^{'\alpha}}{\partial x^{\gamma} \partial x^{\epsilon}}$$

He says the general solution of this equation is the first equation I wrote, and that putting that here would yield the second equation.

I don't see how. What exact mathematical topic I need for this? Or a hint towards actual solution would help too! I am not very much interested in solving this, but even if I can yield the second equation by putting the first one in the third equation, I will be content for now.

Answer 1

I think it would be better to call Poincaré transformation the first equation you wrote. The third equation is basically telling you that the general change of coordinates must be at most linear, so the most general form is that of the first equation, but the second equation needs to be derived from other principles, perhaps requiring the metric in the $x$ and $x'$ coordinates to be the same.