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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

Let \((M, \varphi)\) be a surface of revolution defined as a submantfold of \(\mathbb{E}^{3}\) of the form $$ r=g(u) \cos \theta, \quad y=g(u) \sin \theta, \quad z=h(u) $$ Show that the induced metric (see Example 18.1) is $$ \mathrm{d} s^{2}=\left(g^{\prime}(u)^{2}+h^{\prime}(u)^{2}\right) \mathrm{d} u^{2}+g^{2}(u) \mathrm{d} \theta^{2} $$ Picking the parameter \(u\) such that \(g^{\prime}(u)^{2}+h^{\prime}(u)^{2}=1\) (interpret this choice!), and setting the basis 1-forms to be \(\varepsilon^{\prime}=\mathrm{d} u, \varepsilon^{2}=g d \theta\), calculate the connection I-forms \(\omega^{\prime}\), the curvature 1 -forms \(\rho_{\prime}^{\prime}\), and the curvature tensor component \(R_{1212}\).

Show that the rodiation filled universe, \(P=\frac{1}{3} \rho\) has \(\rho \propto a^{-4}\) and the time evolution for \(k=0\) is given by \(a \propto t^{1 / 2}\). Assuming the radration is black body, \(\rho=a_{3} T^{4}\), where \(a_{5}=7.55 \times\) \(10^{-15} \mathrm{erg} \mathrm{cm}^{-3} \mathrm{~K}^{-4}\), show that the temperature of the unverse evolves with time as $$ T=\left(\frac{3 c^{2}}{32 \pi G a_{\mathrm{s}}}\right)^{1+} t^{-1 / 2}=\frac{1.52}{\sqrt{t}} \mathrm{~K} \quad(t \text { in seconds }) $$

(a) A particle falls radially inwards from rest at in finity in a Schwarzschild solution. Show that it will arrive at \(r=2 m\) m a finite proper time after crossing some fixed reference position \(r_{0}\), but that coordinate time \(t \rightarrow \infty\) as \(r \rightarrow 2 m\). (b) On an infalling extended body compute the tidal force in a radual direction, by parallel propagating a tetrad (only the radial spacelike unit vector need he considered) and calculating \(R_{1414}\). (c) Estimate the total tidal force on a person of height \(1.8 \mathrm{~m}\), weighing \(70 \mathrm{~kg}\), fallang head-first into a solar mass black hole \(\left(M_{3}=2 \times 10^{10} \mathrm{~kg}\right)\), as he crosses \(r=2 \mathrm{~m}\).

Show that every two-dimensional space-time metric (signature 0 ) can be expressed locally in confor mal coontinates $$ \mathrm{ds}^{2}=\mathrm{e}^{2 \varphi}\left(\mathrm{d} x^{2}-\mathrm{d} t^{2}\right) \text { where } \phi=\phi(x, t) $$ Calculate the Rucmann curvature tensor component \(R_{1212}\), and writc out the two-dimensional Enstein vacuum equations \(R_{u j}=0\). What is their general solunon?

(a) Compute the components of the Ricer tensor } R_{\mu v} \text { for a space-tume that has a }\end{array}\( metric of the form $$ \mathrm{d} s^{2}=\mathrm{dx}^{2}+\mathrm{d} v^{2}-2 \mathrm{~d} u \mathrm{~d} v+2 H \mathrm{~d} v^{2} \quad(H=H(\mathrm{x}, y, u, v)) $$(b) Show that the space-time is a vacuum if and only if \)H=\alpha(x, y, v)+f(v) u\( where \)f(v)\( is an arbitrary function and \)\alpha\( sat?sfies the two-dimensional Laplace equation $$ \frac{\partial^{2} \alpha}{\partial x^{2}}+\frac{\partial^{2} \alpha}{\partial y^{2}}=0 $$ and show that it is possible to set \)f(v)=0\( by a coordunate transformation \)u^{\prime}=u g(v), v^{\prime}=h(v)\(. (c) Show that \)R_{\text {taj } 4}=-H_{v}\( for \)i, j=1,2$.
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