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 find a metric of a general observer?

Yes, that's it. How to find a particular metric of an observer in general relativity? Let's say we have a static metric: ...
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98 views

Naturalness of tensor fields in general relativity?

In the third chapter of the book The Large Scale Structure of Space-Time, the authors say regarding the matter fields in general relativity: These fields will obey equations which can be expressed ...
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432 views

Locally flat coordinate and Locally inertial frame

I am having some doubts on myself regarding the above concepts in General Relativity. First, I want to point out how I understand them so far. A male observer follows a timelike worldline ($\gamma$) ...
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94 views

Can I simply find the Christoffel symbols by dividing by $g$?

Given the following equation \begin{equation} g_{\alpha\delta} \Gamma^{\delta}_{\beta\gamma} = \frac{1}{2} \left(\partial_\gamma g_{\alpha\beta} + \partial_\beta g_{\alpha\gamma} - \partial_\alpha ...
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155 views

Generalized spin connection and dreibein in higher spin gravity

I am studying 3D higher spin gravity and I would like to know the mathematical and physical meaning of generalized spin connection and generalized dreibein that appear in this theory. It is well known ...
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487 views

What kind of object is the Landau--Lifshitz pseudotensor?

I understand that it's called a pseudotensor because it's not a tensor. Wikipedia says most pseudotensors are sections of jet bundles, which are perfectly valid objects in GR. ...
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247 views

Lapse function definition

Let $t$ be a time function and $t^a$ the time flow vector such that $t^a\nabla_a t=0$. Let $\Sigma_t$ be a hypersurface of constant $t$ with unit normal $n^a$, $n^a n_a=-1$. Wald (1984), p. 255 ...
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Are there relativistic theories with spacetime modelled on $\mathbb C^4$ rather than real Minkowski space $\mathbb R^4$?

Does anybody know of references to theories where relativity & spacetime is modelled on a (complex/Kähler) manifold which is locally diffeomorphic to $\mathbb C^4$ rather than $\mathbb R^4$, hence ...
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86 views

Physical interpretation of diffeomorphism from $SO(3)$ to $\mathbb R \mathbb P^3$

I am not good at picturing either $SO(3)$ or $\mathbb R \mathbb P^3$, the latter denoting the real projective space. Can someone give me a rough physical understanding of the geometry and implication ...
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250 views

What is an “Einstein transformation” in general relativity?

When introducing the vielbein formalism in general relativity, I came across the use of an infinitesimal general transformation, or Einstein transformation. The latter term seems not to be covered on ...
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3answers
301 views

What besides the metric do you need to set up the EFEs and the geodesic equation?

One of my professors wrote on the board (1) Mass tells spacetime how to curve $\to$ Metric/Einstein Field Equations (2) Spacetime tells mass how to move $\to$ Geodesic equation Suppose I am given ...
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146 views

Is a metric tensor field the same thing as $ds² = -dt² + dx²+ dy² + dz²$?

I am having trouble understanding the nature of the metric tensor field on spacetime manifolds. In particular, a Riemannian manifold $(M,g)$ is defined as a real smooth manifold $M$ equipped with an ...
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107 views

Are Laplace Operator and mean curvature exactly the same thing for 2D function?

Let's assume we study 2D function/surface f(x,y). Then Laplace Operator is defined as: $$\nabla^2 f=\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}$$ And the mean curvature: let ...
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1answer
198 views

Geodesic equation from the proper time integral

This is something that has been bothering me for a little while. The usual procedure that I've seen is to write the proper time as the line integral $$\tau=\int_\gamma d\tau$$ along some curve ...
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3answers
536 views

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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2answers
283 views

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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138 views

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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346 views

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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3answers
189 views

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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2answers
662 views

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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4answers
221 views

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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156 views

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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287 views

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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297 views

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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55 views

Why can you treat coordinates as vector in flat spacetime?

In a manifold there is a distinction between points and vectors, but in flat spacetime this seems to disappear. For example in Minkowski spacetime you can define a coordinate 4-vector ...
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1answer
87 views

Proof of the relation $d^4 \xi = \sqrt{|g|} \,\, d^4x$ switching between local and non-inertial coordinates

Denoting with $d\xi^m$ and $dx^\mu$ respectively flat and non-inertial coordinates, we have the following relation between the volume elements in the two coordinate systems: $$ d^4 \xi = \sqrt{|\det ...
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144 views

Physical visualisation of curvature

I was wondering-how do you visualise curvature in the context of general relativity. The gravity well and trampoline analogies are quite wrong, so I want a more realistic approach to it (say, the way ...
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123 views

Spacetime Metrics and Quantifying Length of a Spacetime Curve

On page 247 in Gravitation by Misner, Thorne, and Wheeler, they state: "No metric means no way to quantify length; nevertheless, parallel transport gives a way to compare length!" Three questions: ...
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78 views

How can we see that the Riemann curvature tensor is covariant?

The Riemann curvature tensor, using the conventions of wikipedia, is written in terms of Christoffel symbols as: $$ \tag{1} R^\lambda_{\,\,\mu \nu \rho} = \partial_\nu \Gamma^\lambda_{\,\,\rho \mu} - ...
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136 views

Kleppner derivation of Lorentz transformation

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, ...
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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 ...
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4answers
809 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 ...
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1answer
179 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 ...
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100 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 ...
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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 ...
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1answer
70 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?
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279 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 ...
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179 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 ...
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1answer
93 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 ...
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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 ...
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150 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 ...
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2answers
307 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 ...
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2answers
159 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 ...
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138 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 ...
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112 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 ...
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208 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 ...
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3answers
544 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 ...
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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 ...