Killing Vectors in Schwarzschild Metric

Question

Given the Schwarzschild metric with $(-,+,+,+)$ signature,

$$\text ds^2=-\left(1-\frac{2M}{r}\right)dt^2+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$$

the lack of dependence of the metric on $t$ and $\phi$ allow us to read off the Killing vectors $K_1=\partial_t$ and $K_2=\partial_{\phi}$. These vectors, in their coordinate representations, are given by

$$K_1=\left(-\left(1-\frac{2M}{r}\right),0,0,0\right)$$

$$K_2=\left(0,0,0,r^2\sin^2\theta\right)$$

How does one immediately read off those vector components for $K_1$ and $K_2$? What is the logic behind reading them off? How would I "read off the Killing vectors" if I, while maintaining no explicit dependence on $t$ or $\phi$, added some off-diagonal terms to the metric? Please help me intuitively understand what's going on here.

Answer 1

If all components of the metric are independent of some particular $x^\nu$, then you have the killing vector $\vec{K}$ with components $K^\mu = \delta^\mu_\nu$. That is, the contravariant form just has a constant in the appropriate slot and zeros elsewhere. In Schwarzschild, you have $K^\mu = (1, 0, 0, 0)$ and $R^\mu = (0, 0, 0, 1)$ ($\vec{K}$ and $\vec{R}$ being your $K_1$ and $K_2$, respectively).

To find the covariant forms, simply lower with the metric. In Schwarzschild we have \begin{align} K_\mu & = g_{\mu\nu} K^\nu = g_{\mu t} = \big({-}(1-2M/r), 0, 0, 0\big) \\ R_\mu & = g_{\mu\nu} R^\nu = g_{\mu\phi} = \big(0, 0, 0, r^2 \sin^2\!\theta\big). \end{align} This is where off-diagonal terms would come in. For example, in Boyer-Lindquist we also have no $t$-dependence, so we have $K^\mu = (1, 0, 0, 0)$ and $$ K_\mu = g_{\mu t} = \big({-}(1-2Mr/\Sigma), 0, 0, -(2Mar/\Sigma)\sin^2\!\theta\big), $$ where the fourth component is precisely $g_{t\phi}$.

Answer 2

Conceptually:

If we leave the mathematical definition aside for a while, we can define the killing vector:

Killing vector $K^\mu(x)$ leaves metric unchanged under infinitesimal coordinate changes

Time coordinate

Change in $t$ does nothing to the metric:

Change in $t\rightarrow t+dt$:

$g_{\mu \nu}=g_{\mu \nu}(t)=g_{\mu \nu}(t+dt)=g_{\mu \nu}$

So one of the killing vectors should be along $t$:

$K^\mu=(1,0,0,0)$

That's it!

Note: The form you have has the indice lowered: $K_\mu=g_{\mu \nu}K^\nu=(g_{\mu0}K^0,0,0,0)=(-(1-2M/r),0,0,0)$