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jameson2
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Homework Statement
Given the the Lagrangian density [tex] L= \frac{1}{2}\partial_\lambda\phi\partial^\lambda\phi + \frac{1}{3}\sigma\phi^3 [/tex]
(a)Work out the equation of motion.
(b)Calculate from L the stress tensor: [tex] T^{\mu\nu}=\frac{\partial L}{\partial(\partial_\mu\phi)}\partial^\nu\phi - g^{\mu\nu}L [/tex] where g is diagonal with matrix entries (1,-1,-1,-1).
(c)Find the 4-divergence of the stress tensor, [tex] \partial_\mu T^{\mu\nu} [/tex]
(d)Show that the stress tensor is conserved by demonstrating its 4-divergence is zero when the scalar field obeys its equation of motion i.e.[tex] \partial_\mu T^{\mu\nu} =0 [/tex]
Homework Equations
Euler Lagrange Equation of Motion
The Attempt at a Solution
(a) I think that I have this right : [tex] \partial_\mu(\partial^\mu\phi)-\sigma \phi^2=0 [/tex]
(b)I have that the first term in the stress tensor is [tex] \frac{\partial L}{\partial(\partial_\mu\phi)}\partial^\nu\phi=\partial^\mu\phi\partial^\nu\phi [/tex] but I don't know how to treat the second part, i.e. [tex] g^{\mu\nu}L=g^{\mu\nu}(\frac{1}{2}\partial_\lambda\phi\partial^\lambda\phi + \frac{1}{3}\sigma\phi^3)=?[/tex]
I just need to know how to treat the metric tensor g.
Obviously I haven't got to (c) or (d) yet, as I need the answer to (b).
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