# Tagged Questions

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

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### How to visualize the gradient as a one-form?

I am reading Sean Carrol's book on General Relativity, and I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I ...
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### Difference between the metric tensor in general relativity and the metric tensor in mathematics?

Is the metric tensor in general relativity the same as the metric tensor in maths, or is there a difference?
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### Why is the cosmological constant a scalar?

Maybe my understanding is just off, but the cosmological constant is just a scalar, right? What are it's units? Why a scalar? - was a tensor 'cosmological constant' ever considered or is it just not ...
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### Is there a physical interpretation of a tensor as a vector with additional qualities?

What is a tensor? has been asked before, with the most highly up-voted answer defining a tensor of rank $k$ as a vector of a tensor of rank $k-1$. But if a scalar is defined as a physical quantity ...
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### Quantum phase space

Classical phase space is defined as a space in which all possible states are represented. Every state corresponds to a unique point in the phase space. On the other hand, in quantum mechanics every ...
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### What is the evidence of interpreting $g_{\mu\nu}$ as the metric of space-time?

I think if we don't mention the meaning of $g_{\mu\nu}$ as the metric of space-time, we can still construct the equation of motion and Einstein field equation in a way such that $g_{\mu\nu}$ is just a ...
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### Feynman's statement of the Einstein Field Equations

In Feynman's Lectures on Physics (Volume 2, chapter 42) he states that Einstein's field equation is equivalent to the statement that in any local inertial coordinate system the scalar curvature of ...
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### Is differential geometry used in solid state?

I'm an undergraduate in physics interested in a career in solid state. While I know that any additional math is helpful--I am on time constraints, and can only take a few supplemental classes. That ...
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### Is there any use for non-orthogonal frames? [closed]

In regular three dimensional space we always limit ourselves to Cartesian (i. e. orthonormal) frames. This has lots of advantages: dot products are easy, no need to distinguish between vectors and ...
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### Hodge dual and the Moyal bracket? Any link? [closed]

I have already asked this on the mathematics Stack exchange but I thought I'd try it here too! The Hodge star operator $\star$ is a linear map between $\bigwedge ^pV$ and $\bigwedge ^{n-p}V$ for an ...
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### Curved space-time VS change of coordinates in Minkowski space

I'm looking for a rather intuitive explanation (or some references) of the difference between the metric of a curved space-time and the metric of non-inertial frames. Consider an inertial reference ...
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### Examples of applications of real-valued closed 1-forms in physics [closed]

Closed 1-forms are well-studied in foliation topology, algebraic geometry, and theory of manifolds. What are examples of their most typical or most interesting applications in physics? I do not mean ...
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I am reading Kleppner.(Lorentz transformations) He said,we take the most general transformation relating the coordinates of a given event in the two systems to be of the form $$x'=Ax +Bt, y'=y, z'=z, ... 2answers 322 views ### Is a local Lorentz frame a coordinate chart on a spacetime manifold? I am just starting to learn GR. I'm alternating between studying physics books and studying math books. I keep seeing the term Lorentz frame and I'm not sure what it means mathematically. Is a ... 4answers 889 views ### If gravitation is due to space-time curvature, how can a body free-fall in a straight line? According to general relativity, Gravity is due to space-time curvature. Then all paths must be curved. If so, how can there be any straight line motion? The body must follow a curved path. So, there ... 1answer 199 views ### Can a spacetime solution in GR have no Killing vector fields? Sometimes Killing vector fields in a given spacetime are described as giving information about a symmetry of that particular spacetime solution. If I look at the requirement of a Killing vector field ... 0answers 102 views ### Time functions in general relativity In my general relativity notes a function f is called time function, if \nabla f is time-like past-pointing. Say that we are in Schwarzschild spacetime and I want to check if f=t is a time ... 1answer 76 views ### Norm of summation of vectors If we have a vector \partial_v and we want o find its norm, we easily say (According to the given metric) that the norm of that vector is: g^{vv}\partial_v\partial_v. My question what if we have ... 1answer 71 views ### Do the concepts of intrinsic and extrinsic curvatures imply that all spaces are embedded in a higher dimensional space? The concepts of intrinsic and extrinsic curvature seem to imply that all spaces must be embedded in a higher dimensional space? What does this imply for physical reality? 1answer 294 views ### Conformal Killing fields on Schwarzschild I am trying to understand which are the conformal Killing Fields on the Schwarzschild spacetime. I say that X is a conformal Killing field on S (S is Schwarzschild) if there exists a function f:... 2answers 183 views ### Akin to gauge field, why GR's lagrangian is not R_{abcd}R^{abcd}? What's the mathematical or physical meaning of R_{abcd}R^{abcd}? For gauge field theory, the Lagrangian of the gauge field is$$\mathcal{L}=-\frac{1}{4}\mathrm{tr}(\mathcal{F}_{\mu\nu}\mathcal{F}^{\mu\nu})=-\frac{1}{8}F_{a\ \mu\nu}F^{a \ \mu\nu}$$The field ... 1answer 99 views ### Why do we need frame-fields to describe fermions in SUGRA? I'm learning about the frame formalism and read that to couple fermions to gravity you need to go to the frame-formalism. As a motivation to learn more about frame-fields would someone sketch me why ... 2answers 1k views ### How does covariant derivative act on Christoffel Symbols? the question is how the covariant derivative acts on the following? \nabla_\nu(\Gamma^\alpha_{\mu\lambda}R^{\beta\lambda})=? and \nabla_\nu(\Gamma^\alpha_{\mu\lambda}R^{\beta\gamma\delta\lambda}... 0answers 163 views ### Euclidean AdS space in Poincaré coordinates I have read anti-de Sitter (AdS) space and its Euclidean version both in Global and Poincaré coordinates. For Lorentzian case it is clear how one Poincaré patch cover only one half of the whole AdS ... 2answers 332 views ### From affine space to a manifold? One of the several definitions of an affine space goes like this. Let M be an arbitrary set whose elements are called points, let \mathcal{V} be a vector space of dimension n, and let \lambda:\... 2answers 163 views ### Geometric interpretation of \vec v \cdot \operatorname{curl} \vec v = 0 In this Math.SE question, I asked a question to which I was hoping to get a simple intuitive answer. Instead I received an otherwise perfectly correct but very mathematical one. Obviously, the words ... 0answers 149 views ### Manifold for Schwarzschild and Bertotti-Robinson In short: what is the manifold in discussion for Schwarzschild metric$$ ds^2 = -(1-\frac {2M}r)dt^2 + \frac1{1-\frac{2M}r} dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2)$$and Bertotti-Robinson ... 1answer 113 views ### Integration and Differentiation of Proper Time My question concerns the general relativity setting. Integration: Proper time is defined by$$\tau = \int_P\sqrt{g_{\mu\nu}dx^\mu dx^\nu}$$but happens when g_{\mu\nu}\neq 0 for \mu\neq \nu ? For ... 0answers 237 views ### Free fall coordinates/Fermi (normal) coordinates It makes sense intuitively given the equivalent principle, and I've seen many times it stated, that for a free fall (geodesic) path in an arbitrary spacetime, we can choose our coordinate system to ... 3answers 573 views ### Magnetic monopole and vector potential Does anyone know how to prove (in a simple way if possible) that it is impossible to define a single-valued globally defined magnetic vector potential \vec{A} on the manifold M=\mathbb{R}^3\... 0answers 109 views ### Can some components of metric be Finslerian while the others be Riemannian? A Finsler metric reduces to a Riemann metric in case it loses its dependence on velocities. Now, my question is this: Can we have a Finsler metric in which some components of the metric have velocity ... 1answer 213 views ### field solutions for covariant derivative of vector field constrained to zero Question: What do the solutions of \nabla_\mu A^\nu = 0  look like? And is it possible for spacetime curvature to somehow restrict the solution to A^\nu = 0? Here is my current ... 0answers 109 views ### Intuition behind U(1)-gauge model of Electrodynamics in a general spacetime As the article Electrodynamics in general spacetime greatly explains, the U(1)-gauge theory is a good base for working in non-simply connected spaces. But I wonder whether there is a deep reason to ... 1answer 143 views ### The relationship between the structure of spacetime and the existence of spinor field? We all know that the existence of spinor fields implies that spacetime must be time-orientable. Thus that spacetime is time-orientable is a necessary condition for existence of spinor fields. Geroch, ... 4answers 620 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, ... 1answer 100 views ### Why do derivatives act on vector fields on a worldsheet? The covariant derivative of a vector A^{\mu} at a point x is defined as$$D_z A^{\mu}=\partial_zA^{\mu}+\Gamma^{\mu}_{\rho\sigma}(x)\partial_{z}x^{\rho}A^{\sigma}$$where Greek symbols are ... 3answers 120 views ### All geodesics are inextendable? I think the title is true, because geodesics has a tangent vector with a constant length parametrized by an affine parameter. Probably, it is easier to think about timelike or spacelike geodesics. ... 2answers 264 views ### Proper time in general relativity For general relativity, Wald's GR states that timelike curves, with the norm g_{ab}T^{a}T^{b} < 0, can be parameterized by the "proper time"$$\tau = \int (-g_{ab}T^{a}T^{b})^{1/2} dt.$$This ... 0answers 90 views ### (Scalar) Ricci flatness of a metric What is the physical meaning to vanishing Ricci scalar R=0 of a metric in general relativity? Note that this is not the same questions as the geometric meaning of R_{\mu\nu}=0 which has been asked ... 2answers 566 views ### Derivation of the Riemann tensor confusion I'm trying to understand the derivation of the Riemann curvature tensor as given in Foster and Nightingale's A Short Course In General Relativity, p. 102. They start by giving the covariant derivative ... 1answer 84 views ### Is this covariant derivative identity true? Trying to work through a textbook derivation of the geodesic deviation equation, I've calculated this identity:$$u_{;\beta}^{\alpha}u_{\alpha}=u_{\alpha;\beta}u^{\alpha}.$$If this is true, I'm ... 0answers 119 views ### Examples of warped product manifolds? Bishop and O'Neil defined warped product manifolds. Space-times are good examples of such warped product manifolds. Is there a famous and important example of space-times I×M where M is itself a ... 0answers 63 views ### Is it possible to build a tensor with the following properties? [closed] I am searching for a tensor in 4-dimensional space-time with two indices that satisfy: \begin{eqnarray} M_{;\mu }^{\mu \nu } &=&0 \\ M^{\mu \nu } &=&-M^{\mu \nu } \nonumber \\ M_{;\... 1answer 233 views ### Total derivative in action of the field theory Consider a classical field theory. When applying the least action I see that a term is considered total derivative. We say that$$\int \partial_\mu (\frac {\partial L}{\partial(\partial_\mu \phi)}\...
According to the answer to this post, the Christoffel symbols in Riemann normal coordinates are approximated by $$\Gamma^{k}_{ij}(x)~\sim~\frac{1}{2} R^k{}_{ilj}(x_0) \xi^l \tag{5.10}$$ which came ...