Questions tagged [differential-geometry]
Mathematical discipline which uses the techniques of calculus to study geometric problems. General relativity is written in this language.
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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" ...
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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 ...
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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 ...
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Why does nature favour the Laplacian?
The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
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What is a manifold?
For complete dummies when it comes to space-time, what is a manifold and how can space-time be modelled using these concepts?
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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 ...
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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.
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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 ...
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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 ...
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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 ...
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What is really curved, spacetime, or simply the coordinate lines?
It is often said that, according to general relativity, spacetime is curved by the presence of matter/energy.
But isn't it simply the coordinate lines of the coordinate system that are curved?
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Is spacetime wholly a mathematical construct and not a real thing? [closed]
Speaking of what I understood, spacetime is three dimensions of space and one of time. Now, if we look at general relativity, spacetime is generally reckoned as a 'fabric'. So my question is, whether ...
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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 ...
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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 ...
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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 ...
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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_{\ ...
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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 ...
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What, to a physicist, are instantons and the Donaldson invariants?
I study gauge theory from a mathematical perspective. To me, one of the most fundamental ideas is the notion of an instanton on a 4-manifold.
To be precise, I have a Riemannian 4-manifold and a ...
user101446
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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 ...
user55867
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Why is this vector field curl-free?
The curl in cylindrical coordinates is defined:
$$\nabla \times \vec{A}=\left({\frac {1}{\rho }}{\frac {\partial A_{z}}{\partial \varphi }}-{\frac {\partial A_{\varphi }}{\partial z}}\right){\hat {\...
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Quantum mechanics on a manifold
In quantum mechanics the state of a free particle in three dimensional space is $L^2(\mathbb R^3)$, more accurately the projective space of that Hilbert space. Here I am ignoring internal degrees of ...
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Why is the covariant derivative of the metric tensor zero?
I've consulted several books for the explanation of why
$$\nabla _{\mu}g_{\alpha \beta} = 0,$$
and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu \beta}...
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Interval preserving transformations are linear in special relativity
In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity:
Prove ...
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The Role of Active and Passive Diffeomorphism Invariance in GR
I'd like some clarification regarding the roles of active and passive diffeomorphism invariance in GR between these possibly conflicting sources.
Wald writes, after explaining that passive ...
user23686
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What is the magnetic flux through a trefoil knot?
Imagine a closed loop in the shape of a trefoil knot (https://en.wikipedia.org/wiki/Trefoil_knot). How should one calculate the flux through this loop? Normally we define an arbitrary smooth surface, ...
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Physical and Geometrical interpretation of Differential Forms
I have a doubt about the physical and geometrical interpretation of differential forms. I've been studying differential forms on Spivak's Calculus on Manifolds, but my real intent is to use those ...
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Why do objects follow geodesics in spacetime?
Trying to teach myself general relativity. I sort of understand the derivation of the geodesic equation $$\frac{d^{2}x^{\alpha}}{d\tau^{2}}+\Gamma_{\gamma\beta}^{\alpha}\frac{dx^{\beta}}{d\tau}\frac{...
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Can general relativity be completely described as a field in a flat space?
Can general relativity be completely described as a field in a flat space? Can it be done already now or requires advances in quantum gravity?
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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 ...
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How exactly is the formalism of thermodynamics based on contact geometry?
There is a famous quote by the mathematician V. I. Arnold that goes like this:
Every mathematician knows it is impossible to understand an elementary
course in thermodynamics.
The source is Contact ...
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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 ...
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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
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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 ...
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Why would one write a vector field as a derivative?
I thought a vector field was a function $\vec{A}:\mathbb{R}^n\to\mathbb{R}^n$, which takes a vector to a vector. This seemed intuitive to me, but in a mathematical physics course I found the ...
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Extended Born relativity, Nambu 3-form and ternary ($n$-ary) symmetry
Background: Classical Mechanics is based on the Poincare-Cartan two-form
$$\omega_2=dx\wedge dp$$
where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. On the other hand, the ...
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Why is de Rham cohomology important in fundamental physics?
I'm currently working on some mathematical aspects of higher-spin gravity theories and de Rham cohomology pops up quite often.
I understand its meaning as the group of closed forms on some space, ...
user20250
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Are black holes perfect spheroids?
What I know about black holes (correct me if I'm wrong) is that they are the most compact objects in the universe that have been discovered. Due to all that gravity, wouldn't black holes be a perfect ...
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Why can we assume torsion is zero in GR?
The first Cartan equation is
$$\mathrm{d}\omega^{a} + \theta^{a}_{b} \wedge \omega^{b} = T^{a}$$
where $\omega^{a}$ is an orthonormal basis, $T^{a}$ is the torsion and $\theta^{a}_{b}$ is the spin ...
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Flux through a Möbius strip
A friend of mine asked me what is the flux of the electric field (or any vector field like
$$
\vec r=(x,y,z)\mapsto \frac{\vec r}{|r|^3}
$$
(where $|r|=(x^2+y^2+z^2)^{1/2}$) through a Möbius strip. ...
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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 ...
user72789
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The geometrical interpretation of the Poisson bracket
"Hamiltonian mechanics is geometry in phase spase."
The Poisson bracket arises naturally in Hamiltonian mechanics, and since this theory has an elegant geometric interpretation, I'm interested in ...
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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, $g_{\mu\nu}$...
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Why do we need tensors in modern physics?
I am wondering why do we really need the concept of tensor. I think it is like vectors, just as a notation of a set of related parameters. I could write the Navier–Stokes equations with scalars, or ...
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What is the connection between a mathematician and physicist's definition of a tensor?
I study mathematics but I have a deep interest in physics as well. I have taken a course in smooth manifolds where a tensor is defined as an alternating multilinear function. Recently I have learned ...
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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 ...
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Why do we select the metric tensor for raising and lowering indices?
Any rank 2 tensor would do the same trick, right?
Then, which is the motivation for choosing the metric one?
Also, if you help me to prove that $g^{kp}g_{ip}=\delta^k_i$, I would be thankful too ^^...
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Where is the Atiyah-Singer index theorem used in physics?
I'm trying to get motivated in learning the Atiyah-Singer index theorem. In most places I read about it, e.g. wikipedia, it is mentioned that the theorem is important in theoretical physics. So my ...
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Maxwell's Equations using Differential Forms
Maxwell's Equations written with usual vector calculus are
$$\nabla \cdot E=\rho/\epsilon_0 \qquad \nabla \cdot B=0$$
$$\nabla\times E=-\dfrac{\partial B}{\partial t} \qquad\nabla\times B=\mu_0j+\...
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What is a dual / cotangent space?
Dual spaces are home to bras in quantum mechanics; cotangent spaces are home to linear maps in the tensor formalism of general relativity. After taking courses in these two subjects, I've still never ...
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When one discusses the "boundary" of Anti-de Sitter space, what do they mean precisely?
The AdS/CFT correspondence refers to the "boundary" of AdS space but I'm a little confused about what this means. Typically, one writes the AdS metric in the form $$ds^2= \frac{L^2}{z^2}(-...
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