Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.

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Akin to gauge field, why GR's lagrangian is not $R_{abcd}R^{abcd}$? What's the mathematical or physical meaning of $R_{abcd}R^{abcd}$?

For gauge field theory, the Lagrangian of the gauge field is $$\mathcal{L}=-\frac{1}{4}\mathrm{tr}(\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu})=-\frac{1}{8}F_{a\ \mu\nu}F^{a \ \mu\nu}$$ The field ...
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1answer
111 views

Does nature of singularity in black hole depend on material that fell in?

Electromagnetic waves have a tracesless stress energy tensor, and therefore if they are the only fields in a region of spacetime, the Ricci curvature scalar $R=0$ according to GR. However $R^{\mu\nu} ...
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730 views

Fourier Transform on a Riemannian Manifold

The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold? I've found that a similar question has been asked at Mathematics.SE but ...
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70 views

Trajectories in Rindler space with zero net time dilation

I've discovered a family of curves in Rindler space that have zero net time dilation. However I struggle to see why this should be so, i.e. what the physical significance of these curves is. My ...
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98 views

Can some components of metric be Finslerian while the others be Riemannian?

A Finsler metric reduces to a Riemann metric in case it loses its dependence on velocities. Now, my question is this: Can we have a Finsler metric in which some components of the metric have velocity ...
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84 views

Is there a null incomplete spacetime which is spacelike and timelike complete?

Geodesic completeness, the fact we can make the domain of the geodesic parametrized with respect an affine parameter the whole real line, is an important concept in GR. Especially, because the lack of ...
6
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155 views

Non-trivial scalar quantity

Is there any scalar quantity made of only the Christoffel symbols, determinant of a metric and tensors, not derivatives? In other words, can we construct a scalar quantity which cannot be written in ...
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90 views

Is it possible to have fermions in Schwarzschild spacetime?

To my understanding Geroch proved that on 4-dimensional non-compact manifold a necessary and sufficient condition for a manifold to have a notion of spinors is to be parallelizabe .1 (General ...
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1answer
508 views

Does the universe obey the holographic principle due to Stokes' theorem?

Does the universe obey the holographic principle due to Stokes' theorem? \begin{equation} \int\limits_{\partial\Omega}\omega = \int\limits_{\Omega}\mathrm{d}\omega. \end{equation} Can this ...
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1answer
565 views

Step by step algorithm to solve Einstein's equations

I cannot completely understand what is a regular method to solve Einstein's equations in GR when there are no handy hints like spherical symmetry or time-independence. E.g. how can one derive ...
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3answers
296 views

Error in books of conformal field theory?

If you look at the book Conformal Field Theory (by Philippe Francesco, Pierre Mathieu and David Senechal) or the lecture notes Applied Conformal Field Theory (by Paul Ginsparg), and many other places: ...
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164 views

Metric tensor in SRT

I just read on this webpage that we have (click me) $g_{\alpha \beta} = g_{\alpha}^{\beta} = g^{\alpha \beta}.$ Now, although I understand that the first and the last one are equal, I don't think ...
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1answer
365 views

Conservative Vector Fields

I was always told that to find whether or not a vector field is conservative, see if the curl is zero. I have now been told that just because the curl is zero does not necessarily mean it is ...
5
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1answer
325 views

Do partial derivatives commute on tensors?

Do partial derivatives commute on tensors? For example, is $$\partial_{\rho}\partial_{\sigma}h_{\mu\nu} - \partial_{\sigma}\partial_{\rho}h_{\mu\nu}=0$$ correct?
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3answers
359 views

Extending General Relativity with Kahler Manifolds?

Standard general relativity is based on Riemannian manifolds. However, the simplest extension of Riemannian manifolds seems to be Kahler manifolds, which have a complex (hermitian) structure, a ...
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3answers
154 views

Integral in different coordinate systems

In Griffiths' electrodynamics book, he uses the equation, $$\nabla^2\mathbf{A}=-\mu_0 \mathbf{J},$$ to state that $$\mathbf{A}(\mathbf{r}) = ...
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3answers
489 views

Integral form of Gauss's law for magnetism from Stokes' theorem?

How can the integral form of Gauss's law for magnetism be described as a version of general Stokes' theorem? How does it follow?
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611 views

Does spacetime in general relativity contain holes?

Are there physical models of spacetimes, which have bounded (four dimensional) holes in them? And do the Einstein equations give restrictions to such phenomena? Here by holes I mean ...
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3answers
526 views

Curvature of Conical spacetime

Inspired by: Angular deficit The 2+1 spacetime is easier for me to visualize, so let's use that here. (so I guess the cosmic string is now just a 'point' in space, but a 'line' in spacetime) Edward ...
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759 views

Confusion about Lie derivative on metric

According to this site, the Lie derivative of a $(0,2)$-tensor is $$ \mathcal{L}_XT_{ab}=\partial_XT_{ab}+T_{cb}\partial_aX^c+T_{ac}\partial_bX^c $$ However, according the same website, the Lie ...
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2answers
306 views

Exterior (covariant) derivatives and electromagnetism

I'm porting Maxwell's equations to curved spacetime and am having trouble reconciling the tensor and forms treatments. I think the problem boils down to a misunderstanding on my part concerning the ...
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3answers
304 views

What are some mechanics examples with a globally non-generic symplecic structure?

In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be ...
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142 views

Momentum is a cotangent vector?

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...
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1answer
116 views

Two expressions for topological instanton number

I have begun to study instantons and I have the following difficulty: $\newcommand{tr}{\operatorname{Tr}}$ I am considering theory with $SU(2)$ gauge group: $S=\frac{1}{2g^{2}}\int \tr ...
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1answer
91 views

how affine connection follows from Two derivative operator

IN wald's GR book in chapter 3 This is stated behind the definition of affine connection : First He showed that if we have two derivative operator $\nabla_a , \tilde\nabla_a$ (both of which ...
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What's the idea behind the Riemann curvature tensor?

The Riemann curvature tensor can be expressed using the Christoffel symbols like this: $R^m{}_{jkl} = \partial_k\Gamma^m{}_{lj} - \partial_l\Gamma^m{}_{kj} + \Gamma^m{}_{ki}\Gamma^i{}_{lj} ...
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1answer
266 views

Kaluza-Klein Christoffel Symbols

I have a question regarding the connection coefficients as they pertain to the following paper: http://www.weylmann.com/kaluza.pdf . When I try to calculate the 4D Christoffel symbols from the 4D part ...
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1answer
926 views

Is 4-volume element a scalar or a pseudoscalar in special relativity?

In general relativity 4-volume element $\mathrm{d}^4 x = \mathrm{d} x^0\mathrm{d} x^1 \mathrm{d} x^2\mathrm{d} x^3$ is clearly a pseudoscalar (or scalar density) of weight 1 since it transforms as ...
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1answer
210 views

Does magnetic monopole violate $U(1)$ gauge symmetry?

Does a magnetic monopole violate $U(1)$ gauge symmetry? In what sense and why? Insofar as I know, there are at least two types of magnetic monopoles. One is the Dirac monopole while the other is the ...
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1answer
183 views

What is the relationship between Cyclic Coordinates and Killing Vector Fields?

My question is related to this question. There are three or four other questions on Killing Vector Fields here, however none of them that I have seen address my question. $\\$ I've been studying ...
5
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1answer
145 views

$\mathcal{N}=2$ supersymmetry and $SU(2)$ holonomy

I was reading this Phys.SE question. I was unable to understand how an $SU(2)$ holonomy would produce $\mathcal{N}=2$ in four dimensions. Could anyone shed some light on this?
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3answers
170 views

How can a Physical law not be invariant?

In Relativity, both the old Galilean theory or Einstein's Special Relativity, one of the most important things is the discussion of whether or not physical laws are invariant. Einstein's theory then ...
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1answer
465 views

Geodesic deviation equation - why does the ordinary second derivative give the correct answer?

I've calculated the correct answer to my problem, but don't understand one of the assumptions I made when doing so. I used the geodesic deviation equation ...
5
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3answers
415 views

Global vs. local gauge group in mathematical sense - physics examples?

Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group: Local gauge group $G$. Corresponds to the fibers of the ...
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4answers
313 views

Kähler and complex manifolds

I was wondering if anyone knows any good references concerning Kähler manifolds and complex manifolds? I am studying supergravity theories and for the simplest $\mathcal{N}=1$ supergravity we will get ...
5
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1answer
159 views

Extent of coordinate freedom to set metric components along a spacetime path

If we describe spacetime with a Lorentzian manifold, it is always possible to choose a coordinate system such that at any particular point $x^\alpha$, the components of the metric are: $$ ...
5
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1answer
257 views

Ricci tensor of the orthogonal space

While reading this article I got stuck with Eq.$(54)$. I've been trying to derive it but I can't get their result. I believe my problem is in understanding their hints. They say that they get the ...
5
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1answer
61 views

What's the definition of incompleteness of a coordinate system and a spacetime?

I always see in GR textbooks that some coordinates or some spacetime is incomplete, such as Rindler spacetime and spacetially flat FRW universe with only positive cosmological constant. This ...
5
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1answer
329 views

Bondi-Metzner-Sachs (BMS) symmetry of asymptotically flat space-times

I started studying the BMS symmetry in connection with the paper: http://arxiv.org/abs/1312.2229 and there are a few strange things I noticed. First of all, from reading the original papers by Bondi, ...
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1answer
225 views

Are conformal, Killing and homothetic vector fields the same in pseudo-riemannian manifolds?

I work in the Lorentzian manifolds, more generally in pseudo Riemannian manifolds and applications to general relativity. I know the definitions of conformal, Killing and homothetic vector fields in ...
5
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1answer
408 views

6 independent Einstein field equations?

I can't understand the comment on page 409, Gravitation, by Misner, Thorne, Wheeler It follows that the ten components $G_{\alpha\beta} =8\pi T_{\alpha\beta}$ of the field equation must not ...
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3answers
327 views

Is there a natural (suitable) definition for functional derivative in Curved space time

If $$\delta S = \int \sqrt g F[\phi] \delta \phi\tag{1}$$ Then is it natural to define the functional derivative as follows, $$\frac{\delta S}{\delta \phi} = F[\phi].\tag{2}$$ In particular does ...
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1answer
118 views

Help with the understanding of boundary conditions on $AdS_3$

So I am trying to reproduce results in this article, precisely the 3rd chapter 'Virasoro algebra for AdS$_3$'. I have the metric in this form: ...
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2answers
906 views

Electromagnetism for Mathematician

I am trying to find a book on electromagnetism for mathematician (so it has to be rigorous). Preferably a book that extensively uses Stoke's theorem for Maxwell's equations (unlike other books that on ...
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90 views

How is foliation of manifolds' theory useful in General Relativity?

I am interested on getting some hints on how Foliations Theory of Manifolds can be used fruitfully on General Relativity. I just started my Ph.D on Mathematics this semester focusing on studying ...
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155 views

Why is the Ricci tensor diagonal for isotropic spacetime?

I'm reading Zee's Einstein Gravity in a Nutshell and while calculating the Ricci tensor for FRW spacetime he claims that because the spacelike slices of constant $t$ are rotationally invariant, the ...
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106 views

Intuitively what's the relationship between forces and connections?

In Einstein's General Relativity we relate the effects of gravity with the curvature of the Levi-Civita connection on the spacetime manifold. Also, when we get the electromagnetic tensor $F = dA$ ...
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165 views

Heat kernel expansion for entanglement entropy

Can somebody please let me know where I can find a reference for calculating heat kernel coefficients on a manifold with conical singularities? I am trying to compute the entanglement entropy for ...
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211 views

An introductory resource for learning AdS space

Can someone please point me to introductory resources about the geometry of Anti DeSitter Space ? What are some examples of other spaces used in theoretical physics ?.I'm learning Differential ...
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7k views

Why does light always travel in a straight line?

No matter the frame light is in, it always moves in a straight line in that frame. Why is that? It doesn't seem like something to me that should necessarily be true. If some one runs forward and sends ...