- #1
GooberGunter
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- Homework Statement
- Prove that the Energy-Momentum Tensor for the Klein-Gordon Equation does not diverge
$$\partial_\mu T^{\mu\nu}=0$$
- Relevant Equations
- $$T^{μν}=∂^μϕ∂^νϕ−η^{μν}L$$
$$L=\frac{1}{2}\partial^2 \phi - \frac{1}{2}m^2 \phi^2$$
I've tried this problem so, so, so so so many times. Given the equations above, the proof starts easily enough:
This solutions manual from a UMD course skips a step between (31) and (32) and too readily removes the 1/2 coefficient in (31). I'm sure its a typo.
This solution from another post just raises the index of the first derivative and lowers the sigma index in the second derivative, but the closest I can replicate is:
$$=\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \frac{1}{2}(\eta^{\mu\nu} \delta^{\sigma}_\nu\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi + \partial^\sigma \phi \cdot \partial_\mu \eta^{\mu\nu} \delta^{\sigma}_\nu \partial_\sigma \phi)$$
$$=\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \frac{1}{2}(\partial^\nu \phi \cdot \partial_\mu \partial^\nu\phi + \partial^\nu \phi \cdot \partial_\mu \partial^\nu \phi)$$
Depending on what you decide to contract, you end up with 2 equations that don't resolve to 0.
$$\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi -\partial^\nu \phi \cdot \partial_\mu \partial^\nu\phi$$
$$\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \partial^\mu \phi \cdot \partial_\mu \partial^\mu \phi$$
My understanding of tensor notation is still a little shaky, I only studied a chapter from a mathematical methods textbook. Where did I go wrong? Any help is greatly appreciated!
$$\partial_\mu T^{\mu\nu}=\partial_\mu (∂^μ ϕ∂^ν ϕ)-\eta^{\mu\nu}\partial_\mu[\frac{1}{2}∂^2ϕ−\frac{1}{2}m^2ϕ^2]$$
apply product rule to all terms
$$=\partial^\nu \phi \cdot \partial_\mu \partial^\mu \phi + \partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \eta^{\mu\nu}[\frac{1}{2}(\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi + \partial^\sigma \phi \cdot \partial_\mu \partial_\sigma \phi)-\frac{1}{2}m^2\partial_\mu(\phi^2)]$$
And the equation of motion starts to appear:
$$=\partial^\nu \phi \cdot \partial_\mu \partial^\mu \phi + \eta^{\mu\nu}(m^2 \phi \partial_\mu \phi) +\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \eta^{\mu\nu}\frac{1}{2}(\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi + \partial^\sigma \phi \cdot \partial_\mu \partial_\sigma \phi)$$
Finally we can eliminate 2 of the 5 terms
$$=\partial^\nu \phi \cdot (\partial_\mu \partial^\mu \phi + m^2 \phi) +\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \eta^{\mu\nu}\frac{1}{2}(\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi+ \partial^\sigma \phi \cdot \partial_\mu \partial_\sigma \phi)$$
$$=\partial^\nu \phi[0] +\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \eta^{\mu\nu}\frac{1}{2}(\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi + \partial^\sigma \phi \cdot \partial_\mu \partial_\sigma \phi)$$
This is the last step I arrive at
$$=\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \eta^{\mu\nu}\frac{1}{2}(\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi + \partial^\sigma \phi \cdot \partial_\mu \partial_\sigma \phi)$$
Every solution I've found online has an error or skips this crucial step.apply product rule to all terms
$$=\partial^\nu \phi \cdot \partial_\mu \partial^\mu \phi + \partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \eta^{\mu\nu}[\frac{1}{2}(\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi + \partial^\sigma \phi \cdot \partial_\mu \partial_\sigma \phi)-\frac{1}{2}m^2\partial_\mu(\phi^2)]$$
And the equation of motion starts to appear:
$$=\partial^\nu \phi \cdot \partial_\mu \partial^\mu \phi + \eta^{\mu\nu}(m^2 \phi \partial_\mu \phi) +\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \eta^{\mu\nu}\frac{1}{2}(\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi + \partial^\sigma \phi \cdot \partial_\mu \partial_\sigma \phi)$$
Finally we can eliminate 2 of the 5 terms
$$=\partial^\nu \phi \cdot (\partial_\mu \partial^\mu \phi + m^2 \phi) +\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \eta^{\mu\nu}\frac{1}{2}(\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi+ \partial^\sigma \phi \cdot \partial_\mu \partial_\sigma \phi)$$
$$=\partial^\nu \phi[0] +\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \eta^{\mu\nu}\frac{1}{2}(\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi + \partial^\sigma \phi \cdot \partial_\mu \partial_\sigma \phi)$$
This is the last step I arrive at
$$=\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \eta^{\mu\nu}\frac{1}{2}(\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi + \partial^\sigma \phi \cdot \partial_\mu \partial_\sigma \phi)$$
This solutions manual from a UMD course skips a step between (31) and (32) and too readily removes the 1/2 coefficient in (31). I'm sure its a typo.
This solution from another post just raises the index of the first derivative and lowers the sigma index in the second derivative, but the closest I can replicate is:
$$=\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \frac{1}{2}(\eta^{\mu\nu} \delta^{\sigma}_\nu\partial_\sigma \phi \cdot \partial_\mu \partial^\sigma \phi + \partial^\sigma \phi \cdot \partial_\mu \eta^{\mu\nu} \delta^{\sigma}_\nu \partial_\sigma \phi)$$
$$=\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \frac{1}{2}(\partial^\nu \phi \cdot \partial_\mu \partial^\nu\phi + \partial^\nu \phi \cdot \partial_\mu \partial^\nu \phi)$$
Depending on what you decide to contract, you end up with 2 equations that don't resolve to 0.
$$\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi -\partial^\nu \phi \cdot \partial_\mu \partial^\nu\phi$$
$$\partial^\mu \phi \cdot \partial_\mu \partial^\nu \phi - \partial^\mu \phi \cdot \partial_\mu \partial^\mu \phi$$
My understanding of tensor notation is still a little shaky, I only studied a chapter from a mathematical methods textbook. Where did I go wrong? Any help is greatly appreciated!