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

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Relationship between those two “exponentials”

Let $G$ be a Lie group and $L(G)$ it's Lie algebra. We know that every left-invariant vector field $X$ in $G$ is complete, and so one can consider the integral curve defined for all $t\in \mathbb{R}$ ...
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57 views

Origin of integral of field strength tensor in path-ordered exponential in gauge field theory

When studying some gauge theories approach to problems in Mechanics, I've found the following integral $$P\exp\left[\oint A \ dt\right]=1+\dfrac{1}{2}\oint_{\partial ...
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65 views

How are these two Riemann tensor equations equivalent?

Poisson in A Relativist's Toolkit defines the Riemann tensor as$$A_{\,;\alpha\beta}^{\mu}-A_{\,;\beta\alpha}^{\mu}=-R_{\phantom{\mu}\nu\alpha\beta}^{\mu}A^{\nu}.$$ Foster and Nightingale's A Short ...
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62 views

How can the D'Alembertian of a field be interpreted intuitively?

The D'Alembertian operator is defined as $$ \Box = g^{\nu\mu}\nabla_\nu\nabla_\mu $$ For the Minkowski metric in Cartesian coordinates that is $$ \Box=\frac{1}{c^2}\frac{\partial^2}{\partial t^2} - ...
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1answer
139 views

How can I make two separate equations for Christoffel symbols give the same answer?

I have been studying the covariant derivative and I'm confused by the calculation of the Christoffel symbols $\Gamma$. The equation for computing $\Gamma$ is given as: $${\Gamma^c}_{ab} = \frac12 ...
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3answers
115 views

$SO(3)$ vs 3-Torus ${(S_1)}^3$

From rigid body rotations point of view, why are $SO(3)$ and 3-Torus not the same. Every rigid rotation is rotation about three axes. So how come $SO(3)$ is not ${(S_1)}^3$? It seems it should be. Is ...
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2answers
65 views

First fundamental form in the Gibbons-Hawking-York boundary term

Let me expose my problem, I am trying to perform the explicit variation of the Gibbons-Hawking-York boundary term, $$S_{GH}=\int_{\partial M} d^{n-1}x\sqrt{\left|h\right|}K$$ The problem I have is ...
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167 views

Technical question about 2-forms

A technical question about the electromagnetic tensor, but before that, it is know if, say, instead of being $$F_{\mu\nu}=\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$$it were ...
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109 views

How does one express a Lagrangian and Action in the language of forms?

In Lipschitzs Classical Mechanics a Lagrangian is defined as: $L(q,q',t)$ for some trajectory $q(t)$ of a particle And the action is defined as: $S:=\int^a_b L(q,q',t) dt$ How does one ...
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74 views

Confusion about two forms of connection coefficients

I am new to GR. In one book I found that the connection coefficient expression is given by $$ \Gamma^\mu_{\nu\lambda} = -\frac{1}{2} g^{\mu\rho} (\partial_\nu g_{\lambda\rho} + \partial_\lambda ...
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79 views

“Shortest” path in general relativity

My professor in mechanics course sneakily teach us some basic idea of general relativity. Which one of the basic assumption is particle walks in shortest world line. I understand shortest path in ...
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103 views

Can Bosons couple to gravity? Why do we need vielbein?

It is said that In theories such as Supergravity where there are fermions coupled to gravity, one must use an auxiliary quantity, the frame field (vielbein). In supergravity, can a boson be coupled ...
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3answers
160 views

In GR, why should the spacetime manifold be differentiable?

In general relativity (GR), spacetime is viewed as a differentiable manifold of dimension $D$ with a metric of Lorentzian signature $(-,+,+,...,+)$. My question is why differentiable?
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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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1answer
51 views

Manifolds, unit 2-sphere and stereographic projection

I am always passing through this example while reading about manifolds that I don't quite get. It is when describing the unit 2-sphere $S^2$ as an example of a manifold. They say, initially it may ...
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1answer
61 views

Can't derive FRW Christoffel symbol [closed]

I'm trying to confirm that the $\Gamma^1_{01}$ Christoffel symbol of the FRW metric is $\dot{a}/a$. I have the FRW metric: $$ds^2=-dt^2+a(t)^2\left[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta\ ...
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1answer
102 views

Classical spin viewed as $SU(2)$

In which sense is the configuration variable of a classical spin $SU(2)$? I can view a classical spin as a unit vector in $\mathbb{S}^2$ (2-dim. sphere), but it seems it is really given by a matrix ...
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64 views

Does Birkhoff's theorem apply to rotating collapsing stars?

Birkhoff's theorem states that every spherically symmetric vacuum solution to $R_{\alpha\beta} = 0$ is static, which greatly assists in the solution to the Schwarzschild solution by eliminating time ...
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1answer
37 views

When does light reach a shell observer in Schwarzschild metric?

I am trying to simulate the trajectory of light in the Schwarzschild metric (as seen by a far away observer) with fixed $\theta = \pi/2$. According to my source (Chapter 18, section 18.5) the ...
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0answers
135 views

Geometric interpretation of quantum Yang-Mills field

In most books\articles review geometric interpretation of classical Yang-Mills field in terms of principal bundle, connections...etc. What are geometric interpretation of quantum Yang-Mills field? ...
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2answers
78 views

Schwarzschild metric: motivations and applications in physics

I have a mathematical background and I have just derived the expression of the Schwarzschild metric. Now I was wondering what were the motivations and applications in physics of this metric. Any info ...
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94 views

Laplace-Beltrami vs d'Alembert operators in flat vs curved space-time

I am confused with the difference between Laplace-Beltrami (LB) and d'Alembert operators in flat/curved space-time. d'Alembert operator in flat space-time (Minkowski) is defined as $$\Box= ...
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1answer
345 views

How does one write density as a form?

In vector calculus, given the density $\rho$ of a body with volume $V$, it's total mass $M$ is simply $M=\int_V \rho dV$. If density $\rho$ is a form, say dm it would need to be a volume form to ...
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70 views

Fibre bundles and space-time

I'm having some trouble understanding the concept for this more than likely due to my lacking mathematical background. I am currently reading Roger Penrose's The Road to Reality page 394 ...
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1answer
114 views

Carroll's derivation of the geodesic equations [duplicate]

In Carroll's derivation of the geodesic equations (page 69, http://preposterousuniverse.com/grnotes/grnotes-three.pdf), he starts with ...
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125 views

Why two different Lagrangians to derive geodesic equations?

I'm trying (very early stages) to understand the derivation of the geodesic equation ...
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52 views

Proof of Schwarzschild metric construction (O'neill chap 13)

I am struggling with a few steps of the proof in O'neill book $\textit{Semi-Riemannian Geometry, with applications to Relativity}$ on the construction of Schwarzschild's metric (chap13, Lemma1). Is ...
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94 views

Divergence Theorem, mathematical approach to Gauss's Law?

Let $D$ be a compact region in $\mathbb{R}^3$ with a smooth boundary $S$. Assume $0 \in \text{Int}(D)$. If an electric charge of magnitude $q$ is placed at $0$, the resulting force field is ...
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1answer
77 views

Calculating Christoffel symbols from Lagrangian

I was given the following metric for a sphere $$g_{\mu\nu} = diag(1, r^2, r^2\sin^2\theta)$$ and tasked to calculate the Christoffel symbols. There are 2 ways that I know of to calculate them. One ...
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1answer
119 views

Differentiating the Lagrangian to find geodesic equations?

I'm stuck pretty much at the first hurdle trying to follow the derivation of the geodesic equations from the Lagrangian ...
2
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1answer
88 views

Spinors and Möbius strips

I asked this question on Math.SE as I thought the perspective of representation theory might be enlightening. But since the question was provoked by a description of Spinors describing the spin of ...
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59 views

Branes wrapping curves in M-theory. What does it mean?

What does it mean that a M5-branes wraps a holomorphic curve in M-theory? In specific a lot of Vafa's paper involve various branes (not only M5) wrapping some cycles. What does this really mean ...
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1answer
139 views

Null geodesic equation

For a null geodesic curve $X^i$, $$0=g_{ij}V^iV^j.$$ When we derive the geodesic equation from E-L equations, will this affine parametrization cause it to blow up? How is it justified to use the ...
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4answers
146 views

Any tips on evaluating Riemann tensor?

I am calculating the Riemann tensor for the Schwarzschild solution. I've calculated all 9 non-vanishing Christoffel symbols already. Now I need to evaluate the Riemann tensor and I find no easy way to ...
14
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4answers
381 views

Lie derivative vs. covariant derivative in the context of Killing vectors

Let me start by saying that I understand the definitions of the Lie and covariant derivatives, and their fundamental differences (at least I think I do). However, when learning about Killing vectors I ...
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2answers
71 views

A couple of questions on the ADM formalism in general relativity

I've been reading up on the ADM formalism in general relativity and have been stuck on a couple of concepts. The first is to do with the foliation of spacetime into space-like hypersurfaces. I ...
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1answer
37 views

Determinant of the curved space scalar wave operator

I am reading a paper titled 'Analogue Gravity' (http://www.livingreviews.org/lrr-2011-3 or http://arxiv.org/abs/gr-qc/0505065) In the paper (page 15/159) they say this: $$\det(\sqrt{-g} g^{\mu \nu}) ...
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2answers
96 views

Making sense out of covariance and contravariance

I just read about co- and contravariant vectors and I am not sure that I got it right: If we imagine that we have a n-dimensional manifold $M$ then a tangent space is spanned by the vectors ...
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1answer
154 views

Why a timelike geodesic maximizes path length?

I'm studying some GR and my book says that in Pseudo-Riemannian manifolds geodesics may even maximize the path locally. That's what happen to the timelike geodesics, for example. My first question: Is ...
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0answers
76 views

Lagrangian, geodesics and relativity [closed]

My background is in maths, but I have been studying some basic physics with occasional input from a friend who is studying for a physics PhD. Due to my background, I am keen to visualize things ...
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1answer
65 views

General Relativity - Four Velocity Derivative Question

I am trying to get my head around a small point used in a book I am reading about General Relativity. The book states that because $u_au^a = c^2$ it follows that $u_a \nabla_b u^a = 0 $ The first ...
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1answer
88 views

Spinor notation in general relativity

I have a somewhat broad/big question, and I know that there are many references for it available out there. However, so far I couldn't find anything that I can really understand, that's why here is my ...
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1answer
50 views

Parallel Transported Orthonormal Basis

The following argument results in a conclusion that I find strange, and makes me suspect there is something wrong with the reasoning. First, consider a timelike geodesic $\gamma$ with normalized ...
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2answers
89 views

Deriving the geodesic equation [closed]

I having been reading a general relativity book, but when in comes to the geodesic equation, it is not derived. How does one go about doing this?
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0answers
91 views

metric determinant and its partial and covariant derivative

question : $\nabla_a \nabla_b \sqrt{g} \phi =\partial_a \sqrt{g} \partial_b \phi$ is true ? because $\nabla_a \sqrt{g}=0$ so we can write $\sqrt{g} \nabla_a \nabla_b \phi$ , but because metric ...
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1answer
72 views

Varying wrt metric [closed]

I saw people write $\frac{\partial( F^{ab} F_{ab})}{\partial g^{ef}}$ as $\frac {\partial (g^{ca}g^{db}F_{cd}F_{ab})}{\partial g^{ef}}$ in a way that exposes the dependence on the metric. but ...
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40 views

Books or papers recommendation on orbifold and CFT [duplicate]

Could you recommend some references on orbifold CFT? I have found this paper "The conformal field theory of orbifolds"(1987)(http://inspirehep.net/record/230342) is very useful for me, so I want to ...
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1answer
30 views

Why does this allegedly Hermitian Kähler metric have non-zero diagonal terms?

In string theory, the Kähler potential of Kähler moduli (e.g. - the volume of a Calabi-Yau manifold) is given by (see, for instance, Becker, Becker, Schwarz: "String Theory and M Theory" p. 498) $$K ...
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1answer
57 views

Show that a solenoidal field is always a curl of a vector field [closed]

Can someone prove that: $$\nabla \cdot \mathbf{B} = 0 \implies \mathbf{B} = \nabla \times \mathbf{A}~?$$ I know that $$\nabla \cdot (\nabla \times \mathbf{A}) = 0$$ identically. But can one prove ...
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2answers
152 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 ...