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

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Time functions in general relativity

In my general relativity notes a function $f$ is called time function, if $\nabla f$ is time-like past-pointing. Say that we are in Schwarzschild spacetime and I want to check if $f=t$ is a time ...
3
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1answer
52 views

Why do we need frame-fields to describe fermions in SUGRA?

I'm learning about the frame formalism and read that to couple fermions to gravity you need to go to the frame-formalism. As a motivation to learn more about frame-fields would someone sketch me why ...
33
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9answers
2k views

Why are differential equations for fields in physics of order two?

What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations? If someone on the street would flat out ask ...
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1answer
46 views

Norm of summation of vectors

If we have a vector $\partial_v$ and we want o find its norm, we easily say (According to the given metric) that the norm of that vector is:$ g^{vv}\partial_v\partial_v$. My question what if we have ...
2
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1answer
81 views

Relationship between Connection and Material Derivative

Suppose $D\subset \Bbb R^3$ contains a fluid and that $f : D\times \mathbb{R}\to \mathbb{R}$ is a time dependent function defined on the fluid region. In that case, the material derivative is defined ...
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2answers
140 views

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
47 views

Do the concepts of intrinsic and extrinsic curvatures imply that all spaces are embedded in a higher dimensional space?

The concepts of intrinsic and extrinsic curvature seem to imply that all spaces must be embedded in a higher dimensional space? What does this imply for physical reality?
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1answer
65 views

Conformal Killing fields on Schwarzschild

I am trying to understand which are the conformal Killing Fields on the Schwarzschild spacetime. I say that $X$ is a conformal Killing field on $S$ ($S$ is Schwarzschild) if there exists a function ...
5
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4answers
241 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 ...
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2answers
82 views

How does covariant derivative act on Christoffel Symbols?

the question is how the covariant derivative acts on the following? $\nabla_\nu(\Gamma^\alpha_{\mu\lambda}R^{\beta\lambda})=?$ and ...
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0answers
14 views

Reference Request: Resources on Affine Geometry [closed]

I am reading Gravitation by Misner, Thorne, and Wheeler and they mention the term affine geometry. Like for many math topics, I was hoping there was just a go-to textbook on the subject but I haven't ...
4
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1answer
76 views

field solutions for covariant derivative of vector field constrained to zero

Question: What do the solutions of $\nabla_\mu A^\nu = 0 $ look like? And is it possible for spacetime curvature to somehow restrict the solution to $A^\nu = 0$? Here is my current ...
4
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2answers
197 views

Can Minkowski spacetime be redefined as a non-flat riemannian manifold?

Minkowski space time is defined in terms of a flat pseudo-Riemannian manifold. I have wondered if it can be redefined as Riamannian manifold and in the case what type of curvature would there appear. ...
7
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4answers
832 views

Connection between Poisson Brackets and Symplectic Form

Jose and Saletan say the matrix elements of the Poisson Brackets (PB) in the $ {q,p} $ basis are the same as those of the inverse of the symplectic matrix $ \Omega^{-1} $, whereas the matrix elements ...
2
votes
2answers
119 views

Geometric interpretation of $\vec v \cdot \operatorname{curl} \vec v = 0$

In this Math.SE question, I asked a question to which I was hoping to get a simple intuitive answer. Instead I received an otherwise perfectly correct but very mathematical one. Obviously, the words ...
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0answers
27 views

Euclidean AdS space in Poincaré coordinates

I have read anti-de Sitter (AdS) space and its Euclidean version both in Global and Poincaré coordinates. For Lorentzian case it is clear how one Poincaré patch cover only one half of the whole AdS ...
2
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1answer
69 views

From affine space to a manifold?

One of the several definitions of an affine space goes like this. Let $M$ be an arbitrary set whose elements are called points, let $\mathcal{V}$ be a vector space of dimension $n$, and let ...
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0answers
38 views

Maps on manifolds

I am reading the SUPERGRAVITY textbook of Freedman and Van Proyen. I am reading that in the sphere we can introduce two patches that their union covers the whole sphere. Ok, I understand why we need ...
6
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0answers
77 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 ...
8
votes
1answer
85 views

How to Visualize 1-forms and $p$-forms?

I am having trouble understanding the common way of visualizing one-forms. Example of the visualization: On Wikipedia and in several math and physics texts books, I have come across visualizations ...
0
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0answers
49 views

Coordinate Symbol confusion in general relativity

In a previous post (Finding the metric tensor from the Einstein field equation?), the equation used lambda, rho mu and nu (not sure of the names of the letters!) for the Ricci tensor and swapped to a, ...
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0answers
59 views

Manifold for Schwarzschild and Bertotti-Robinson

In short: what is the manifold in discussion for Schwarzschild metric $$ ds^2 = -(1-\frac {2M}r)dt^2 + \frac1{1-\frac{2M}r} dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2)$$ and Bertotti-Robinson ...
2
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1answer
49 views

Integration and Differentiation of Proper Time

My question concerns the general relativity setting. Integration: Proper time is defined by $$\tau = \int_P\sqrt{g_{\mu\nu}dx^\mu dx^\nu}$$ but happens when $g_{\mu\nu}\neq 0$ for $\mu\neq \nu$ ? For ...
15
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2answers
346 views

Global Properties of Spacetime Manifolds

When solving the Einstein field equations, $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R = 8\pi GT_{\mu\nu}$$ for a particular stress-energy tensor, we obtain the metric of the spacetime manifold, ...
3
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0answers
60 views

Is the metric-induced topology relevant at all in a (psuedo) Riemannian manifold? [duplicate]

A (pseudo) Riemannian manifold is a tuple: $$(M,g)$$ where $M$ is a smooth manifold (in particular, a topological space with an atlas) and $g$ is a (pseudo) Riemannian metric tensor. It is apparent ...
2
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2answers
337 views

Local inertial coordinates/Fermi normal coordinates

It is said that we can introduce local inertial coordinates/Fermi normal coordinates for any timelike geodesic. But why only for timelike geodesics? What about null geodesics? Perhaps it has to do ...
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0answers
36 views

Free fall coordinates/Fermi (normal) coordinates

It makes sense intuitively given the equivalent principle, and I've seen many times it stated, that for a free fall (geodesic) path in an arbitrary spacetime, we can choose our coordinate system to ...
2
votes
4answers
264 views

Magnetic monopole and vector potential

Does anyone know how to prove (in a simple way if possible) that it is impossible to define a single-valued globally defined magnetic vector potential $\vec{A}$ on the manifold ...
3
votes
1answer
431 views

Can one raise indices on covariant derivative and products thereof?

Can the following be true? $g^{\sigma\rho}\nabla_{\rho}\nabla_{\mu} = \nabla^{\sigma}\nabla_{\mu}$ $g^{\sigma\rho}\nabla_{\nu}\nabla_{\sigma} = \nabla_{\nu}\nabla^{\rho}$ ...
8
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4answers
3k views

Maxwell's Equations using Differential Forms

Maxwell's Equations written with usual vector calculus are $$\nabla \cdot E=\rho/\epsilon_0 \qquad \nabla \cdot B=0$$ $$\nabla\times E=-\dfrac{\partial B}{\partial t} \qquad\nabla\times ...
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3answers
77 views

All geodesics are inextendable?

I think the title is true, because geodesics has a tangent vector with a constant length parametrized by an affine parameter. Probably, it is easier to think about timelike or spacelike geodesics. ...
2
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1answer
77 views

The relationship between the structure of spacetime and the existence of spinor field?

We all know that the existence of spinor fields implies that spacetime must be time-orientable. Thus that spacetime is time-orientable is a necessary condition for existence of spinor fields. Geroch, ...
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0answers
50 views

Intuition behind $U(1)$-gauge model of Electrodynamics in a general spacetime

As the article Electrodynamics in general spacetime greatly explains, the $U(1)$-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to ...
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3answers
2k views

Ricci scalar for a diagonal metric tensor

I was wondering if there is a general formula for calculating Ricci scalar for any diagonal $n\times n$ metric tensor?
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1answer
83 views

If a point r lies in the boundary of the chronological future of another point p, why does the chronological future of r belong to that of p?

I am studying the global causality of the spacetime. Here, I come across a problem. Suppose a point $r\in \partial I^+(p)$. $I^+(p)$ is the chronological future of a different point $p$ in ...
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0answers
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Determination of Ricci tensor and Scalar curvature from vielbeins [migrated]

Consider the following metric: $ds^2=h(r)\bigg(dr^2+r^2\big(d\theta^2+\sin^2\theta ~d\phi^2+(d\psi+\cos\theta ~d\phi)^2\big)\bigg)$ We can try to compute the Ricci scalar of this metric by using ...
4
votes
3answers
81 views

Geodesic Equation from variation: Is the squared lagrangian equivalent?

It is well known that geodesics on some manifold $M$, covered by some coordinates ${x_\mu}$, say with a Riemannian metric can be obtained by an action principle . Let $C$ be curve $\mathbb{R} \to M$, ...
2
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0answers
72 views

Expectation of 2-form field $B_{MN}$ in string theory

In the context of string theory, in particular when we are dealing with a low energy effective action, if we have an effective action of the form, $$S_{\mathrm{eff}} \sim S^{(0)} + \alpha S^{(1)} + ...
2
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1answer
69 views

Why do derivatives act on vector fields on a worldsheet?

The covariant derivative of a vector $A^{\mu}$ at a point $x$ is defined as $$D_z A^{\mu}=\partial_zA^{\mu}+\Gamma^{\mu}_{\rho\sigma}(x)\partial_{z}x^{\rho}A^{\sigma}$$ where Greek symbols are ...
3
votes
2answers
343 views

Why don't global coordinates always exist for a manifold?

Let $M$ be a manifold and $(\phi,U)$ a patch. Then $\phi(P)=\bar{x}=\begin{bmatrix} x^1\\ x^2\\ \vdots\\ x^n \end{bmatrix}$ for each $P$ in $U$. But each $P$ in $M$ is in some patch, so this ...
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0answers
60 views

(Scalar) Ricci flatness of a metric

What is the physical meaning to vanishing Ricci scalar $R=0$ of a metric in general relativity? Note that this is not the same questions as the geometric meaning of $R_{\mu\nu}=0$ which has been asked ...
6
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1answer
2k views

Physical significance of Killing vector field along geodesic

Let us denote by $X^i=(1,\vec 0)$ the Killing vector field and by $u^i(s)$ a tangent vector field of a geodesic, where $s$ is some affine parameter. What physical significance do the scalar quantity ...
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0answers
57 views

Covariant Derivative Chain rule?

I want to prove that a covariant derivative of a vector $A^{\mu}(x(z))$ at the point $x(z)$ in general would be defined as $$D_z ...
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2answers
65 views

Proper time in general relativity

For general relativity, Wald's GR states that timelike curves, with the norm $g_{ab}T^{a}T^{b} < 0$, can be parameterized by the "proper time" $$\tau = \int (-g_{ab}T^{a}T^{b})^{1/2} dt.$$ This ...
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Uniqueness of the Einstein tensor

This is related with an exercise 17.4-a in MTW Here what i want to show is the Einstein tensor $G_{\alpha\beta} = R_{\alpha\beta} - \frac{1}{2} R g_{\alpha \beta}$ is the only second-rank, symmetric ...
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1answer
104 views

How can we derive the gauge field Lagrangian?

I learned the gauge field Lagrangian is given in this form: $$\mathcal{L} = -\frac{1}{4} \mathrm{Tr}(F_{\mu \nu}F^{\mu \nu}).$$ But how one can derive this equation starting from defining the ...
5
votes
1answer
260 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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2answers
90 views

Derivation of the Riemann tensor confusion

I'm trying to understand the derivation of the Riemann curvature tensor as given in Foster and Nightingale's A Short Course In General Relativity, p. 102. They start by giving the covariant derivative ...
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0answers
43 views

Background field expansion in normal coordinates

Background field expansion following form $Y= X+\pi$ where $X$ is my background field and $\pi$ is the fluctuation. From the Normal coordinates we have the expansion of $\pi^{\mu} = ...