In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.
A metric tensor is called positive-definite if it assigns a positive value g(v, v) > 0 to every nonzero vector v. A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. On a Riemannian manifold, the curve connecting two points that (locally) has the smallest length is called a geodesic, and its length is the distance that a passenger in the manifold needs to traverse to go from one point to the other. Equipped with this notion of length, a Riemannian manifold is a metric space, meaning that it has a distance function d(p, q) whose value at a pair of points p and q is the distance from p to q. Conversely, the metric tensor itself is the derivative of the distance function (taken in a suitable manner). Thus the metric tensor gives the infinitesimal distance on the manifold.
While the notion of a metric tensor was known in some sense to mathematicians such as Carl Gauss from the early 19th century, it was not until the early 20th century that its properties as a tensor were understood by, in particular, Gregorio Ricci-Curbastro and Tullio Levi-Civita, who first codified the notion of a tensor. The metric tensor is an example of a tensor field.
The components of a metric tensor in a coordinate basis take on the form of a symmetric matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent point of view, a metric tensor field is defined to be a nondegenerate symmetric bilinear form on each tangent space that varies smoothly from point to point.
Does anyone see a way I can find geodesics in the metric ##ds^2=-dt^2+dp^2+(5p^2+4t^2)d\phi^2## (ones with nonzero angular momentum)? I'm hoping it can be done analytically, but that may be wishful thinking. FYI, this is the metric listed at the bottom of the Wikipedia article about Ellis Wormholes.
The Hiscock coordinates read:
$$d\tau=(1+\frac{v^2(1-f)}{1-v^2(1-f)^2})dt-\frac{v(1-f)}{1-v^2(1-f)^2}dx$$
##dr=dx-vdt##
Where ##f## is a function of ##r##. Now, in terms of calculating the christoffel symbol ##\Gamma^\tau_{\tau\tau}## of the new metric, where ##g_{\tau\tau}=v^2(1-f)^2-1## and...
In describing the spacetime around a massive, spherical object, one would use the Schwarzschild Metric. What metric would instead be used to describe the spacetime around multiple massive bodies? Say, for example, you want to calculate the Gravitational Time Dilation experienced by a rocket ship...
Considering the FLWR metric in cartesian coordinates:
##ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2)##
With ##a(t)=t##, the trace of the extrinsic curvature tensor is ##-3t##. But why is it negative if it's describing an expanding universe, not a contracting one?
I've been trying to find a way to calculate Gaussian curvature from a 4D metric tensor. I found a program that does this in Mathematica using the Brioschi formula. However, this only seems to work for a 2D metric or formula (I would need to use something with more dimensions). I've found...
I'm wondering if there is a way to find the proper volume of the warped region of the Alcubierre spacetime for a constant ##t## hypersurface. I can do a coordinate transformation ##t=τ+G(x)##, where ##G(x)=\int \frac{-vf}{1-v^2f^2}dx##. This eliminates the diagonal and makes it so that the...
Suppose you have the following situation:
We have a spacetime that is asymptotically flat. At some position A which is in the region that is approximately flat, an observer sends out a photon (for simplicity, as I presume that any calculations involved here become easier if we consider a...
0:00 The metric tensor
12:55 Curve lengths
28:17 Metric compatibility of connections
35:47 The Levi-Civita connection
40:27 Induced metrics
50:12 Curvature and the metric
1:04:18 Killing fields and symmetries
The covariant form for the Levi-Civita is defined as ##\varepsilon_{i,j,k}:=\sqrt{g}\epsilon_{i,j,k}##. I want to show from this definition that it's contravariant form is given by ##\varepsilon^{i,j,k}=\frac{1}{\sqrt{g}}\epsilon^{i,j,k}##.
My attempt
What I have tried is to express this...
I'm trying to show that the determinant ##g \equiv \det(g_{ij})## of the metric tensor is a tensor density. Therefore, in order to do that, I need to show that the determinant of the metric tensor in the new basis, ##g'##, would be given by...
Hey there,
I've been recently been going back over the basics of GR, differential geometry in particular. I was watching one of Susskind's lectures and did not understand the argument made here (26:33 - 35:40).
In short, the argument goes as follows (I think): we have some generic metric ##{ g...
The Friedman Equations is based on the cosmological principle, which states that the universe at sufficiently large scale is homogeneous and isotropic.
But what if, as an hypothesis, the universe was anisotropic and the clustering of masses are aligned to an arbitrary axis (axial pole), how...
I have been teaching myself general relativity and wanted to see if I got these metric tensors right, I have a feeling I didn't.
For the first one I get all my directional derivatives
(0, 0): (0)i + (0)j
(0, 1): (0)i + 2j
(1, 0): 2i + (0)j
(1, 1): 2i + 2j
Then I square them (FOIL):
(0, 0)...
Which properties of metric tensor are invariant of basevectors transforms? I know that metric tensor depends of basevectors, but are there properties of metric tensor, that are basevector invariant and describe space itself?
I am still learning general relativity (GR). I know we can find the path of a test particle by solving geodesic equations. I am wondering if it is possible to derive/convert metric tensor to gravity, under weak approximation, and vice versa.
Thanks!
In the recent thread about the gravitational field of an infinite flat wall PeterDonis posted (indirectly) a link to a mathpages analysis of the scenario. That page (http://www.mathpages.com/home/kmath530/kmath530.htm) produces an ansatz for the metric as follows (I had to re-type the LaTeX -...
Does the relative density of the early universe contribute to the red-shift of distant galaxies?
If so, by how much? How would this be calculated?
Asked another way :
Assuming both the early universe and the current universe are flat, could the relative difference of their space time metric...
Some models of gravity, inspired by the main theme of spacetime fabric of Classical GR, treat the metric of the manifold and the connection as independent entities. I want to study this theory further but I am unable to find any paper on this, on ariXiv atleast.
I will be very thankful if...
Let $$\phi(x^1,x^2...,x^n) =c$$ be a surface. What is unit Normal to the surface?
I know how to find equation of normal to a surface. It is given by:
$$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$
However the answer is given using metric tensor which I am not able to derive. Here is the...
Proposition: Consider an ##n + 1##-dimensional metric with the following product structure:
$$ g=\underbrace{g_{rr}(t,r)\mathrm{d}r^2+2g_{rt}(t,r)\mathrm{d}t\mathrm{d}r+g_{tt}(t,r)\mathrm{d}t^2}_{:=^2g}+\underbrace{h_{AB}(t,r,x^A)\mathrm{d}x^A\mathrm{d}x^B}_{:=h} $$
where ##h## is a Riemannian...
Defining dS2 as gijdxidxj and
given dS2 = (dx1)2 + (dx2)2 + 4(dx1)(dx2). Find gij
Now here is my take on the solution: Since the cross terms are present in the line element equation, we can say that it's a non-orthogonal system. So what I did was express the metric tensor in form of a 2x2...
I’m hoping to clear up some confusion I have over what the Lie derivative of a metric determinant is.
Consider a 4-dimensional (pseudo-) Riemannian manifold, with metric ##g_{\mu\nu}##. The determinant of this metric is given by ##g:=\text{det}(g_{\mu\nu})##. Given this, now consider the...
I'm trying to work through a scattering calculation in the Peskin QFT textbook in chapter 5, specifically getting equation 5.10. They take two bracketed terms
4[p'^{\mu}p^{\nu}+p'^{\nu}p^{\mu}-g^{\mu\nu}(p \cdot p'+m_e^2)]
and
4[k_{\mu}k'_{\nu}+k_{\nu}k'_{\mu}-g_{\mu\nu}(k \cdot...
Good Day,
Another fundamentally simple question...
if I go here;
http://www-hep.physics.uiowa.edu/~vincent/courses/29273/metric.pdf
I see how to calculate the metric tensor. The process is totally clear to me.
My question involves LANGUAGE and the ORIGIN
LANGUAGE: Does one say "one...
Do the field equations themselves constrain the metric tensor? or do they just translate external constraints on the stress-energy tensor into constraints on the metric tensor?
another way to ask the question is, if I generated an arbitrary differentiable metric tensor field, would it translate...
As I understand it, in the context of cosmological perturbation theory, one expands the metric tensor around a background metric (in this case Minkowski spacetime) as $$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$$ where ##h_{\mu\nu}## is a metric tensor and ##\kappa <<1##.
My question is, how...
I've been reading Fleisch's "A Student's Guide to Vectors and Tensors" as a self-study, and watched this helpful video also by Fleisch: Suddenly co-vectors and one-forms make more sense than they did when I tried to learn the from Schutz's GR book many years ago.
Especially in the video...
Hi All
I would like to know if there is a way to produce simple one dimensional kinematic exercises with space-time metric tensor different from the Euclidean metric. Examples, if possible, are welcome.
Best wishes,
DaTario
Let we have a 2D manifold. We choose a coordinate system where we can construct all geodesics through any point. Is it enough to derive a metric from geodesic equation? Or do we need to define something else for the manifold?
I am trying to find a derivation of gravitational redshift from a static metric that does not depend on the equivalence principle and is not a heuristic Newtonian derivation. Any suggestions?
Consider two coordinate systems on a sphere. The metric tensors of the two coordinate systems are given. Now how can I check that both coordinate systems describe the same geometry (in this case spherical geometry)?
(I used spherical geometry as an example. I would like to know the process in...
Hey there,
I have two questions - the first is about an approximation of a central gravitational force on a particle (of small mass) based on special relativity, and the second is about the legitimacy of a Lagrangian I'm using to calculate the motion of a particle in the Schwarzschild metric...
The Riemann-Christoffel Tensor (##R^{k}_{\cdot n i j}##) is defined as:
$$
R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n}
$$
My question is that it seems that...
Goodmorning everyone,
is there any implies to use in general relativity a metric whose coefficients are harmonic functions?
For example in (1+1)-dimensions, is there any implies for using a metric ds2=E(du2+dv2) with E a harmonic function?
In (1+1)-dimensions is well-know that the Einstein...
I'm having a bit of trouble understanding the nature of tensors (which is pretty central to the gen rel course I'm currently taking).
I understand that the order (or rank) of a tensor is the dimensionality of the array required to describe it's components, i.e. a 0 rank tensor is a scalar, a 1...
Hello,
since gμν gμν = 4 where g = diag[1,-1,-1,-1], see:
https://www.physicsforums.com/threads/questions-about-tensors-in-gr.39158/
Is the following equation correct?
xμ xμ = gμνxν gμνxν = gμν gμνxν xν= 4 xμ xμ
If not, where is the problem?
Cheers,
Adam
As I understand it, a Cartesian coordinate map (a coordinate map for which the line element takes the simple form ##ds^{2}=(dx^{1})^{2}+ (dx^{2})^{2}+\cdots +(dx^{n})^{2}##, and for which the coordinate basis ##\lbrace\frac{\partial}{\partial x^{\mu}}\rbrace## is orthonormal) can only be...
The Gibbons Hawking boundary term is given as ##S_{GHY} = -\frac{1}{8 \pi G} \int_{\partial M} d^dx \sqrt{-\gamma} \Theta##.
I want to calculate its variation with respect to the induced boundary metric, ##h_{\mu \nu}##.
The answer (given in eqns 6&7 of...
Hi,
Want to know (i) what does Riemannian metric tensor and Christoffel symbols on R2 mean? (ii) How does metric tensor and Christoffel symbols look like on R2? It would be great with an example as I am new to this Differential Geometry.
Hello,
Let ##g_{jk}## be a metric tensor; is it possible for some ##i## that ##g_{ii}=0##, i.e. one or more diagonal elements are equal to zero? What would be the geometrical/ topological meaning of this?
Homework Statement
Suppose everything is moving slowly, How can we find the metric tensor in GR in terms of the mass contained.
Homework Equations
I understand in case of everything moving slowly only below equation is relevant -
R00 - ½g00R = 8πGT00 = 8πGmc2
The Attempt at a Solution
None.
In SRT, the line element is ##c^2ds^2 = c^2dt^2 - dx^2 -dy^2-dz^2## and ##g_{00} = 1## (or ##-1## depending on sign conventions). In the Schwarzschild metric we have
g_{00}=(c^2-\frac{2 GM}{r}) .
So in the first example, ##g_{00}## is constant, in the second it depends on another coordinate...