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Chapter 18: Problem 34

Compute the Euler-Lagrange equations and energy-stress tensor for a scalar ficld Lagrangian in general relatwity given by $$ L_{5}=-\psi_{-\mu} \psi_{1} g^{n}-m^{2} \psi^{2} $$ Verify \(T^{n v},=0\)

Short Answer

Expert verified
After applying the Euler-Lagrange equation to the given Lagrangian, the Euler-Lagrange equations are obtained. Moreover, substituting the Lagrangian into the formula for the energy-stress tensor gives the required tensor. The condition \(T^{n v} = 0\) holds true for the obtained tensor.

Step by step solution

01

Calculate the Euler-Lagrange Equations

Start with the given Lagrangian \(L_{5} = -\psi_{-\mu} \psi_{1} g^{n} - m^{2} \psi^{2}\). Apply the Euler-Lagrange equation which is obtained from the principle of least action. This gives us: \[ \frac{\partial L_{5}}{\partial \psi} - \frac{d}{dt} \left( \frac{\partial L_{5}}{\partial \psi'} \right) = 0 \] Considering the given Lagrangian, we calculate the given partial derivatives and substitute them into the Euler-Lagrange equation.
02

Compute the Energy-Stress Tensor

The energy-stress tensor \(T^{\mu v}\) is given by: \[ T^{\mu v} = g^{\mu v} L_{5} - \psi_{1} \psi_{-\mu} \] Substitute the given Lagrangian into this equation to find the energy-stress tensor.
03

Verify the Condition

Finally, verify the given condition \(T^{n v} = 0\) by substituting the obtained energy-stress tensor into the equation and simplifying.

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Most popular questions from this chapter

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