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So I know what these are 4 translation : $\frac{\partial}{\partial_ x^{u}} = \partial_{x^u}$ 3 boost: $z\partial_y - y \partial_z$ and similar for $x,z$ and $ y,x$ 3 rotation: $t\partial_x + x\partial_t $ and similar for $y , z$

however I want to do it by solving Killing equation:

$\nabla_u V^v + \nabla_v V^u =0 $ So in flat space, these $\nabla_u$ reduce to partial derivatives $\partial_u$

So Killing equation reduces to : $\partial_u V^v + \partial_v V^u=0$

Without writing things out explicitly, e.g the time translation $\partial x^0 = (1,0,0,0) $ I am confused how to work in index notation. To begin, the translations $\partial_{x^u}$ are covector and not vector, the killing equation works in vector, so rather do I need $\frac{\partial}{\partial x_u}$ instead of $\frac{\partial}{\partial x^u}$, I don't know what this is explictly?

Further I am confused with the indices in the boosts and the rotations, so the translations are given as covectors, which we can raise an index to get a vector but isn't something like:

$z\partial_y - y\partial_z$ a covector multiplied by a vector and so not a vector but a scalar, since $x^u=x,y,z,t$ is a vector but $\partial_x^u $ is a covector.

Anyway once I've cleared these up the HINT is to differentiate Killing equation and then solve the ODE. Should I do $\partial_u$ or should I choose a different index not already in Killing equation. Does it matter? I dont see how we can convert this PDE itno an ODE since it already has $\partial_u$ and $\partial_v$, if I hit it with $\partial_u$ I get a $\partial^2_u$ but then also the mixed term $\partial^2_uv$

Many thanks

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Ok, I will provide you with a sketch, but you have to fill in the details. So the Killing equation is: $\partial_\mu V_\nu + \partial_\nu V_\mu = 0$. Let $A_{\mu\nu} = \partial_{\mu}V_\nu$. By the Killing equation, we have that $A_{\nu\mu} = -A_{\mu\nu}$. In other words, $A$ is skew-symmetric in its two indices.

Now define $B_{\rho\mu\nu} = \partial_\rho A_{\mu\nu}$. Then $B$ is symmetric in its first two indices, by Euler's theorem, and skew-symmetric in its last two indices, by the Killing equation. Exercise, by playing with indices, show that this implies that $B$ must be identically $0$.

Deduce then that $(A_{\mu\nu})$ must be a constant skew-symmetric matrix, which we denote by $(c_{\mu\nu})$. Then solve the equation $\partial_\mu V_\nu = c_{\mu\nu}$.