# Tagged Questions

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

5answers
28k views

### 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" ...
6answers
4k 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 ...
2answers
4k views

### Intuitively, why are bundles so important in Physics?

I've seem the notion of bundles, fiber bundles, connections on bundles and so on being used in many different places on Physics. Now, in mathematics a bundle is introduced to generalize the ...
9answers
4k 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 ...
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 ...
4answers
14k 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 ...
8answers
5k 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 ...
4answers
7k views

1answer
390 views

3answers
2k 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 ...
4answers
2k 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 ...
1answer
2k views

### How to prove that a spacetime is maximally symmetric?

In Carroll's book on general relativity ("Spacetime and Geometry"), I found the following remark: In two dimensions, finding that $R$ is a constant suffices to prove that the space is maximally ...
2answers
2k 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 ...
2answers
700 views

### Stokes theorem in Lorentzian manifolds

I've fallen accross the following curious property (in p.10 of these lectures): in order to be able to apply Stokes theorem in Lorentzian manifolds, we must take normals to the boundary of the volume ...
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 ...
1answer
946 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 ...
1answer
804 views

### What's the physical intuition for symplectic structures?

I always thought about symplectic forms as elements of areas in little subspaces because of the Darboux theorem, however I cannot get the physical intuition for it and for the hamiltonian vector field....
1answer
1k views

### Equation of a torus

In the recent paper http://arxiv.org/abs/1509.03612, page 37. They say that a torus can be described by the equation $$y^2=x(z-x)(1-x)$$ where $x$ is a coordinate on the base $\mathbb{P}_1$. Could ...
1answer
495 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 ...
4answers
366 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 ...
3answers
2k views

### Book covering differential geometry and topology required for physics and applications

I am a physics undergrad, and interested to learn Topology so far as it has use in Physics. Currently I am trying to study Topological solitons but bogged down by some topological concepts. I am not ...
2answers
867 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 ...
4answers
415 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 ...
1answer
657 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 ...
3answers
542 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?
2answers
1k 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 ...
3answers
1k 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 ...