Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.
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1answer
113 views
what is wrong with the following argument about stokes law in compact universes?
I want to understand what is wrong with the following argument:
in a topologically compact spacetime, a closed 3D boundary separates the spacetime in two connected components, because of this ...
1
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3answers
166 views
Where 2 comes from in formula for Schwarzschild radius?
In general theory of relativity I've seen several times this factor:
$$(1-\frac{2GM}{rc^2}),$$
e.g. in the Schwarzschild metric for a black hole, but I still don't know in this factor where 2 comes ...
3
votes
3answers
355 views
Equations of fluid dynamics and differential geometry
Where can I look for equations of fluid motion written in terms of nifty things from differential geometry like exterior derivative, Hodge dual, musical isomorphism?
Preferably both with and without ...
4
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1answer
531 views
Covariant derivative and Leibniz rule
I read the Wikipedia page about the covariant derivative, my main problem is in this part:
http://en.wikipedia.org/wiki/Covariant_derivative#Coordinate_description
Some of the formulas seem to lead ...
2
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1answer
103 views
What is the physical intepretation of harmonic coordinates?
When I see harmonic coordinates used somewhere, what should my association be?
Is there some general use or need to consider the harmonic cooridnate condition?
I don't really see what's ...
6
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5answers
547 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 ...
7
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4answers
348 views
Hamiltonian and the space-time structure
I'm reading Arnold's "Mathematical Methods of Classical Mechanics" but I failed to find rigorous development for the allowed forms of Hamiltonian.
Space-time structure dictates the form of ...
4
votes
5answers
580 views
Introduction to differential forms in thermodynamics
I've studied differential geometry just enough to be confident with differential forms. Now I want to see application of this formalism in thermodynamics.
I'm looking for a small reference, to learn ...
4
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1answer
285 views
Is 4-volume element a scalar or a pseudoscalar in special relativity?
In general relativity 4-volume element $\mathrm{d}^4 x = \mathrm{d} x^0\mathrm{d} x^1 \mathrm{d} x^2\mathrm{d} x^3$ is clearly a pseudoscalar (or scalar density) of weight 1 since it transforms as ...
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1answer
68 views
Smooth trajectory on a smooth manifold
Physicists talk about a smooth trajectory of a particle on a smooth manifold and they label it as q(t) where q_1(t)....q_n(t) are component functions coming from the homeomorphism. I don't see how we ...
4
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3answers
182 views
What are some mechanics examples with a globally non-generic symplecic structure?
In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be ...
7
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6answers
860 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 ...
3
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4answers
211 views
Formulation of general relativity
EDIT: I think I can pinpoint my confusion a bit better. Here comes my updated question (I'm not sure what the standard way of doing things is - please let me know if I should delete the old version). ...
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0answers
279 views
de Sitter and anti de Sitter metric
Is the following correct for the distance $d$ from the origin $(0,0)$ to point $(t,x)$ in the 2-dimensional
de-Sitter and anti de-Sitter spaces? Here, $t$ is time and the distance may be called the ...
4
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1answer
110 views
How should one interpret the de Sitter slicings?
When 'constructing' the usual de Sitter space in $\mathcal{M^5}$ by invoking the contraint $-X^{2}_{0} +X^{2}_{1} +X^{2}_{2} +X^{2}_{3} + X^{2}_{4} = \alpha^2$ we quickly see that we end up with a ...
3
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3answers
228 views
How to connect Einstein's Special Relativity(SR) with General Relativity(GR)?
How Einstein's SR becomes GR?
$$ds^2=dr^2-c^2dt^2,$$
$$ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu}.$$
When the $s$ is constant $ds^2=0$, isn't it true?
How to connect Einstein's SR with GR?
What is the ...
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votes
4answers
478 views
Topology needed for Differential Geometry [duplicate]
I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know. I know some basic concepts reading from ...
6
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1answer
146 views
If a fundamental theory exibits e.g. a mirror symmetry, in what sense it the underlying geometry real?
Are the more recently discovered symmetries in string theory such that the theories based on mirroring geometries are absolutely the same from an observable point of view?
I have mirror symmetry ...
3
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0answers
60 views
Relating the deformation of Calabi-Yau metrics and the conformal quantum field theories
(v2)
As I read e.g. in this question, the nice holonomy group features of Calabi-Yau manifolds are valuable regarding supersymmetry (I suspect because it's a symmetry involving the target manifold, ...
6
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1answer
532 views
What is the stress energy tensor?
I'm trying to understand the Einstein Field equation equipped only with training in Riemannian geometry. My question is very simple although I cant extract the answer from the wikipedia page:
Is the ...
2
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0answers
129 views
What are the topics of string theory that are comprehensible with only a mathematical background on Manifolds and Algebraic Topology?
What are the topics of string theory that are comprehensible with only a mathematical background on manifolds and algebraic topology? Also, I have read only the first four chapters in Peskin & ...
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0answers
132 views
An introductory resource for learning AdS space
Can someone please point me to introductory resources about the geometry of Anti DeSitter Space ? What are some examples of other spaces used in theoretical physics ?.I'm learning Differential ...
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2answers
373 views
Where do I start with Non-Euclidean Geometry?
I've been trying to grok General Relativity for a while now, and I've been having some trouble. Many physics textbooks gloss over the subject with an "it's too advanced for this medium", and many ...
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2answers
402 views
Why do objects follow geodesics in spacetime?
Trying to teach myself general relativity. I sort of understand the derivation of the geodesic equation ...
2
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1answer
170 views
A question regarding particle trajectories in the symplectic manifold formalism
How to solve a free particle on a 2-sphere using symplectic manifold formalism of classical mechanics ?
Is there a way to get coriolis effect directly, without going into Newton mechanics?
And is ...
4
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1answer
628 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 ...
2
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1answer
152 views
How to express the heat capacity in terms of heat?
The first law of thermodynamics divides the internal energy change into contributions of heat and work.
$$\text dU=\omega_Q-\omega_W,$$
Here I chose the notation to emphasise that the two parts are ...
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2answers
163 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 ...
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1answer
1k views
Conversion of motion equation from Cartesian to Polar coordinates: Is covariant differentiation necessary?
I have earlier posted the same question here on math stackexchange but without any answer. As the question concerns tensors, I guess that I have come to the right ...
4
votes
2answers
346 views
Does spacetime in general relativity contain holes?
Are there physical models of spacetimes, which have bounded (four dimensional) holes in them?
And do the Einstein equations give restrictions to such phenomena?
Here by holes I mean ...
3
votes
2answers
352 views
Lorentz invariance of the 3 + 1 decomposition of spacetime
Why is allowed decompose the spacetime metric into a spatial part + temporal part like this for example
$$ds^2 ~=~ (-N^2 + N_aN^a)dt^2 + 2N_adtdx^a + q_{ab}dx^adx^b$$
($N$ is called lapse, $N_a$ is ...
2
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1answer
132 views
Derivative of quantities with internal indices
In the context of the 3 + 1 decomposition of spacetime needed for a Hamiltionian formulation of general relativity, quantities with so called internal indices are introduced (in the book I am reading ...
2
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1answer
90 views
Question from Schutz's
In q. 22 in page 141, I am asked to show that if $U^{\alpha}\nabla_{\alpha} V^{\beta} = W^{\beta}$, then $U^{\alpha}\nabla_{\alpha}V_{\beta}=W_{\beta}$.
Here's what I have done:
$V_{\beta}=g_{\beta ...
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1answer
343 views
Stokes' theorem in GR
I read this formula in Sean Carroll's book of GR:
$$\int_{\Sigma}\nabla_{\mu}V^{\mu}\sqrt{g}d^nx~=~\int_{\partial\Sigma}n_{\mu}V^{\mu}\sqrt{\gamma}d^{n-1}x$$
where n is the 4-vector orthogonal to ...
2
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1answer
157 views
Contraction of indices
We use contraction of indices method to manipulate Tensors. However, I cannot relate that manipulation visually. We can change covariant tensor to contravariant tensor and vice versa by contracting ...
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2answers
541 views
Visualizing Ricci Tensor
By definition Ricci Tensor is a Tensor formed by contracting two indices of Riemann Tensor. Riemann Tensor can be visualized in terms of a curve, a vector is moving and orientation of the initial and ...
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2answers
1k views
Riemann Tensor Calculation trick(number of element)
When we calculate Riemann Tensor for different curvature we have lots of components. However, there are many components that are zero. How can we argue, based on the symmetry of connection , that ...
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5answers
2k 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 ...
2
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1answer
281 views
Superposition of Ricci scalars [closed]
Suppose I have two point/line singularities in spacetime (what is important to me is that they are localized). Also suppose I have some fields in spacetime and that the two singularities interact with ...
13
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4answers
2k 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:
1-Are matrices and second rank tensors ...
0
votes
1answer
54 views
Why is developable surface developable (ie. can be flattened onto a plane without distortion)?
The course Differential Geometry told me that developable surfaces, of which the Gaussian curvature is $0$, can be flattened onto a plane without distortion.
Some says this is because a developable ...
6
votes
1answer
258 views
What is the information geometry of 1D Ising model for a complex magnetic field?
Consider the one-dimensional Ising model with constant magnetic field and node-dependent interaction on a finite lattice, given by
$$H(\sigma) = -\sum_{i = 1}^N J_i\sigma_i\sigma_{i + 1} - h\sum_{i = ...
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3answers
273 views
Need some basic help with notation and the Christoffel symbols
Apologies in advance if some of the questions below seem overly simple.
In an introductory GR book, I find the following expression for the autoparallel of the affine connection (the upper bound of ...
6
votes
1answer
43 views
How does a geodesic equation on an n-manifold deal with singularities?
My general premise is that I want to investigate the transformations between two distinct sets of vertices on n-dimensional manifolds and then find applications to theoretical physics by:
...
18
votes
7answers
856 views
Why are differential equations for fields in physics of order two?
What is the reason for the observation that across the board fields in physics are generally governed by second order (partial) differential equations?
If someone on the street would flat out ask ...
2
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1answer
213 views
Vanishing Ricci flow on a curved manifold
If I understand this right the Ricci flow on a compact manifold given by
$\partial g_{\mu \nu} = - 2R_{\mu \nu} + \frac{2}{n}\!R_{\alpha}^{\alpha} \,g_{\mu \nu}$
tends to expand negatively curved ...
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2answers
123 views
Equivalence between Differential Geometry and Mechanics?
Given a metric
$$ ds^{2}~=~ g_{a,b}dx^{a}dx^{b}. $$
Here Einstein's summation convention is assumed for $a$ and $b$.
Then given the Laplacian over that metric, can then we find a metric $ ...
6
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3answers
130 views
From Manifold to Manifold?
Tensor equations are supposed to stay invariant in form wrt coordinate transformations where the metric is preserved. It is important to take note of the fact that invariance in form of the tensor ...
1
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0answers
81 views
Self-organizing maps
I'm currently interested in this subject but all I can see is about neural networks and I'm more interested on the Theoretical point of view: "how can a system (Lagrangian/Hamiltonian) alter it's ...
6
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0answers
247 views
Classical mechanics: Generating function of lagrangian submanifold
I have a short question regarding the geometrical interpretation of the Hamilton-Jacobi-equation.
One has the geometric version of $H \circ dS = E$ as an lagrangian submanifold $L=im(dS)$, which is ...
