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Energy-Momentum Tensor for the Klein-Gordon Lagrangian

  • #1
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Homework Statement



The energy-momentum tensor ##T^{\mu\nu}## of the Klein-Gordon Lagrangian ##\mathcal{L}_{KG} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}## is given by

$$T^{\mu\nu}~=~\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG}.$$
Show that ##\partial_{\mu}T^{\mu\nu}=0##.

Homework Equations



The Attempt at a Solution



$$\partial_{\mu}T^{\mu\nu} \\
=\partial_{\mu}[\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG}]\\
=\partial_{\mu}(\partial^{\mu}\phi\partial^{\nu}\phi-\partial^{\nu}\mathcal{L}_{KG})\\
=(\partial_{\mu}\partial^{\mu}\phi)(\partial^{\nu}\phi)-\partial^{\nu}(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2})\\
=(\partial_{\mu}\partial^{\mu}\phi)(\partial^{\nu}\phi)-\partial^{\nu}(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi)+m^{2}\phi(\partial^{\nu}\phi)$$

Where do I go from here? I know I need to use the Klein-Gordon equation, but using the KG equation cancels the first and third terms and leaves the second term.
 

Answers and Replies

  • #2
TSny
Homework Helper
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$$\partial_{\mu}T^{\mu\nu} \\
=\partial_{\mu}[\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG}]\\
=\partial_{\mu}(\partial^{\mu}\phi\partial^{\nu}\phi-\partial^{\nu}\mathcal{L}_{KG})$$
The right parenthesis is misplaced in last line above.

$$=(\partial_{\mu}\partial^{\mu}\phi)(\partial^{\nu}\phi)-\partial^{\nu}(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2})$$
Did you miss a term when writing out ##\partial_{\mu}(\partial^{\mu}\phi\partial^{\nu}\phi)##?
 
  • #3
203
83
μνμνφ + m2φ = 0
 
  • #4
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Ok. Let me start again.

##\partial_{\rho}T^{\mu\nu} = \partial_{\rho}(\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG})##

##=(\partial_{\rho}\partial^{\mu}\phi)(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial_{\rho}\partial^{\nu}\phi)-\eta^{\mu\nu}\partial_{\rho}\mathcal{L}_{KG}##

##=(\partial_{\rho}\partial^{\mu}\phi)(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial_{\rho}\partial^{\nu}\phi)-\eta^{\mu\nu}\partial_{\rho}(\frac{1}{2}\partial_{\sigma}\phi\partial^{\sigma}\phi-\frac{1}{2}m^{2}\phi^{2})##

##=(\partial_{\rho}\partial^{\mu}\phi)(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial_{\rho}\partial^{\nu}\phi)-\frac{1}{2}\eta^{\mu\nu}(\partial_{\rho}\partial_{\sigma}\phi)(\partial^{\sigma}\phi)-\frac{1}{2}\eta^{\mu\nu}(\partial_{\sigma}\phi)(\partial_{\rho}\partial^{\sigma}\phi)+\eta^{\mu\nu}m^{2}\phi(\partial_{\rho}\phi)##

Where do I go from here?
 
  • #5
TSny
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##=(\partial_{\rho}\partial^{\mu}\phi)(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial_{\rho}\partial^{\nu}\phi)-\frac{1}{2}\eta^{\mu\nu}(\partial_{\rho}\partial_{\sigma}\phi)(\partial^{\sigma}\phi)-\frac{1}{2}\eta^{\mu\nu}(\partial_{\sigma}\phi)(\partial_{\rho}\partial^{\sigma}\phi)+\eta^{\mu\nu}m^{2}\phi(\partial_{\rho}\phi)##

Where do I go from here?
Are the 3rd and 4th terms related?

You want to investigate the divergence ##\partial_{\mu}T^{\mu \nu}## rather than ##\partial_{\rho}T^{\mu \nu}##
 
  • #6
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Are the 3rd and 4th terms related?
Yes. The third and fourth terms are identical as follows:

##(\partial_{\rho}\partial^{\mu}\phi)(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial_{\rho}\partial^{\nu}\phi)-\frac{1}{2}\eta^{\mu\nu}(\partial_{\rho}\partial_{\sigma}\phi)(\partial^{\sigma}\phi)-\frac{1}{2}\eta^{\mu\nu}(\partial_{\sigma}\phi)(\partial_{\rho}\partial^{\sigma}\phi)+\eta^{\mu\nu}m^{2}\phi(\partial_{\rho}\phi)##

##=(\partial_{\rho}\partial^{\mu}\phi)(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial_{\rho}\partial^{\nu}\phi)-\frac{1}{2}\eta^{\mu\nu}(\partial_{\rho}\partial^{\sigma}\phi)(\partial_{\sigma}\phi)-\frac{1}{2}\eta^{\mu\nu}(\partial_{\sigma}\phi)(\partial_{\rho}\partial^{\sigma}\phi)+\eta^{\mu\nu}m^{2}\phi(\partial_{\rho}\phi)##

##=(\partial_{\rho}\partial^{\mu}\phi)(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial_{\rho}\partial^{\nu}\phi)-\frac{1}{2}\eta^{\mu\nu}(\partial_{\sigma}\phi)(\partial_{\rho}\partial^{\sigma}\phi)-\frac{1}{2}\eta^{\mu\nu}(\partial_{\sigma}\phi)(\partial_{\rho}\partial^{\sigma}\phi)+\eta^{\mu\nu}m^{2}\phi(\partial_{\rho}\phi)##

##=(\partial_{\rho}\partial^{\mu}\phi)(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial_{\rho}\partial^{\nu}\phi)-\eta^{\mu\nu}(\partial_{\sigma}\phi)(\partial_{\rho}\partial^{\sigma}\phi)+\eta^{\mu\nu}m^{2}\phi(\partial_{\rho}\phi)##

You want to investigate the divergence ##\partial_{\mu}T^{\mu \nu}## rather than ##\partial_{\rho}T^{\mu \nu}##
Okay. Let me relabel ##\rho## to ##\mu##. Then, I have

##=(\partial_{\mu}\partial^{\mu}\phi)(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial_{\mu}\partial^{\nu}\phi)-\eta^{\mu\nu}(\partial_{\sigma}\phi)(\partial_{\mu}\partial^{\sigma}\phi)+\eta^{\mu\nu}m^{2}\phi(\partial_{\mu}\phi)##

##=(\partial_{\mu}\partial^{\mu}\phi)(\partial^{\nu}\phi)+(\partial_{\mu}\phi)(\partial^{\mu}\partial^{\nu}\phi)-(\partial_{\sigma}\phi)(\partial^{\nu}\partial^{\sigma}\phi)+m^{2}\phi(\partial^{\nu}\phi)##

##=(\partial_{\mu}\partial^{\mu}\phi)(\partial^{\nu}\phi)+(\partial_{\sigma}\phi)(\partial^{\sigma}\partial^{\nu}\phi)-(\partial_{\sigma}\phi)(\partial^{\nu}\partial^{\sigma}\phi)+m^{2}\phi(\partial^{\nu}\phi)##, where the second and third terms will now cancel

##=(\partial_{\mu}\partial^{\mu}\phi)(\partial^{\nu}\phi)+m^{2}\phi(\partial^{\nu}\phi)##

##=(\partial_{\mu}\partial^{\mu}\phi+m^{2}\phi)(\partial^{\nu}\phi)##, where we will now use the Klein-Gordon equation

##=0##.

I think it's all correct now, isn't it?
 
  • #7
TSny
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Looks nice.
 
  • #8
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Thanks!
 

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