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

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Ricci tensor of direct product of manifolds

Imagine I have a (Lorentzian) manifold with a metric $\left[ {\begin{array}{cc} g_{\mu\nu} &0\\ 0&g_{mn}\\ \end{array} } \right]$ Will the Ricci tensor be also block diagonal ...
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1answer
161 views

Deriving the transformation under Weyl rescaling in Polchinski eq. (1.2.31)

I have another question in Polchinski's string theory book volume 1, namely how to derive Eq. (1.2.32)? $$(-\gamma')^{1/2} R'=(-\gamma)^{1/2} (R-2 \nabla^2 \omega) \tag{1.2.32}$$ I have awared his ...
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53 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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0answers
44 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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2answers
96 views

What is the notion of a spatial angle in general relativity?

Is there a notion of spatial angles in general relativity? Example: The world line of a photon is given by $x^{\mu}(\lambda)$. Suppose it flies into my lab where I have a mirror. I align the mirror ...
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1answer
60 views

The Spin Connection

Why do we need to introduce the spin connection coefficients $\omega_{\mu \space \space b}^{\space \space a} $ in General Relativity? To me, they just look (mathematically) like the Christoffel ...
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40 views

Rigid rectangle in Schwarzschild

Say I build a perfect rectangle. Side lengths $l_1$ and $l_2$ and perfect right angles. I am on earth and the metric is given by the Schwarzschild metric. Setting $dt=0$ leads to the spatial ...
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1answer
67 views

Metric for infinite straight cosmic string

A string theory question on my general relativity problem set: Metric is given as $$\mathrm{d}s^2 = -A(r)\mathrm{d}t^2 + B(r)\mathrm{d}r^2 + r^2 \mathrm{d}\theta^2.$$ a) Solve the vacuum equations ...
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2answers
294 views

Orthochronous Lorentz transformations are time-preserving and $SL(2,\mathbb{R})$

Let's consider the psuedosphere/hyperboloid in $\mathbb{R}^{1,2}$ given by $$x^2+y^2-z^2=-R^2.$$ We know that the Lorentz group $$O(1,2)=\{ A \in Mat(3,\mathbb{R}): A^tGA=G \},$$ where ...
3
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2answers
272 views

Restriction of a Lagrangian

I'm wondering if anyone could help me with the following questions. Let $M$ be the Minkowski spacetime, given $f\in C^{\infty}(M) ; f(m)=x^{0}(m)$, with $\{x^{\mu}\}$ being a global Cartesian ...
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1answer
47 views

How to calculate spatial distance in space-time?

Pinning two test particles at two different points in space, how can I calculate their spatial distance, when the geometry is given by the Schwarzschild metric? Let's say particle 1 is pinned at ...
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5answers
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What does a frame of reference mean in terms of manifolds?

Because of my mathematical background, I've been finding it hard to relate the physics-talk I've been reading, with mathematical objects. In (say special) relativity, we have a Lorentzian manifold, ...
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1answer
48 views

Covariant derivative of a vanishing tensor component [closed]

Is the covariant derivative of a vanishing tensor component necessarily zero?
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0answers
31 views

Geodesic trajectories of 3D hyperbola [migrated]

Consider a 3-dimensional space given by the set of points {(x,y,z),x∈R,y∈R,z>0} with the metric ds2=a/z2(dx2+dy2+dz2). b) Consider two geodesic trajectories with initial conditions ...
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1answer
74 views

Geodesic devation on a two sphere

So the geodesic deviation equation gives the relative acceleration between two geodesics in motion. But given a pair of geodesic (let's say on the two sphere) that start at the equator, separated by ...
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3answers
89 views

Aside from experimental evidence, is there any reason to model space as Euclidean?

Obviously experiment is the end-all-be-all of any science, but I'm curious if there's any a priori reason to model space as Euclidean three-space (from a pre-relativity viewpoint, of course; I'm ...
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3answers
123 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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2answers
44 views

Killing vector contractions along isometric curves

Imagine $\xi_{\nu}$ is a Killing vector field on a manifold. Does $\xi_{\nu}\xi^{\nu}$ remain constant along any isometric curve defined by the Killing vector field? My guess is that yes since as ...
2
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1answer
63 views

What are the spaces over spacetime points in which a field takes its values? Is it always the same?

When it comes to the fibrations encountered in field theories of physics, are the fibers over the base space always the same?
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0answers
45 views

Riemann curvature tensor in first order perturbation theory as a Lie derivative of Riemann curvature tensor in zero order

I am having a difficulty solving my homework so I was hoping I could get some help, so here it is. It is about gravitational waves and first order gravitational perturbation theory, I have to prove ...
2
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0answers
38 views

Can we define a induced metric like this? [migrated]

Let $\Sigma_r$ be a topological sphere in a 3-dimensional asymptotically flat Riemannian manifold $M$ with metric $g$, $\{\frac{\partial}{\partial x^i}\}, 1\leq i\leq3$ is the standard coordinate ...
13
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4answers
281 views

The Lagrangian as a metric

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
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1answer
49 views

Deriving Cartan formula

I have trouble deriving Cartan formula of the form: $$ \mathrm{d} \omega (X,Y) = X[\omega(Y)] - Y[\omega(X)] - \omega([X,Y]) \tag{1} $$ where $\mathrm{d}$ is the exterior derivative, $\omega$ is a ...
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2answers
117 views

Hilbert action's invariance under general coordinate changes

In an article, when considering invariance of the Hilbert action under a general coordinate change this formula appears for how the metric changes ...
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3answers
318 views

Geometric interpretation of Electromagnetism

For gravity, we have General Relativity, which is a geometric theory for gravitation. Is there a similar analog for Electromagnetism?
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89 views

Tensor fields and vector bundles [migrated]

Let $M$ be a differentiable manifold, $TM$ and $T^*M$ a tangent and cotangent bundle of $M$ and let $\Gamma (TM),\ \Gamma (T^*M)$ be spaces of smooth sections of $TM$ and $T^*M$. Let $T_s^r (M)$ ...
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2answers
147 views

Why do we require manifolds to be a topological space?

Roughly speaking, we define a manifold $M$ to be covered by a set of charts $\{(U_i , \varphi_i)\}$ such that locally the $n$-dimensional manifolds looks like $\mathbb{R}^n$. One of the conditions is ...
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6answers
390 views

Physical meaning of non-trivial solutions of vacuum Einstein's field equations

According to Einstein, the space-time is curved and the origin of the curvature is the presence of matter i.e. the presence of the energy-momentum tensor $T_{ab}$ in Einstein's field equations. If our ...
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1answer
96 views

Why is $D$ a $2$-form and $E$ a $1$-form?

Usually in electrostatics we start by introducing the vector field $\mathbf{E}$ representing the electric field due to some charge distribution. Later when we study fields in materials we consider the ...
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2answers
99 views

Definition of a spinor and applications to GR

I understand the construction of the Clifford algebra $C(r,s)$ and in turn the corresponding $Pin$ and $Spin$ groups. I would like first to clarify that $Spin(r,s)^e$ is the universal covering group ...
12
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1answer
133 views

Soliton Moduli Spaces and Homotopy Theory

The four-dimensional $SU(N)$ Yang-Mills Lagrangian is given by $$\mathcal{L}=\frac{1}{2e^2}\mathrm{Tr}F_{\mu\nu}F^{\mu\nu}$$ and gives rise to the Euclidean equations of motion $\mathcal{D}_\mu ...
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1answer
88 views

What are type system examples of local gauge transformation- and field strength-like objects?

This is essentially a follow up motivated by this answer to my question about the gauge transformation interpretation of identity types. A field $$\psi:\mathcal M\to\mathbb C^n$$ is a section of the ...
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1answer
55 views

Covariant derivative as a tensor

$$\nabla_{j} v^{i}~=~g^{ik}\nabla_{j}v_{k}.$$ Does this equality involve an intermediate step, where I take the metric inside the derivative, and then use the fact that covariant derivative of the ...
6
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1answer
235 views

Why is the Hodge dual so essential?

It seems unnatural to me that it is so often worthwhile to replace physical objects with their Hodge duals. For instance, if the magnetic field is properly thought of as a 2-form and the electric ...
9
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1answer
378 views

Can masses move in 2+1 gravity?

I would like to understand basic concepts of the general relativity in 2+1 spacetime. As far as I know, GR predicts that such a spacetime is flat everywhere except for the point masses which create ...
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0answers
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Why such hypersurface orthogonal vector leading to $g_{0i}=0$ for $i=1,2,3$?

Suppose that the hypersurface orthogonal co-vector $W$ us perpendicular to the family of hypersurface defined by a function $\varphi$ with $\varphi=constant$. If we choose a coordinate in which ...
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0answers
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Questions about deduction the dual form of Frobenius's Theorem

I am reading Page 435, General Relativity by Wald. Let $T^*\subset V^*$ be a subspace of the dual tangent space of a manifold, $W\subset V$ be the subspace of the tangent space annihilated by $T^*$, ...
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2answers
273 views

Why isn't general relativity the obvious thing to try after special relativity?

To preface my question, I ask this as a mathematics student, so I don't have a very good sense of how physicists think. Here is the historical context I'm imagining (in particular taking into account ...
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2answers
106 views

Frames, Tetrads and GR

Given a general metric, $g_{ab}$ I can select an orthonormal basis $\omega^{a}$ such that, $$g_{ab} = \eta_{ab}\omega^a \otimes \omega^b$$ where $\eta_{ab}$ = $\mathrm{diag}(1,-1,-1,-1).$ We may ...
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2answers
239 views

Parallel Transport of a 4-vector

Why does the parallel transported $4$-vector change from $X^a(x)$ to $X^a(x) + \bar{\delta}X^a(x)$ ? This is also discussed in D'Inverno's relativity book [page - 72]; but the reason is not clear.
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3answers
188 views

Resources showing how to use differential forms in Physics

I've been learning for a while about multivectors and forms and how they simplify many things that in simple vector calculus seems to be complicated. The only problem until now is that differently ...
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1answer
66 views

Why can a killing vector field be determined globally by its value and first derivative at one point?

It is said in Weinberg's Book, Gravitation and Cosmology, page 377, that a killing vector field (which we a priori assume exists globally) can be uniquely determined by its value and first derivative ...
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Tetrad formalism: getting back to coordinate basis

Let $\omega^{\hat{a}}$ be an orthonormal basis, and $\theta^{\hat{a}}_{\hat{b}}$ be the associated connections. From Cartan's second structure equation, we may compute the curvature 2-form, i.e. ...
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0answers
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Computing the Einstein tensor for a spherically symmetrical metric using the tetrad formalism

I am having some trouble understanding how to use the tetrad formalism. I will start with what I have so far, my question will be after that. I begin with the metric $$ \text{d}s^2 = e^{2a} \text{ ...
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1answer
56 views

Null Geodesics in flat 2+1 dimensional Minkowski space

For a given line element in flat 2+1 dimensional Minkowski space $$ g = ds^{2} = − dz \otimes dz + dx \otimes dx + dy \otimes dy .$$ The null geodesics are supposedly given by: $$ x = lu + l' $$ ...
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Einstein frame vs. Matter frame

What is the difference between Einstein frame and Matter frame in General Relativity? -A brief comment on each could be useful too. These two frames were used in this manuscript ...
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Asymptotic flatness implies existence of rotation axis

Suppose we have an asymptotically flat, globally hyperbolic spacetime $M$ endowed with two one-parameter isometry groups $\sigma_t$ and $\chi_{\phi}$ which commute (i.e. $\sigma_t \circ \chi_{\phi}= ...
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1answer
49 views

Gauge field with flat connection

Consider a gauge field $A_z^a$ with a flat connection $$F_{z{\bar z}}^a = \partial_z A_{\bar z} ^a - \partial_{\bar z} A_z^a + f_{bc}{}^a A_z^b A_{\bar z}^c = 0$$ where $f_{bc}{}^a$ is the structure ...
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2answers
204 views

Dimensions of strings in string theory

In the above image taken from wikipedia, at the string level the strings have been shown as some loops, the article in wikipedia says that in string theory the particles at lower level are broken ...
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Topology of spacetime in 2+1 dimension

In the book Quantum Gravity in 2+1 dimension by S. Carlip, in the second chapter (section 2.1), he comments that a compact 3-manifold with a flat time orientable Lorentzian metric and a purely ...