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

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73
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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" ...
55
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6answers
3k 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 ...
41
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2answers
2k views

Intuitively, why are bundles so important in Physics?

This question probably seems silly and I don't really know if it fits properly here, but the point is the following: I've seem the notion of bundles, fiber bundles, connections on bundles and so on ...
37
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9answers
3k 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 ...
32
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5answers
2k views

Why do we need coordinate-free descriptions?

I was reading a book on differential geometry in which it said that a problem early physicists such as Einstein faced was coordinates and they realized that physics does not obey man's coordinate ...
29
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4answers
9k 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 ...
25
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4answers
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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_{\ ...
25
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4answers
2k views

Physical and Geometrical interpretation of Differential Forms

I have a doubt about the physical and geometrical interpretation of differential forms. I've been studying differential forms on Spivak's Calculus on Manifolds, but my real intent is to use those ...
25
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4answers
2k views

Is topology of universe observable?

There is an idea that the geometry of physical space is not observable(i.e. it can't be fixed by mere observation). It was introduced by H. Poincare. In brief it says that we can formulate our ...
23
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7answers
10k 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 ...
23
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4answers
4k 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 ...
20
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5answers
4k views

What exactly is a dimension?

How do you exactly define what is and isn't a dimension? I heard somewhere that it is "anything you can move through" but if that is right, why wasn't time and space considered a dimension before ...
20
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7answers
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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 ...
20
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2answers
158 views

Kerr Geometry, Separability and Twistors

One of the remarkable properties of the Kerr black hole geometry is that scalar field equations separate and are exactly solvable (reducible to quadrature), even though naively it does not have enough ...
18
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1answer
2k views

Flux through a Mobius strip

I was sent here from mathoverflow, hoping for a complete answer to this: === A friend of mine asked me what is the flux of the electric field (or any vector field like $$ \vec r=(x,y,z)\mapsto ...
17
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7answers
3k views

What is a tensor?

I have a pretty good knowledge of physics but couldn't understand what a tensor is. I just couldn't understand it, and the wiki page is very hard to understand as well. Can someone refer me to a good ...
17
votes
2answers
489 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, ...
16
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6answers
1k views

Proving that interval preserving transformations are linear

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 ...
16
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3answers
2k views

Why do objects follow geodesics in spacetime?

Trying to teach myself general relativity. I sort of understand the derivation of the geodesic equation ...
16
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3answers
1k views

Representing forces as one-forms

First of all, sorry if any of those things are silly or nonsense, I'm just trying to understand better how the concepts of forms, exterior derivative and so on can be used in physics. This question ...
16
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2answers
608 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 ...
15
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4answers
826 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 ...
15
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1answer
325 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 ...
14
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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, ...
14
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1answer
904 views

How to prove that a spacetime is maximally symmetric?

In Carroll's book on general relativity, I found the following remark: In two dimensions, finding that $R$ is a constant suffices to prove that the space is maximally symmetric [...] In higher ...
14
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4answers
378 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 ...
13
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4answers
394 views

Discontinuities and nondifferentiability in thermodynamics

In physics and engineering sources, calculus-based formalisms - whether differential forms on a manifold, or "differentials" of functions of several variables - are presented as a way of modeling and ...
13
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1answer
530 views

Physical Interpretation of EM Field Lagrangian

Using differential forms and their picture interpretations, I wonder if it's possible to give a nice geometric & physical motivation for the form of the Electromagnetic Lagrangian density? The ...
12
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1answer
417 views

Covariant derivatives

I need correctly define covariant derivatives on the coset space $G/H$, where a group $G \equiv \{X_i, Y_a\}$ ($X$ and $Y$ are generators) have a subrgroup $H \equiv \{X_i\}$ Lie algebra of $G$ has ...
12
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2answers
899 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 ...
12
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4answers
248 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 ...
12
votes
4answers
1k 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 ...
12
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2answers
731 views

Is spacetime simply connected?

As I've stated in a prior question of mine, I am a mathematician with very little knowledge of Physics, and I ask here things I'm curious about/things that will help me learn. This falls into the ...
12
votes
1answer
2k views

What is Euler Density?

Can someone please explain to me what Euler Density is? I have encountered it in Weyl anomaly related issues in various articles. Most of them assumes that its familiar, but I couldn't find any ...
12
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2answers
986 views

The Role of Active and Passive Diffeomorphism Invariance in GR

I'd like some clarification regarding the roles of active and passive diffeomorphism invariance in GR between these possibly conflicting sources. 1) Wald writes, after explaining that passive ...
12
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1answer
731 views

Is there a “covariant derivative” for conformal transformation?

A primary field is defined by its behavior under a conformal transformation $x\rightarrow x'(x)$: $$\phi(x)\rightarrow\phi'(x')=\left|\frac{\partial x'}{\partial x}\right|^{-h}\phi(x)$$ It's fairly ...
12
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3answers
558 views

Is force a contravariant vector or a covariant vector (or either)?

I don't understand whether something physical, like velocity for example, has a single correct classification as either a contravariant vector or a covariant vector. I have seen texts indicate that ...
12
votes
1answer
268 views

Conformal/trace anomaly and index theorem

I am reading the chapters on characteristic classes and the index theorems in Nakahara. It is proven in the text that any chiral or gravitational anomaly $\mathcal{A}$ is given by $$\mathcal{A}=\int ...
12
votes
1answer
564 views

Asymptotic symmetry algebra

So after a lot of research, and tons and tons of papers that I've went through, I finally have some idea how to solve the equations that will give me candidates for the asymptotic symmetry group for ...
11
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4answers
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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 ...
11
votes
3answers
451 views

What kind of manifold can be the phase space of a Hamiltonian system?

Of course it should have dimension $2n$. But any more conditions? For example, can a genus-2 surface be the phase space of a Hamiltonian system?
11
votes
4answers
570 views

Is partial derivative a vector or dual vector?

The textbook(Introduction to the Classical Theory of Particles and Fields, by Boris Kosyakov) defines a hypersurface by $$F(x)~=~c,$$ where $F\in C^\infty[\mathbb M_4,\mathbb R]$. Differentiating ...
11
votes
3answers
1k views

What is a dual / cotangent space?

Dual spaces are home to bras in quantum mechanics; cotangent spaces are home to linear maps in the tensor formalism of general relativity. After taking courses in these two subjects, I've still never ...
11
votes
2answers
868 views

Equivalence of definitions of ADM Mass

ADM Mass is a useful measure of a system. It is often defined (Wald 293) $$M_{ADM}=\frac{1}{16\pi} \lim_{r \to \infty} \oint_{s_r} (h_{\mu\nu,\mu}-h_{\mu\mu,\nu})N^{\nu} dA$$ Where $s_r$ is two ...
11
votes
7answers
280 views

How can a set of components fail to make up a vector?

Many books in Physics insist to define vectors are objects with components with the property that the components transform in a proper way under a change of coordinates. Now, in mathematics, on the ...
11
votes
1answer
823 views

Can GR be derived by postulating a maximum force?

This paper was published in a peer review journal, and claims the answer is yes. http://arxiv.org/abs/physics/0607090 The derivation in the paper seems more like dimensional analysis hand-waving in ...
11
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2answers
325 views

Differential geometry of Lie groups

In Weinberg's Classical Solutions of Quantum Field Theory, he states whilst introducing homotopy that groups, such as $SU(2)$, may be endowed with the structure of a smooth manifold after which they ...
11
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1answer
528 views

Formulation of Transformation optics using a Material Manifold

Dear Community, recently, Transformation optics celebrates some sort of scientific revival due to its (possible) applications for cloaking, see e.g. Broadband Invisibility by Non-Euclidean Cloaking ...
10
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3answers
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What does the dual of a tensor mean (e.g. dual stress tensor in relativistic ED)?

I know what the dual of a vector means (as a map to its field), and I am also aware of of the definition a dual of a tensor as, $$F^{*ij} = \frac{1}{2} \epsilon^{ijkl} F_{kl}\tag{1}$$ I just don't ...
10
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3answers
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How many times can light revolve around a black hole?

Take a light ray approaching a black hole from infinity which goes out again to infinity. What is the maximum finite rotation it can describe? (I know it can loop around indefinitely in the ...