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

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7answers
1k 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 ...
5
votes
1answer
708 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 ...
1
vote
1answer
86 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 ...
5
votes
3answers
270 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 ...
12
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6answers
2k 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 ...
6
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1answer
1k views

Maxwell's equations in microscopic and macroscopic forms, and quantization

The macroscopic Maxwell's equations can be put in terms of differential forms as $$\mathrm{d}\mathrm{F}=0,\quad\delta \mathrm{D}=j\implies \delta j=0,\quad \mathrm{D}=\mathrm{F}+\mathrm{P}.$$ ...
3
votes
4answers
358 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). ...
2
votes
0answers
517 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
157 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
votes
3answers
303 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 ...
3
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4answers
766 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
votes
2answers
924 views

Book covering 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 ...
6
votes
1answer
167 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
votes
0answers
86 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, ...
7
votes
1answer
1k 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 ...
3
votes
0answers
191 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 & ...
5
votes
0answers
194 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 ...
4
votes
2answers
610 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 ...
13
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3answers
1k 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
241 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 ...
5
votes
1answer
2k 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
votes
1answer
229 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 ...
10
votes
2answers
581 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 ...
0
votes
1answer
3k 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 ...
5
votes
3answers
589 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 ...
7
votes
2answers
647 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
votes
1answer
210 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
votes
1answer
137 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 ...
4
votes
1answer
685 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
votes
1answer
264 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 ...
8
votes
2answers
774 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 ...
3
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2answers
2k 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 ...
20
votes
7answers
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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
votes
1answer
303 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 ...
26
votes
4answers
6k 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 ...
0
votes
1answer
84 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
321 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 = ...
0
votes
3answers
525 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
93 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: ...
4
votes
1answer
485 views

How do we know the geodesic is a minimum?

The geodesic equation is derived from the Euler-Lagrange equation, which (as I understand it) is a necessary but not sufficient condition to ensure that the geodesic is a minimum. The introductory GR ...
31
votes
8answers
2k 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
votes
1answer
265 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 ...
1
vote
2answers
136 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
votes
3answers
191 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 ...
2
votes
0answers
92 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 ...
9
votes
0answers
375 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 ...
12
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1answer
367 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 ...
21
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4answers
3k 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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2answers
115 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 ...
4
votes
1answer
451 views

How does (or can) SR/GR extend to phase space or symplectic manifolds?

I'm asking this question because of an article in New Scientist about a recent preprint by a group including Lee Smolin. I haven't taken the time to comprehend the paper completely. My knowledge of ...