Metric Definition and 1000 Threads

Here is the second paragraph from the article on metric expansion of space from Wikipedia Metric expansion is a key feature of Big Bang cosmology, is modeled mathematically with the FLRW metric, and is a generic property of the Universe we inhabit. However, the model is valid only on large...

Hi, I'm having trouble answering Question 9.20 in Hobson's book (Link: http://tinyurl.com/pjsymtd). This asks to prove that a photon will just graze the surface of a massive sphere if the impact parameter is b = r(\frac{r}{r-2\mu})^\frac{1}{2} So far I have used the geodeisic equations...

How do we calculate the 4 velocity of a particle that is projected radially downwards at velocity u at a radius ra? The condition on 4 velocity is that gμνvμvν = 1 which implies that at radius ra we have ga00(v0)2 + ga11(v1)2 = 1 (eq 1) So if we start from xμ = (t,r) we get vμ = (1/√g00 ...

Let ##\mathbb{X}## be the set of all sequences in ##\mathbb{R}## that converge to ##0##. For any sequences ##\{x_n\},\{y_n\}\in\mathbb{X}##, define the metric ##d(\{x_n\},\{y_n\})=\sup_{n}{|x_n−y_n|}##. Show the metric space ##(\mathbb{X},d)## is separable. I understand that I perhaps need to...

This question is motivated by one on stack exchange, and on this paper (which comes across a bit student-y but it claims to have been reviewed, and in any case I have reproduced its results in ctensor and gnuplot). So: the KS (abbreviation!) conveys an overview of curvature at a given point in...

I want to derive Reissner Nordstrom solution using this paper as a guide: http://gmammado.mysite.syr.edu/notes/RN_Metric.pdf, but I get confused by the Eq. (8). Why the Enstein's equation can be rewritten in that form and what is the physical meaning of the Eq. (8)?

I want to calculate two things (This is not a homework question so I am posting here or actually I don't have homework like this) First question is finding universe volume using spacetime metric approach.The second thing is find a smallest volume of a spacetime metric (related to plank...

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.

So, i need to proof the triangle inequality ( d(x,y)<=d(x,z)+d(z,y) ) for the distance below But I'm stuck at In those fractions i need Xk-Zk and Zk-Yk in the denominators, not Xk-Yk and Xk-Yk. Thanks in advance

How can I figure out ##\partial_\mu x^2## on the manifold ##(M,g)##? I thought that it should be ##2x_\mu##, but I think I'm wrong and the answer is ##2x_\mu+x^\nu x^\lambda \partial_\mu g_{\nu\lambda}##, right?! In particular, it seems to me, we can't write...

I read in many books the metric tensor is rank (0,2), its inverse is (2,0) and has some property such as ##g^{\mu\nu}g_{\nu\sigma}=\delta^\mu_\sigma## etc. My question is: what does ##g^\mu_\nu## mean?! This tensor really confuses me! At first, I simply thought that...

Hi, just curious. Sorry I am trying to get a handle on this , will try to make it more precise: I am trying to see if the following has a categorical parallel/counterpart. Consider the case of measure spaces (X,S,m) : X any space, S a sigma algebra, m a measure and that of metric spaces (Y,d)...

So, I've been reading through "Exploring Black Holes: Introduction to General Relativity" by Wheeler and Taylor, and I've had some ideas I wanted to pursue and do some research in regarding trajectories within the event horizon. In this, I'd like to have the mathematical tools to investigate...

Refusing to give an inch: Why America is anti-metric (cnn.com) http://www.cnn.com/interactive/2015/07/us/metric-road-american-story/

I have managed to work out parts a and b of Exercise 13.7 from MTW (attached), but can't see how part c works. I can see how it could work in (say) the example of taking a radar measurement of the distance to Venus, where we have the Euclidian distance prediction and the result of the radar...

I have recently derived both the purely covariant Riemann tensor as well as the purely covariant Weyl tensor for the Gödel solution to Einstein's field equations. Here is a wiki for the Gödel metric if you need it: http://en.wikipedia.org/wiki/Gödel_metric There you can see the line element I...

I am trying with no luck to prove: Let (X,d) be a metric space and A a non-empty subset of X. For x,y in X, prove that d(x,A) ≤ d(x,y) + d(y,A)d(x,A)=infz∈Ad(x,z). Now, say z0∈A and y∈X. Then d(x,z0)≤d(x,y)+d(y,z0). Taking infimum over all z∈A of the left hand side, we obtain...

When we define line element of Minkowski space, we also define the metric tensor of the equation. What actually is the function of the tensor with the line element.

So I have been following various derivations of the FRW metric and have a bit of confusion due to varying convention... Would it be correct to say that curvature K can be expressed as both K = \frac{k}{a(t)^2} and K = \frac{k}{R(t)^2} where k is the curvature parameter? If so, is it correct to...

Would it be possible to write the torsion tensor in terms of the metric? I know that a symmetric Christoffel Symbol can be written in terms of the partial derivatives of the metric. This definition of the christoffel symbols does not apply if they are not symmetric. Is it possible to write a...

Here is the Robertson Walker metric: ds2= (cdt)2 - R2(t)[dr2/(1- kr2) + r2(dθ2 + sin2(θ)dΦ2)] This metric is seen and discussed in this link: http://burro.cwru.edu/Academics/Astr328/Notes/Metrics/metrics.html Now I am in the process of deriving the general relativistic mathematical objects...

How could I proof this ##ds^2=cos^2(v)du^2+dv^2## is bilinear?

I have ##ds^2=\cos^2(v)du^2 + dv^2## , i take a coordinate transformation x=u and cos(v)=##\frac{1}{(cosh(y))}##, I have to find a new metric with this coordinate transformation and proof it is in agreement with Theorema Egregium. ##ds^2=\frac{dx^2}{(cosh^2(y)}) +\frac{dy^2 }{(y^2(1-y^2))}##...

Homework Statement Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Rudin: Principles of Mathematical Analysis, 2nd ed.) Homework Equations Every separable metric space has a countable base. The...

Hello! Given a metric in terms of the Veirbein, ##g_{\mu\nu}=e^{a}_{\mu}e^{b}_{\nu}{\eta}_{ab}## , how would you go about varying it with respect to ##e^{a}_{\mu}## ? I know that ##{\delta}g_{\mu\nu}={\delta}e^{a}_{\mu}e^{b}_{\nu}{\eta}_{ab}+e^{a}_{\mu}{\delta}e^{b}_{\nu}{\eta}_{ab}## , with...

So I was wondering, is the variation of the Minkowski Metric zero? As in ##{\delta}{\eta}_{\mu\nu}=0## . I would think this is the case because the components of the Minkowski Metric are just numbers (either +1 or -1), so varying it gives you 0. Is this correct?

What is the difference between General metric gαβ and flat metric ηβα in GR? Elaborate answers are appreciated.

The metric tensor has the property that it can raise and lower indices, but this is on the assumption that it (the metric) is symmetric. If we were to construct a metric tensor that was non-symmetric, would it still raise and lower indices?

So i am studying GR at the moment, and I've been trying to figure out what the derivative (not covarient) of the mixed metric tensor $$\delta^\mu_\nu$$ would be, since this tensor is just the identity matrix surely its derivative should be zero. Yet at the same time $$\delta^\mu_\nu =...

Apologies for not doing too much research prior to asking this question; I suppose actually delving into the mathematics would reveal the answer I'm looking for but I haven't taken the time just yet. Considering the concept of GR where matter/energy tells space how to curve and space tells how...

Is it true that for a free falling observer in a non homogeneuos gravitational field, the metric according to his reference frame is always Minkowski? If it is true, Is it valid only locally?

Homework Statement If ##A## is the subspace of ##l^\infty## consisting of all sequences of zeros and ones, what is the induced metric on ##A##? Homework EquationsThe Attempt at a Solution The metric imposed on ##l^\infty## is ##d(x,y) = \underset{i \in \mathbb{N}}{\sup} |x_i - y_i|##. I...

Just started self teaching myself differential geometry and tried to find the christoffel symbols of the second kind for 2d polar coordinates. I am checking to see if I did everything correctly. With a line element of: therefore the metric should be: The christoffel symbols of the second kind...

Homework Statement Let g_{\mu\nu} be a static metric, \partial_t g_{\mu\nu}=0 where t is coordinate time. Show that the metric induced on a spacelike hypersurface t=\textrm{const} is given by \gamma_{ij} = g_{ij} - \frac{g_{ti} g_{tj}}{g_{tt}} . Homework Equations Let y^i be the coordinates...

Starting from the general expression of the metric in coordinates, what is the time misured by a clock in a non inertial reference sistem?

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?

Hi, I was wondering if there's any way to plot/visualize a metric (mostly the spatial part). I want to see how the robertson-walker metric differs from a rotating rw-metric; \begin{align} ds^2=-(1-\omega^2a^2r^2\sin^2\theta)dt^2+a^2[\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2\sin^2\theta...

Homework Statement The schwarzschild metric is given by ##ds^2 = -Ac^2 dt^2 + \frac{1}{A} dr^2 + r^2\left( d\theta^2 + sin^2\theta d\phi^2 \right)##. A particle is orbiting in circular motion at radius ##r##. (a) Find the frequency of photon at infinity ##\omega_{\infty}## in terms of when it...

Homework Statement I'm just wondering if I'm doing this calculation correct? eta and f are both tensors Homework EquationsThe Attempt at a Solution \frac{\delta \left ( \gamma_{3}f{_{\lambda}}^{k}f{_{k}}^{\sigma}f{_{\sigma}}^{\lambda} \right )}{\delta f^{\mu\nu}}=\frac{\delta\left (\gamma_{3}...

In most GR textbooks, the general form of a spherically symmetric metric is obtained by inspection which is acceptable. But in the textbook I'm reading, the author does that with a mathematical analysis just to illustrate the method. But I can't follow his calculations. In fact he omits much of...

Homework Statement The metric near Earth is ##ds^2 = -c^2 \left(1-\frac{2GM}{rc^2} \right)dt^2 + \left(1+\frac{2GM}{rc^2} \right)\left( dx^2+dy^2+dz^2\right)##. (a) Find all non-zero christoffel symbols for this metric. (b) Find satellite's period. (c) Why does ##R^i_{0j0} \simeq \partial_j...

What is the meaning of r in the Schwartzschild metric?. ds^2 = \frac{{dr^2 }}{{1 - \frac{{2GM}}{{c^2 r}}}} + r^2 (d\theta ^2 + \sin ^2 \theta d\varphi ^2 ) - c^2 \left( {1 - \frac{{2GM}}{{c^2 r}}} \right)dt^2 If you were to actually measure the radius, your observation would be affected by...

http://en.wikipedia.org/wiki/G%C3%B6del_metric Could someone please plug the line element for the Godel metric (seen on the above wiki page) into some software to see what comes out for the Einstein tensor in a coordinate basis (preferably the covariant version rather than the contravariant...

Homework Statement Two rockets are orbiting a Schwarzschild black hole of mass M, in a circular path at some location R in the equatorial plane θ=π/2. The first (rocket A) is orbiting with an angular velocity Ω=dΦ/dt and the second (rocket B) with -Ω (they orbit in opposite directions). Find...

I'm hoping someone can clarify for me, I have seen the following used: \frac{\partial}{\partial g^{ab}}\left( g^{cd} \right) = \frac{1}{2} \left( \delta_a^c \delta_b^d + \delta_b^c \delta_a^d\right) I understand the two half terms are used to account for the symmetry of the metric tensor...

So from a killing tensor the FRW metric is known to possess, for a massless particle we find the well known result that as the universe expands the frequency of the photons decreases . But , what does this do for gr ? Was this known to happen before gr ? Thanks a lot. (I know it is used to...

Homework Statement for example: ##\frac{\partial(F^{ab}F_{ab})}{ \partial g^{ab}} ## where F_{ab} is electromagnetic tensor. or ##\frac{\partial N_{a}}{\partial g^{ab}}## where ##N_{a}(x^{b}) ## is a vector field. Homework EquationsThe Attempt at a Solution i saw people write ##F^{ab}F_{ab}##...

What is the cosmological constant for the Godel metric?

Homework Statement i want to find the variation of this action with respect to ## g^{\mu\nu}## , where ##N_\mu(x^\nu)## is unit time like four velocity and ##\phi## is scalar field. ##I_{total}=I_{BD}+I_{N}## ## I_{BD}=\frac{1}{16\pi}\int dx^4\sqrt{g}\left\{\phi...

I'm looking at 2 sources. One has it defined as ##T^{c}_{ab}=-\Gamma^{c}_{ab}+\Gamma^{c}_{ba}## And the other has ##T^{c}_{ab}=\Gamma^{c}_{ab}-\Gamma^{c}_{ba}## ##T## the torsion tensor and ##\Gamma^{c}_{ab}## the connection. Or is it more that different texts use different conventions? Thanks.