Questions tagged [differential-geometry]
Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.
688
questions
99
votes
12
answers
12k
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 me ...
- 8,255
157
votes
6
answers
52k
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" ...
- 1,866
20
votes
4
answers
7k
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 $\mathbb{R} \to M$, $...
- 2,707
36
votes
8
answers
5k
views
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 ...
- 3,592
70
votes
12
answers
22k
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.
- 3,216
117
votes
6
answers
10k
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 ...
- 1,644
12
votes
1
answer
2k
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 ...
- 657
53
votes
5
answers
14k
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 ...
68
votes
5
answers
20k
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 ...
- 1,076
87
votes
9
answers
46k
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 ...
- 16.5k
11
votes
1
answer
2k
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 $(-,+,\ldots,+)$...
- 439
49
votes
8
answers
14k
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
15k
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_{\ ...
- 13.3k
32
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 ...
- 33.3k
15
votes
2
answers
6k
views
Geodesics equations via variational principle
I would like to recover the (timelike) geodesics equations via the variational principle of the following action:
$$
\mathcal{S}[x] = -m \int d\tau = -m \int \sqrt{-g_{\mu\nu}\,dx^{\mu}\,dx^{\nu}}
$$
...
- 499
36
votes
5
answers
40k
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}...
- 889
19
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 ...
- 3,372
16
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?
- 445
6
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 ...
- 3,924
43
votes
4
answers
23k
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 ...
33
votes
10
answers
9k
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{...
- 2,999
22
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 ...
- 513
15
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 ...
- 67
21
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 ...
- 9,581
20
votes
4
answers
1k
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 ...
- 2,620
9
votes
1
answer
605
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 ...
29
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 ...
- 34.2k
28
votes
5
answers
5k
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 ...
- 411
18
votes
2
answers
8k
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 ...
- 545
17
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 ...
- 2,839
6
votes
1
answer
1k
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 ...
5
votes
3
answers
2k
views
Clarifying what metric counts as flat space
In (2D) Cartesian coordinates, the Euclidean metric...
$$\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}$$
...is flat space. If the diagonal elements are exchanged for other real numbers ...
- 1,487
5
votes
2
answers
1k
views
Confusion with Virtual Displacement
I have just been introduced to the notion of virtual displacement and I am quite confused. My professor simply defined a virtual displacement as an infinitesimal displacement that occurs ...
- 348
59
votes
6
answers
43k
views
Laplace operator's interpretation
What is your interpretation of Laplace operator? When evaluating Laplacian of some scalar field at a given point one can get a value. What does this value tell us about the field or it's behaviour in ...
- 3,107
20
votes
1
answer
3k
views
Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory
In Einstein-Cartan gravity, the action is the usual Einstein-Hilbert action but now the Torsion tensor is allowed to vary as well (in usual GR, it is just set to zero).
Variation with respect to the ...
- 877
12
votes
5
answers
2k
views
Geodesics: Straightest or Shortest? When and Why?
In classical General Relativity (meaning not modified) one can think of geodesics in two ways.
One way is to say that a geodesic is the curve which is the straightest (in analogy with the flat case) ...
user78618
9
votes
1
answer
3k
views
Use partial or covariant derivatives when deriving equations of a field theory?
I feel like this question has been asked before but I can't find it. would the Euler Lagrange equation for, say, the standard model Lagrangian be $$\frac{\partial L}{\partial \phi}=\partial_\mu \frac{\...
- 367
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 ...
- 996
6
votes
1
answer
664
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 ...
- 442
2
votes
4
answers
11k
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 ...
- 31
37
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 ...
- 3,675
26
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 ...
user1532
23
votes
2
answers
8k
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 ...
- 477
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 ...
- 45.5k
21
votes
5
answers
5k
views
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, $...
- 817
20
votes
1
answer
3k
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.
...
- 5,741
16
votes
5
answers
2k
views
Can general relativity be explained by equations describing a fabric of space embedded in a flat 5-dimensional Minkowski space?
Does such a set of equations exist or does our universe have an intrinsic curvature that can't be explained by an embedding in a flat Minkowski space of 1 higher dimension? Even if general relativity ...
- 1,640
13
votes
5
answers
4k
views
What does symplecticity imply?
Symplectic systems are a common object of studies in classical physics and nonlinearity sciences.
At first I assumed it was just another way of saying Hamiltonian, but I also heard it in the context ...
- 930
12
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 ...
- 1,439
11
votes
3
answers
2k
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'...
- 1,064