Are Killing Vectors the Key to Solving Complex Equations?

[TimeFall]

See below. I screwed up the edit and the use of tex.

 

[TimeFall]

EDIT: Used proper tex (hopefully!)

Hello! I'm working through Weinberg's book Gravitation and Cosmology, and I'm currently in chapter 13, symmetric spaces. I'm trying to follow his derivation of the Killing condition, and I simply cannot, for the life of me, get from equation 13.1.2 to equation 13.1.4. I plugged 13.1.3 into 13.1.2 as he says to, but what I get is very different.
13.1.2: $$g_{\mu\nu} (x) = \frac{\partial x'^\rho}{\partial x^\mu} \frac{\partial x'^\sigma}{\partial x^\nu}g_{\rho\sigma} (x')$$
And 13.1.3: $$x'^\mu = x^\mu + \epsilon \zeta^\mu (x)$$

Then, only keep the result of the substitution to first order in epsilon. When I do this, I get:
$$g_{\mu\nu} (x) = \frac{\partial x^\rho}{\partial x^\mu} \frac{\partial x^\sigma}{\partial x^\nu} g_{\rho\sigma} (x') + \epsilon \left [ \frac{\partial \zeta^\sigma (x) }{\partial x^\nu } \frac{\partial x^\rho }{\partial x^\mu } g_{\rho\sigma} (x') + \frac{\partial \zeta^\rho (x)}{\partial x^\mu } \frac{\partial x^\sigma }{\partial x^\nu } g_{\rho\sigma} (x') \right ]$$.

It's supposed to be 13.1.4: $$0 = \frac{\partial \zeta^\mu (x)}{\partial x^\rho} g_{\mu\sigma}(x) + \frac{\partial \zeta^\nu (x)}{\partial x^\sigma} g_{\rho\nu} (x) + \zeta^\mu (x) \frac{\partial g_{\rho\sigma} (x)}{\partial x^\mu}$$

All of his metrics are functions of x, not x', and he has no epsilon in the equation. That makes it seem to me that the first term on the right hand side of the equation I got has to equal the left hand side, so that they cancel and equal 0. Then the epsilon can divide out. The problem is that then there are only two terms left, as opposed to the three that he has. I'm guessing it has something to do with switching from g(x) to g(x'), but I don't see it. Any help would be greatly, greatly appreciated! Thank you very much!

 

[WannabeNewton]

Your expression reduces to $g_{\mu\nu}(x) = g_{\mu\nu}(x) + \epsilon (\zeta^{\rho}\partial_{\rho}g_{\mu\nu}(x) + g_{\mu\sigma}(x)\partial_{\nu}\zeta^{\sigma} + g_{\rho \nu}\partial_{\mu}\zeta^{\rho})$ after using the fact that $g_{\rho\sigma}(x') = g_{\rho\sigma}(x) + \epsilon \zeta^{\gamma}\partial_{\gamma}g_{\rho\sigma}(x) + O(\epsilon^2)$ hence $\zeta^{\mu}\partial_{\mu}g_{\rho\sigma}(x) + g_{\rho\nu}(x)\partial_{\sigma}\zeta^{\nu} + g_{\mu\sigma}\partial_{\rho}\zeta^{\mu} = 0$ after appropriately relabeling the indices. Don't forget that $\partial_{\nu}x^{\mu} = \delta^{\mu}_{\nu}$.

 

[TimeFall]

Thank you very much! I totally forgot about expanding the metric, as well as the delta condition.

 

[sardar]

Killing vectors are a powerful tool in the field of differential geometry and can be used to simplify complex equations in certain cases. However, they are not the sole key to solving all types of complex equations.

Killing vectors are vector fields that preserve the metric of a manifold, which is a mathematical structure that describes the properties of space. This means that when a Killing vector is applied to a metric, the resulting metric remains unchanged. This property makes Killing vectors useful in solving equations that involve the geometry of a space.

In particular, Killing vectors are often used in solving Einstein's field equations, which describe the behavior of gravity in the theory of general relativity. The equations involve the metric of spacetime, and Killing vectors can be used to simplify the equations by reducing the number of variables and making them more manageable.

However, not all equations involve the geometry of a space, and therefore, Killing vectors may not be applicable in all cases. In addition, even in cases where Killing vectors can be used, they may not be the only tool needed to solve the equations. Other mathematical techniques and concepts may also be required.

In conclusion, while Killing vectors are a valuable tool in solving certain types of complex equations, they are not the only key to solving all complex equations. A combination of different techniques and approaches may be necessary to effectively solve a complex problem in science.