# I Derivation geodesics equation from EM tensor of point particle

#### sergiokapone

The energy-momentum tensor of a free particle with mass $m$ moving along its worldline $x^\mu (\tau )$ is

T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}.\tag{2}

The covariant derivative of tensor gives

\nabla_{\mu} T^{\mu\nu} = \frac{1}{\sqrt{-g}} \frac{\partial \left( \sqrt{-g} T^{\mu\nu}\right) }{\partial y^{\mu}} + \Gamma^{\nu}_{\mu\lambda}T^{\mu\lambda}

And a derivation of geodesics equation from energy-momentum tensor of point particle in GR $\nabla_{\mu} T^{\mu\nu} = 0$ looks like:

\begin{align}

0 &~~~~~~=~ \frac{1}{\sqrt{-g(y)}}\partial^{(y)}_{\mu} \left(\sqrt{-g(y)}T^{\mu\nu}(y)\right)
+\Gamma^{\nu}_{\mu\lambda}(y) T^{\mu\lambda}(y) \cr
&~~~~~~=~\frac{m}{\sqrt{-g(y)}} \int \!\mathrm{d}\tau ~\dot{x}^{\nu}\dot{x}^{\mu}\partial^{(y)}_{\mu}\delta^4(y\!-\!x(\tau )) ~+~\Gamma^{\nu}_{\mu\lambda}(y) T^{\mu\lambda}(y) \cr
&~~~~~~=~\underbrace{-\frac{m}{\sqrt{-g(y)}} \int \!\mathrm{d}\tau ~\dot{x}^{\nu} \frac{d}{d\tau}\delta^4(y\!-\!x(\tau ))}_\text{this term we integrating by parts}
~+~\Gamma^{\nu}_{\mu\lambda}(y) T^{\mu\lambda}(y) \cr
&\stackrel{\text{int. by parts}}{=}~\frac{m}{\sqrt{-g(y)}}\underbrace{ \dot{x}^{\nu}\delta^{(4)}(y−x(τ))}_\text{shoul be =0! Why?} + \cr
&~~~~~\frac{m}{\sqrt{-g(y)}} \int \!\mathrm{d}\tau\underbrace{\left[\ddot{x}^{\nu}+ \Gamma^{\nu}_{\mu\lambda}(x(\tau))\dot{x}^{\mu}\dot{x}^{\lambda} \right]}_\text{=0 geodesics equation}\delta^4(y\!-\!x(\tau )) .
\end{align}

Can any one explain where did the term $uv=\dot{x}^{\nu}\delta^{(4)}(y−x(τ))$ go after integration by parts $\int udv=uv−\int vdu$ in the last term (5)?

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#### Gaussian97

Well, you can think that when evaluating the limits of the integral $\delta^{(4)}(y-x(\tau))$ will become 0 because it's basically 0 everywhere. For more rigorous ways you should study generalized functions and it's derivatives. You can look at the Gel'Fand and Shilov book "Generalized Functions Volume I".

#### sergiokapone

Well, you can think that when evaluating the limits of the integral $\delta^{(4)}(y-x(\tau))$ will become 0 because it's basically 0 everywhere. For more rigorous ways you should study generalized functions and it's derivatives. You can look at the Gel'Fand and Shilov book "Generalized Functions Volume I".
Can you write appropriate formulae from Gel'Fand and Shilov book's here?

#### Gaussian97

Well, is not a formula, is an entire book dedicated to generalized functions, but maybe you need the equation of Example 7 of Chapter 2:

$$\int \delta'(x-h)\varphi(x)\text{d}x=-\varphi'(h)$$

#### sergiokapone

Yes, this formula explain the transition from (4) to (6), but why then mentioned integration by parts?

#### Gaussian97

Maybe because the author knows that the lector may not be familiarized with derivatives of generalized functions and then uses the argument I said before of "integrating by parts and supposing that $\delta$ outside an integral is 0."

#### sergiokapone

Well, you can think that when evaluating the limits of the integral δ(4)(y−x(τ))δ(4)(y−x(τ))\delta^{(4)}(y-x(\tau)) will become 0 because it's basically 0 everywhere.
"integrating by parts and supposing that δδ\delta outside an integral is 0."
Then I conclude,
$\dot{x}\delta (x) = 0$ everywrehe, even in $x=0$. Does it right?

#### Gaussian97

Well, you have to be very careful because $\delta$ is not a usual function, but most of the times you can think of it as a function that is 0 everywhere and is not defined at $x=0$. Usually having a $\delta$ outside an integral is not a good idea.

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