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Questions tagged [differential-geometry]

Mathematical discipline which studies some properties of smooth manifolds, which allow to generalize calculus to beyond $\mathbb{R}^n$. General relativity is written in this language.

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161 votes
6 answers
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Why would spacetime curvature cause gravity?

It is fine to say that for an object flying past a massive object, the spacetime is curved by the massive object, and so the object flying past follows the curved path of the geodesic, so it "appears" ...
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20 votes
4 answers
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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 $\mathbb{R} \to M$, $...
zzz's user avatar
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37 votes
8 answers
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Interval preserving transformations are linear in special relativity

In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity: Prove ...
a06e's user avatar
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74 votes
12 answers
24k views

What is a tensor?

I have a pretty good knowledge of physics, but couldn't deeply understand what a tensor is and why it is so fundamental.
0x90's user avatar
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56 votes
5 answers
16k views

Mathematically-oriented Treatment of General Relativity

Can someone suggest a textbook that treats general relativity from a rigorous mathematical perspective? Ideally, such a book would Prove all theorems used. Use modern "mathematical notation" as ...
125 votes
6 answers
11k views

What is known about the topological structure of spacetime?

General relativity says that spacetime is a Lorentzian 4-manifold $M$ whose metric satisfies Einstein's field equations. I have two questions: What topological restrictions do Einstein's equations ...
Eric's user avatar
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12 votes
1 answer
3k views

Difference Between Algebra of Infinitesimal Conformal Transformations & Conformal Algebra

in Blumenhagen Book on conformal field theory, It is mentioned that the algebra of infinitesimal conformal transformation is different from the conformal algebra and on page 11, conformal algebra is ...
QGravity's user avatar
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93 votes
9 answers
51k views

Are matrices and second rank tensors the same thing?

Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified: Are matrices and second rank tensors ...
Revo's user avatar
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71 votes
5 answers
23k views

Conformal transformation/ Weyl scaling are they two different things? Confused!

I see that the weyl transformation is $g_{ab} \to \Omega(x)g_{ab}$ under which Ricci scalar is not invariant. I am a bit puzzled when conformal transformation is defined as those coordinate ...
vishmay's user avatar
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12 votes
1 answer
3k views

Infinitesimal transformations for a relativistic particle

The action of a free relativistic particles can be given by $$S=\frac{1}{2}\int d\tau \left(e^{-1}(\tau)g_{\mu\nu}(X)X^\mu(\tau)X^\nu(\tau)-e(\tau)m^2\right),\tag{1.8}$$ with signature $(-,+,\ldots,+)$...
Natanael's user avatar
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49 votes
8 answers
15k views

Classical mechanics without coordinates book

I am a graduate student in mathematics who would like to learn some classical mechanics. However, there is one caveat: I am not interested in the standard coordinate approach. I can't help but think ...
46 votes
4 answers
16k views

What is the physical meaning of the connection and the curvature tensor?

Regarding general relativity: What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)? What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ ...
Sklivvz's user avatar
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15 votes
2 answers
8k views

Geodesics equations via variational principle

I would like to recover the (timelike) geodesics equations via the variational principle of the following action: $$ \mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}} $$ ...
nerdizzle's user avatar
  • 509
9 votes
1 answer
788 views

Topological/Geometrical justification for $\text{CFT}_2$ being special

It is known as a fact that conformal maps on $\mathbb{R}^n \rightarrow \mathbb{R}^n$ for $n>2$ are rotations, dilations, translations, and special transformations while conformal maps for $n=2$ are ...
heaven-of-intensity's user avatar
34 votes
5 answers
4k views

Can Lagrangian be thought of 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 ...
N. Virgo's user avatar
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49 votes
5 answers
27k views

Book covering differential geometry and topology for physics

I'm interested in learning how to use geometry and topology in physics. Could anyone recommend a book that covers these topics, preferably with some proofs, physical applications, and emphasis on ...
39 votes
5 answers
45k views

Why is the covariant derivative of the metric tensor zero?

I've consulted several books for the explanation of why $$\nabla _{\mu}g_{\alpha \beta} = 0,$$ and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu \beta}...
Aftnix's user avatar
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23 votes
8 answers
5k views

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

According to Einstein, 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 ...
Dubious's user avatar
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21 votes
1 answer
3k 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 ...
xiaohuamao's user avatar
  • 3,701
17 votes
2 answers
4k 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?
Mahl-Deneb's user avatar
16 votes
2 answers
4k 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 ...
Pritam Bemis's user avatar
7 votes
2 answers
2k views

Lagrange multiplier and constraint force

The Lagrangian with Lagrange multiplier in the form $$L= T- V + \lambda f(q, \dot{q},t).$$ But there are different ways of writing the constraint $f = 0$. Will that lead to different EOMs? Let me ...
velut luna's user avatar
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33 votes
10 answers
10k views

Why do objects follow geodesics in spacetime?

Trying to teach myself general relativity. I sort of understand the derivation of the geodesic equation $$\frac{d^{2}x^{\alpha}}{d\tau^{2}}+\Gamma_{\gamma\beta}^{\alpha}\frac{dx^{\beta}}{d\tau}\frac{...
Peter4075's user avatar
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6 votes
1 answer
2k views

Diffeomorphism & Weyl transformations in the 2D worldsheet of string theory and the existence of conformal gauge

D. Tong's notes on string theory (PDF), subsection 5.1.1, feature the following in introducing the symmetries used in the Faddeev-Popov method: We have two gauge symmetries: diffeomorphisms and Weyl ...
Colliding Particles's user avatar
31 votes
3 answers
4k views

Representing forces as one-forms

This question arose because of my first question Interpreting Vector fields as Derivations on Physics. The point here is: if some force $F$ is conservative, then there's some scalar field $U$ which is ...
Gold's user avatar
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23 votes
5 answers
3k views

Is there a topological difference between an electric monopole and an magnetic monopole?

When we introduce magnetic monopoles, we have duality, i.e. invariance under the exchange of electric and magnetic fields. Magnetic (Dirac) monopoles are usually discussed using topological ...
jak's user avatar
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21 votes
4 answers
2k views

Difference between matrix representations of tensors and $\delta^{i}_{j}$ and $\delta_{ij}$?

My question basically is, is Kronecker delta $\delta_{ij}$ or $\delta^{i}_{j}$. Many tensor calculus books (including the one which I use) state it to be the latter, whereas I have also read many ...
GRrocks's user avatar
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5 votes
2 answers
2k views

Confusion with Virtual Displacement

I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs ...
J_Psi's user avatar
  • 348
29 votes
5 answers
6k views

Does curved spacetime change the volume of the space?

Mass (which can here be considered equivalent to energy) curves spacetime, so a body with mass makes the spacetime around it curved. But we live in 3 spatial dimensions, so this curving could only be ...
Erick Weil's user avatar
23 votes
3 answers
2k views

Why does the analogy between electromagnetism and general relativity differ if you consider them as gauge theories or fiber bundles?

Electromagnetism and general relativity can both be thought of as gauge theories, in which case there is a natural analogy between them: (Strictly speaking, the gauge symmetry of diffeomorphism ...
tparker's user avatar
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18 votes
2 answers
2k views

Can a non-Euclidean space be descripted through an Euclidean space of higher dimension? So why use non-Euclidean?

If you draw a big triangle in Earth 2D surface you will have an approximated spherical triangle, this will be a non euclidean geometry. but from a 3D perspective, for example the same triangle from ...
HDE's user avatar
  • 2,909
18 votes
2 answers
9k views

Why is light described by a null geodesic?

I'm trying to wrap my head around how geodesics describe trajectories at the moment. I get that for events to be causally connected, they must be connected by a timelike curve, so free objects must ...
P O'Conbhui's user avatar
14 votes
5 answers
2k views

Geodesics: Straightest or Shortest? When and Why?

In classical General Relativity (meaning not modified) one can think of geodesics in two ways. One way is to say that a geodesic is the curve which is the straightest (in analogy with the flat case) ...
user avatar
5 votes
3 answers
3k views

Clarifying what metric counts as flat space

In (2D) Cartesian coordinates, the Euclidean metric... $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ ...is flat space. If the diagonal elements are exchanged for other real numbers ...
ben's user avatar
  • 1,517
69 votes
6 answers
47k views

Laplace operator's interpretation

What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in ...
Džuris's user avatar
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29 votes
4 answers
5k views

Can spacetime be non-orientable?

This question asks what constraints there are on the global topology of spacetime from the Einstein equations. It seems to me the quotient of any global solution can in turn be a global solution. In ...
user avatar
22 votes
1 answer
3k views

Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory

In Einstein-Cartan gravity, the action is the usual Einstein-Hilbert action but now the Torsion tensor is allowed to vary as well (in usual GR, it is just set to zero). Variation with respect to the ...
CuriousKev's user avatar
13 votes
1 answer
2k views

Geometric meaning of spin connection

A very short question: Does the spin connection that we encounter in General Relativity $$\omega_{\mu,ab}$$ have a geometric meaning? I am supposing it does because it comes from mathematical terms ...
PhilosophicalPhysics's user avatar
12 votes
3 answers
8k views

A helpful proof in contracting the Christoffel symbol?

Out of all of my time learning General relativity, this is the one identity that I cannot get around. $$ \Gamma_{\alpha \beta}^{\alpha} = \partial_{\beta}\ln\sqrt{-g} \tag{1}$$ where $g$ is the ...
iron2man's user avatar
  • 485
12 votes
3 answers
3k views

Conformal transformation vs diffeomorphisms

I am reading Di Francesco's "Conformal Field Theory" and in page 95 he defines a conformal transformation as a mapping $x \mapsto x'$ such that the metric is invariant up to scale: $$g'_{\mu \nu}(x'...
MBolin's user avatar
  • 1,154
11 votes
5 answers
3k views

Open source computer algebra systems for general relativity

I would like to use an open source computer algebra system (CAS) for research in general relativity. Kindly suggest a good choice between Ipython and Sage. I am more interested in the theoretical ...
10 votes
1 answer
3k views

Use partial or covariant derivatives when deriving equations of a field theory?

I feel like this question has been asked before but I can't find it. would the Euler Lagrange equation for, say, the standard model Lagrangian be $$\frac{\partial L}{\partial \phi}=\partial_\mu \frac{\...
Eben Cowley's user avatar
7 votes
2 answers
6k views

Gauge theory for mathematicians?

I'm looking for a textbook or set of lecture notes on gauge theory for mathematicians that assumes only minimal background in physics. I'd prefer a text that uses more sophisticated mathematical ...
7 votes
1 answer
2k views

Physical interpretation of 2-forms dual to pseudovectors

Mathematically for every 3D pseudovector $x^i$ there is a 2-form $F_{ij}=\epsilon_{ijk}x^k$ such that the 2-form transforms properly under all orthogonal transformations. Therefore I would expect it ...
Akerai's user avatar
  • 1,047
6 votes
1 answer
707 views

When can an autonomous system be written using a Hamiltonian?

If I have an autonomous series of differential equations $$\tag{1} \frac{dx_i}{dt} ~=~ A_i(x_1,...,x_n)$$ with the condition that $$\tag{2} \sum_{i=1}^n\frac{\partial A_i}{\partial x_i}~=~0$$ in all ...
djbinder's user avatar
  • 452
2 votes
4 answers
14k views

Non-zero components of the Riemann tensor for the Schwarzschild metric

Can anyone tell me which are the non-zero components of the Riemann tensor for the Schwarzschild metric? I've been searching for these components for about 2 weeks, and I've found a few sites, but ...
Omar M.'s user avatar
  • 31
39 votes
7 answers
8k views

Quantum mechanics on a manifold

In quantum mechanics the state of a free particle in three dimensional space is $L^2(\mathbb R^3)$, more accurately the projective space of that Hilbert space. Here I am ignoring internal degrees of ...
MBN's user avatar
  • 3,825
32 votes
4 answers
4k views

Can general relativity be completely described as a field in a flat space?

Can general relativity be completely described as a field in a flat space? Can it be done already now or requires advances in quantum gravity?
Anixx's user avatar
  • 11.2k
24 votes
2 answers
9k views

Explicit Variation of Gibbons-Hawking-York Boundary Term

Are there any references that present the explicit variation of the Hilbert-Einstein action plus the Hawking-Gibbons-York boundary term, and demonstrate the cancellation of the normal derivatives of ...
Michael Shaw's user avatar
22 votes
1 answer
4k views

Why is it so coincident that Palatini variation of Einstein-Hilbert action will obtain an equation that connection is Levi-Civita connection?

There are two ways to do the variation of Einstein-Hilbert action. First one is Einstein formalism which takes only metric independent. After variation of action, we get the Einstein field equation. ...
346699's user avatar
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