Metric tensor Definition and 206 Threads

Why cosmological constant term ##\Lambda g_{uv}## in Einstein equation is proportional to ##g_{uv}##. Why it is even proportional to ##g_{uv}## in spacetime of MInkowski?

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?

The principle of least action states that the evolution of a physical system - how a system progresses from one state to another- is given by a stationary point of the action. So I think this is varying the path and keeping two points fixed- the points of the initial and final state I know...

Hello Say, the metric tensor is diagonal, ##g=\mbox{diag}(g_{11}, g_{22},...,g_{NN})##. The (null) geodesic equations are ##\frac{d}{ds}(2g_{ri} \frac{dx^{i}}{ds})-\frac{\partial g_{jk}}{\partial x^{r}}\frac{dx^{j}}{ds}\frac{dx^{k}}{ds} = 0## These are ##N## equations containing ##N## partial...

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.

Hi guys. I am trying to understand einstein field equation and thus have started on learning tensor. For metric tensor, is it just comprised of two contra/covariant vectors tensor product among each other alone or does it requires an additional kronecker delta? I am confused about the idea...

Does anyone know a reference with a discussion on the experimental determination of the metric tensor of spacetime? I only know the one in "The theory of relativity" by Møller, pages 237-240. https://archive.org/details/theoryofrelativi029229mbp

Some subtleties of the metric tensor are just becoming clear to me now. If I take ##g_{\mu\nu}=diag(+1,-1,-1,-1)## and want to write ##\partial_\mu\phi^\mu##, it would be ##\partial_0\phi^0 -\partial_i\phi^i##, correct? ##\phi## is a 4-vector.

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...

\mathcal{L}_M(g_{kn}) = -\frac{1}{4\mu{0}}g_{kj} g_{nl} F^{kn} F^{jl} \\ \frac{\partial{\mathcal{L}_M}}{\partial{g_{kn}}}=-\frac{1}{4\mu_0}F^{pq}F^{jl} \frac{\partial}{\partial{g_{kn}}}(g_{pj}g_{ql})=+\frac{1}{4\mu_0} F^{pq} F^{lj} 2 \delta^k_p \delta^n_j g_{ql} I need to know how...

Suppose, I know the metric tensor of a 2D space. for example, the metric tensor of a sphere of radius R, gij = ##\begin{pmatrix} R^2 & 0 \\ 0 & R^2\cdot sin^2\theta \end{pmatrix}## ,and I just know the metric tensor, but don't know that it is of a sphere. Now I want to draw a 2D space(surface)...

Hello! I'd appreciate any help or pokes in the right direction. Homework Statement Show that a co-tensor of rank 2, ##T_{\mu\nu}##, is obtained from the tensor of rank 2 ##T^{\mu\nu}## by using a metric to lower the indices: $$T_{\mu\nu} = g_{\mu\alpha}g_{\nu\beta}T^{\alpha\beta}$$ Homework...

After my recent studies of the curvature of the 2- sphere, I would like to move on to Minkowski space. However, I can not seem to find the metric tensor of the 4 sphere on line, nor can I seem to think of the vector of transformation properties that I would use to derive the metric tensor of the...

I have one question, which I don't know if I should post here again, but I found it in GR... When you have a metric tensor with components: g_{\mu \nu} = \eta _{\mu \nu} + h_{\mu \nu}, ~~ |h|<<1 (weak field approximation). Then the inverse is: g^{\mu \nu} = \eta^{\mu \nu} - h^{\mu \nu}...

Alright, so I was reading up on tensors and such with non-Cartesian coordinate systems all day but now I'm a bit tired an confused so you'll have to forgive me if it's a stupid question. So to express the dot product in some coordinate system, it's: g(\vec{A}\,,\vec{B})=A^aB^bg_{ab} And, if...

Definition/Summary The metric tensor g_{\mu\nu} is a 4x4 matrix that is determined by the curvature and coordinate system of the spacetime Equations The proper time is given by the equation d\tau^2=dx^{\mu}dx^{\nu}g_{\mu\nu} using the Einstein summation convention It is a symmetric...

I recently derived a matrix which I believe to be the metric tensor in spherical polar coordinates in 3-D. Here were the components of the tensor that I derived. I will show my work afterwards: g11 = sin2(ø) + cos2(θ) g12 = -rsin(θ)cos(θ) g13 = rsin(ø)cos(ø) g21 = -rsin(θ)cos(θ)...

Suppose we have a foliation of leaves (hypersurfaces) with codimension one of some Riemannian manifold ##M## with metric ##g##. For any point ##p## in ##M## we can then find some flat coordinate chart ##(U,\phi) = (U, (x^\mu, y))## such that setting ##y## to a constant locally labels each leaf...

From what I've understood, 1) the metric is a bilinear form on a space 2) the metric tensor is basically the same thing Is this correct? If so, how is the metric related to/different from the distance function in that space? Some other sources state that the metric is defined as the...

Hello All, Sorry if my question seems to be elementary. I am trying to find out a little bit details of the Riemann metric tensor but not too much in details. I found out the metric (g11, g12, g13, g14...). It provides information on the manifold and those parameters have the information...

I am trying to write the Einstein field equations $$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} R=\frac{8\pi G}{c^4}T_{\mu\nu}$$ in such a way that the Ricci curvature tensor $$R_{\mu\nu}$$ and scalar curvature $$R$$ are replaced with an explicit expression involving the metric tensor $$g_{\mu\nu}$$...

I've been watching the Stanford lectures on GR by Leonard Susskind and according to him the metric tensor is not constant in polar coordinates. To me this seems wrong as I thought the metric tensor would be given by: g^{\mu \nu} = \begin{pmatrix} 1 & 0\\ 0 & 0\\ \end{pmatrix} Since...

How do you find the inverse of metric tensor when there are off-diagonals? More specifivally, given the (Kerr) metric, $$ d \tau^2 = g_{tt} dt^2 + 2g_{t \phi} dt d\phi +g_{rr} dr^2 + g_{\theta \theta} d \theta^2 + g_{\phi \phi} d \phi^2 + $$ we have the metric tensor; $$ g_{\mu \nu} =...

Suppose we have some two-dimensional Riemannian manifold ##M^2## with a metric tensor ##g##. Initially it is always locally possible to transform away the off-diagonal elements of ##g##. Suppose now by choosing the appropriate equivalence relation and with a corresponding surjection we construct...

I'm working on a problem that requires me to take the cartesian metric in 2D [1 0;0 1] and convert (using the transformation equations b/w polar and cartesian coords) it to the polar metric. I have done this without issue using the partial derivatives of the transformation equations and have...

I have been reading an introductory book to General Relativity by H Hobson. I have been following it step by step and now I am stuck. It is stated in the book that: "It is straightforward to show that the coordinate and dual basis vectors themselves are related... "ea = gabeb ..." I have...

Our galaxy is rotating and is charged therefore the choice for the metric is the Kerr-Newman Metric. I want to solve for the Kerr-Newman Metric Tensor. There are a few questions. 1-What is the value for Q in the equation: ##r_Q^2=\frac{Q^2*G}{4*\pi*\epsilon_0*c^4}## where ##G=6.674E-20...

I am looking for the Metric Tensor of the Reissner–Nordström Metric.g_{μv} I have searched the web: Wiki and Bing but I can not find the metric tensor derivations. Thanks in advance!

I want to find the ricci tensor and ricci scalar for the space-time curvature at the Earth surface. Ignoring the moon and the sun. I have used the scwharzschilds metric, but then the ricci tensor and the scalar where equal to zero.

If I have a 4x4 Covarient Metric Tensor g_{ik}. I can find the determinant: G = det(g_{ik}) How do I find the 4x4 Cofactor of g_ik? G^{ik} then g^{ik}=G^{ik}/G

I was bored, so I tried to do something to occupy myself. I started going through withdrawal, so I finally just gave in and tried to do some math. Three months of no school is going to be painful. I think I have problems. MATH problems. :-p Atrocious comedy aside, Spivak provides a parametric...

Is it the value of the metric tensor that determines the strength of a gravitational field at a specific point in spacetime?

hello, Do I have the right to perform the following : gjo,i + g0i,j = (gj0 δij + g0i ),j = (2 g0i ),j Thank you, Clear skies,

Suppose, I have the next metric: g = du^1 \otimes du^1 - du^2 \otimes du^2 And I want to calculate g(W,W), where for example W=\partial_1 + \partial_2 How would I calculate it? Thanks.

How to calculate something relating to the determinant of metric tensor? for example, its derivative ∂_{λ}g. and how to calculate1/g* ∂_{λ}g, which is from (3.33) in the book Spacetime and Geometry, in which the author says that it can be related to the Christoffel connection.

We are stating with equivalence principle that passing locally to non inertial frame would be analogous to the presence of gravitational field at that point, so g^'_{ij}=A g_{nm} A^{-1} where g' is the galilean metric and g is the metric in curved space, and A is the transformation which...

my exploration of relativity followed by first reading various books which failed to explain to me how relativity worked but built a strong feel of how one can think about it. after which i decided to take the mathematical way of understanding it for which i am going on with the prof susskind's...

Homework Statement This is not homework but more like self-study - thought I'd post it here anyway. I'm taking the variation of the determinant of the metric tensor: \delta(det[g\mu\nu]). Homework Equations The answer is \delta(det[g\mu\nu]) =det[g\mu\nu] g\mu\nu...

If you know that {{x}^{a}}{{g}_{ab}}={{x}_{b}} is it proper to say that you also know {{g}_{ab}}=\frac{{{x}_{b}}}{{{x}^{a}}}

I've read that the metric tensor is defined as {{g}^{ab}}={{e}^{a}}\cdot {{e}^{b}} so does that imply that? {{g}^{ab}}{{g}_{cd}}={{e}^{a}}{{e}^{b}}{{e}_{c}}{{e}_{d}}={{e}^{a}}{{e}_{c}}{{e}^{b}}{{e}_{d}}=g_{c}^{a}g_{d}^{b}

Hello, can anyone suggest a geometric interpretation of the metric tensor? I am also interested to know how we could "derive" the metric tensor (i.e. the matrix <ai,aj>) from some geometric considerations that we impose.

Barbour writes: the metric tensor g. Being symmetric (g_uv = g_vu) it has ten independent components, corresponding to the four values the indices u and v can each take: 0 (for the time direction) and 1; 2; 3 for the three spatial directions. Of the ten components, four merely...

H is a contravariant transformation matrix, M is a covariant transformation matrix, G is the metric tensor and G-1 is its inverse. Consider an oblique coordinates system with angle between the axes = α I have G = 1/sin2α{(1 -cosα),(-cosα 1)} <- 2 x 2 matrix I compute H = G*M where M =...

Hello, So, given two points, x and x', in a Lorentzian manifold (although I think it's the same for a Riemannian one). If in x the determinant of the metric is g and in the point x' is g'. How are g and g' related?By any means can g=g'? In what conditions? I'm sorry if this is a dumb...

I have a very limited knowledge of tensor calculus, and I've never had proper exposure to general relativity, but I hope that the people reading this forum are able to help out. So I'm doing some stat. mech. and a part of a system's free energy is \mathcal{F} = \int V(\rho)\nabla^2\rho dx I'd...

Now let's say I have the metric for some curved two surface ds^2=G(u,v)du^2+P(u,v)dv^2 ( the G and P functions being the 00 and 11 components, assuming the metric is diagonal) Now my question is, since the metric defines the scalar product of two vectors, let's say (1,0) and (0,1), for...

Hi all, In flat space-time the metric is ds^2=-dt^2+dr^2+r^2\Omega^2 The Schwarzschild metric is ds^2=-(1-\frac{2MG}{r})dt^2+\frac{dr^2}{(1-\frac{2MG}{r})}+r^2d\Omega^2 Very far from the planet, assuming it is symmetrical and non-spinning, the Schwarzschild metric reduces to the...

Hi all, I am wondering if it is possible to derive the definition of a Christoffel symbols using the Covariant Derivative of the Metric Tensor. If yes, can I get a step-by-step solution? Thanks! Joe W.

I am working on the weak gravitational field by using linearized einstein field equation. What if the metric tensor, hαβ turn out to be a complex numbers? What is the physical meaning of the complex metric tensor? Can I just take it's real part? Or there is no such thing as complex metric...

I have looked at the definition of the metric tensor, and my sources state that to calculate it, one must first calculate the components of the position vector and compute it's Jacobian. The metric tensor is then the transpose of the Jacobian multiplied by the Jacobian. My problem with this...