5 votes

Energy-Momentum tensor in the non-relativistic limit of Klein-Gordon Field

We can ignore the cross-terms in the Klein-Gordon (KG) action from OP's expansion $$\Phi(\vec{x},t)~=~\frac{\hbar}{\sqrt{2m}}\left(\exp\left(-\frac{imc^2t}{\hbar}\right)\psi(\vec{x},t) +\exp\left(\...
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3 votes

Understanding EFE: RHS linear, LHS not?

Consider the equation \begin{equation} x^2 + 1 = x \end{equation} The left hand side is nonlinear in $x$, while the right hand side is linear, but this is still a valid equation. Actually the ...
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2 votes
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Show Energy-momentum operator transforms as a tensor under Lorentz transformations

In regards to your first question, that is, in fact, a definition. To appreciate it, let's recall what a representation is. Let $G$ be some group. A representation of $G$ is a pair $(V,D)$ where $V$ ...
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2 votes
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Lower vs Upper indices in stress energy tensor

The functional derivative flips the up/downdedness of the index, which has to be the case, because if you have an expression like [for any two arbitrary tensors $A$ and $B$]: $$\frac{\delta}{\delta A_{...
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2 votes

Does the stress energy tensor scale with the metric tensor?

What you're asking about is a specific case of Weyl transformations of the form $$ g_{ab} \rightarrow e^{-2\phi(x)}g_{ab} $$ where $\phi$ is a constant. Under these, the Ricci scalar is not invariant (...
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2 votes

Understanding EFE: RHS linear, LHS not?

This is just a small comment to buttress Andrew's answer (since it was a long string of comments on his answer!). Let us review the definitions of basic geometric gadgets used in Einstein's field ...
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How can any spatially extended object have 4-momentum assigned?

Consider a particular inertial reference frame with coordinates $\{t, x, y, z\}$, and let $t^a$, $x^a$, $y^a$, and $z^a$ be their corresponding orthonormal basis vectors. The set of events in ...
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1 vote
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Linear Momentum in General Relativity

Is there a metric that describes this case? Yes. It is the Schwarzschild metric (valid outside of the gravitating body if we are talking about something like a star). When written in the form where ...
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1 vote

Does the stress energy tensor scale with the metric tensor?

To supplement Eletie's answer, note that under the transformation $g\mapsto \lambda g$ with $\lambda$ an $\mathbb R$-valued constant, the Christoffel symbols remain unchanged because $\Gamma \sim g^{-...
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Do Einstein field equations only relate local spacetime curvature to local energy-momentum of matter?

Yes, they cannot be extended to relate global curvature to global energy-momentum, not in general at least. You can see this by noting that the Einstein Field Equations can be derived by demanding ...
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1 vote
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Energy-momentum tensor of a perfect fluid flowing at the speed of light?

Once the radiation fluid is traveling at the speed of light in a particular spatial direction, the situation effectively becomes equivalent to a mixture of plane waves of light traveling in the same ...
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1 vote

Covariant vs. contravariant definition of the Energy-Momentum tensor

The metric $g_{\mu\nu}$ can be used to raise and lower indices. In this case you are lowering the indices on the SEM tensor, that is converting it from contravariant to covariant. So the answer is ...
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1 vote

Understanding the meaning of (and using) Maxwell Stress Tensor

Field momentum: $\mu_0 \epsilon_0 \vec{S}$ is the momentum density of the field. Integrating this about a volume, finds the TOTAL field momentum in that volume. The rate of change of this, is the ...
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