# Tag Info

4

To find points where local flatness breaks down, a general strategy is to calculate the curvature tensor $R_{\mu\nu\rho\sigma}$, and find the locus of various singularities (points where $R_{\mu\nu\rho\sigma}$ becomes unbounded). Except in rare cases (i.e. unusual cancellation), singular behavior of $R_{\mu\nu\rho\sigma}$ is apparent in the Ricci scalar ...

3

Although the radiation-dominated (RD) era is long in comparison to the matter-dominated (MD) and $\Lambda$-dominated ($\Lambda$D) eras, it is nice to have an answer which can be adapted easily for any cosmological era. If we assume that the Universe is permeated by a perfect fluid we may use the equation of state w = \frac{P}{\rho}, ...

3

Take a future-directed timelike curve $\gamma= \gamma(\tau)$, $\tau$ being the proper time along $\gamma$ in the spacetime $M$. Assume that $p = \gamma(0)$ is the initial point of $\gamma$. Fermi coordinates adapted to $\gamma$ are constructed this way. Consider an orthonormal basis of $T_pM$ with $e_0$ parallel to $\dot{\gamma}$. Transport the basis ...

3

Indeed, $f$ is a symmetric form, since $\omega$ and $\omega '$ are Grassmann-even: $$(\text dx \wedge \text d y)\wedge (\text d z \wedge \text d t)=(\text d z \wedge \text d t)\wedge(\text dx \wedge \text d y)$$etc.. Now, to calculate the signature, you should find a basis which diagonalizes $\omega$, the dimension of the space is $6$. A basis is given ...

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Diffeomorphism Invariance Let $M$ be a smooth manifold. Let $\phi: M \to M$ be a diffeomorphism. A simple property of the Einstein equations is $$g \in \otimes^2 TM \text{ is solution to vacuum Einstein equation} \implies \text{ so is } \phi^*g$$ To see that this is true, simply pull back both sides of the Einstein equation by $\phi$, and use the ...

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The conditions about (i) differentiability of the functions and (ii) the maximal rank of the corresponding rectangular Jacobian matrix are regularization conditions imposed to simplify the mathematical analysis of the physical problem, in particular to legitimate the possible future use of the inverse function theorem. In the affirmative case, the ...

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The commutator of two vector fields $n^a$ and $X^b$ is $[n,X]^a = n^b \nabla_b X^a - X^b \nabla_b n^a$. Since this vanishes, it follows that $n^b \nabla_b X^a = X^b \nabla_b n^a$, and the second step follows from there. I don't have my copy of Wald in front of me, but I'm 99% sure that the commutator of two vector fields is defined in terms of the ...

2

I will answer this with a simple example. Let us consider the metric for weak gravity, $$ds^2 = \left(1 - \frac{2GM}{rc^2}\right)c^2dt^2 - dr^2 -r^2d\Omega^2.$$ The $g_{tt}$ metric element is largest by a factor of $c^2$ and we have $$\Gamma^r_{tt} = \frac{1}{2}g^{rr}\partial_r g_{tt} = \frac{GM}{r^2}.$$ Now let us work with the geodesic equation that is ...

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Torsion is not frame dragging. Torsion is having an anti-symmetric spacetime connection. As you do parallel transport in general relativity (GR) you drag frames the frames roll as they move. With torsion they would twist. The connection is GR is the Christopher symbols, symmetric in the two bottom indices. The torsion is an anti-symmetric tensor. It will ...

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The acceleration should be $$a = \frac{G\cdot M}{r^2 \cdot \sqrt{1-r_s/r}}$$ with $r$ as the height above the center of mass and the Schwarzschildradius $$r_s = \frac{2\cdot G\cdot M}{c^2}$$ The force to hold the ball at rest is $$F=m\cdot a$$ As one can see it now takes an infinite force and energy to keep a body at a fixed height when $r=r_s$.

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A standard reference packed to the brim with other references to everything you ever wanted to know about anomalies is "Anomalies in Quantum Field Theory" by Bertlmann. This particular topic is what comprises part of chapter 11 there. I'll highlight the main points, but this is a technical topic for which you'll have to go to the references and follow all of ...

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The solution that you wrote in your last (not numbered) equation is not a basis of a Hilbert space of sections because the phase factor: $(-1)^n e^{2i\pi A}$ depends on $n$. The phase factor should not depend on $n$. Please see your (correct) equation (2) defining the boundary conditions, in which the phase factor does not depend on $n$. Thus there is no ...

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A general diffeomorphism does not map geodesics to geodesics. Some simple counter examples You can a build diffeomorphism on the Euclidean plane by imagining putting one finger on a tablecloth at point $x$ and dragging it. This map is clearly smooth, a smooth inverse is constructed by dragging your finger back. Any geodesic on the plane (a line) passing ...

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Therefore you need to calculate the future light cone $$LC_{proper}=\int_{a(t_0)}^{a(t_1)} \frac{c\cdot a(t_1)}{\alpha^2\cdot H(\alpha )} \, \text{d}\alpha$$ In comoving coordinates you divide that by the scale factor of the time at absorption $$LC_{comoving}=\frac{LC_{proper}}{a(t_1)}$$ with H as the Hubble parameter $$H(a)=H_0\cdot ... 2 In the geometrical optics approximation light ray is represented by a null geodesic. Therefore you only need to find a null geodesic connecting points (t_0,0,0,0) and (t_1,x,0,0) for some t_1 (and this condition will determine t_1 uniquely). This is probably quite easy to do directly in this case, but in general for investigation of null curves in ... 2 You don't. Two given spacetimes can have their metrics written in the same way but may have different coordinate ranges. A simple example is just a spacetime with spatial coordinate identified , such as the cylinder spacetime :$$ds^2 = -dt^2 + d\theta Identical to Minkowski space, which is its universal cover. Of course, two things to watch out for : ...

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