Energy-Momentum Tensor for a particle

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PML
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Hello everyone,

I was studying how to define, formally, an energy-momentum tensor for a point particle.

I was reading this two references:http://academic.reed.edu/physics/courses/Physics411/html/page2/files/Lecture.19.pdf , page 1; and http://th-www.if.uj.edu.pl/acta/vol29/pdf/v29p1033.pdf page 1038.

They both start from an action that I understand, it's just the action for a particle moving in space time. They, then, differentiate the action with respect to the metric to get the energy-momentum tensor, but then, somehow, they arrive at an expression that has the dirac delta in it...
Can anyone help me out?

Thank you
 

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They, then, differentiate the action with respect to the metric to get the energy-momentum tensor, but then, somehow, they arrive at an expression that has the dirac delta in it...

That's because, as noted on page 2 of your first reference, the action in question needs to be a Lagrangian density, i.e., something you integrate over a 4-volume; but equation 19.1 of that paper, the usual action for a point particle in SR, is not a Lagrangian density. The delta function comes in when you try to make it one (basically because you have to pick out just the points in the 4-volume that lie on the point particle's worldline).
 

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