Metric Definition and 1000 Threads

Homework Statement Let X be a metric space and let E be a subset of X. Show that E is bounded if and only if there exists M>0 s.t. for all p,q in E, we have d(p,q)<M. Homework Equations Use the definition of bounded which states that a subset E of a metric space X is bounded if there exists...

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...

If every continuous function on M is bounded, what does this mean? I am not sure what this function actually is... is it a mapping from M -> M or some other mapping? Is the image of the function in M? Any help would be greatly appreciated!

In Topology: is the multiplicative inverse of a metric, a metric? How do we define the Schwarz inequality then? if ##d(x,z) ≤ d(x,y) + d(y,z)## the inverse ##1/d(x,z)## would give the opposite?

Homework Statement can someone help me to solve these problems in details?? Consider A =(0,1)× R. Is A open w.r.t. the topology induced by the French railway metric in R2? how about B=(-1,1)× R? 2. The attempt at a solution I know A is open in the topology induced by d if and only...

Homework Statement Find the non zero Christoffel symbols of the following metric ds^2 = -dt^2 + \frac{a(t)^2}{(1+\frac{k}{4}(x^2+y^2+z^2))^2} (dx^2 + dy^2 + dz^2 ) and find the non zero Christoffel symbols and Ricci tensor coefficients when k = 0 Homework Equations The...

Could somebody please explain something regarding the Nordstrom metric? In particular, I am referring to the last part of question 3 on this sheet -- http://www.hep.man.ac.uk/u/pilaftsi/GR/example3.pdf about the freely falling massive bodies. My thoughts: The gravitational effects...

Hello, I was wondering if it was possible (or advisable) to read Chapter 7 of Munkres (Complete Metric Spaces and Function Spaces) without having done Tietze Extension Theorem, the Imbeddings of Manifolds section, the entirety of Chapter 5 (Tychonoff Theorem) and the entirety of Chapter 6...

In Minkowski spactime (Flat), if the coordinate system makes a rotation e.g. around y-axis (centred) , for the metric ds^2, how to make the tertad (flat spacetime) as the coordinate system rotats?

Hello Everybody, Instead of solving the geodesic equations for the Schwarzschild metric, in many books (nearly in all books that I consulted), conserved quantities are looked at instead. So take for eg. Carroll, he looks at the killing equation and extracts the equation K_\mu...

I think I remember reading somewhere that all the machinery of manifolds and a metric needed to be established first before the integral and the differential of calculus had any meaning. Am I remembering wrong? Is there such a thing as coordinate independent integration or differentiation? Thanks.

How can find components of tetrads from metric ? i know the relation between tetrads and metric g_{μ \nu}=η_{ab}e^{a}_{μ}e^{b}_{\nu} where e^{b}_{\nu} are component of tetrads , in the case of Schwarzschild that metric is diagonal , it is a easy problem but what about non-diagonal metric like...

Let X be a set, and let fn : X---> R be a sequence of functions. Let ρ be the uniform metric on the space RX. Show that the sequence (fn) converges uniformly to the function f:X--> R if and only if the sequence (fn) converges to f as elements of the metric space (RX, ρ). [Note: the ρ's should...

The question comes from the Munkres text, p. 133 #3. Let Xn be a metric space with metric dn, for n ε Z+. Part (a) defines a metric by the equation ρ(x,y)=max{d1(x,y),...,dn(x,y)}. Then, the problem askes to show that ρ is a metric for the product space X1 x ... x Xn. When I originally...

Hello, I have read that, in a freely-falling frame, the metric/ interval will be of the form: ds2 = -c2dt2(1 + R0i0jxixj) - 2cdtdxi(\frac{2}{3} R0jikxjxk) + (dxidxj(δij - \frac{1}{3} Rikjlxkxl) to second order. Does anyone know where I could find a derivation of this result?

Let $A\subset X$ for $(X,d)$ metric space, then prove that $\text{diam}(A)=\text{diam}(\overline A).$ I know that $\text{diam}(A)=\displaystyle\sup_{x,y\in A}d(x,y),$ but I don't see how to start the proof. The thing I have is to let $\text{diam}(\overline A)=\displaystyle\sup_{x,y\in \overline...

Homework Statement I nedd some help to write a left-invariant and right invariant metric on SU(2) Homework Equations The Attempt at a Solution

Say we were given an expression for the energy-momentum tensor (also assuming a perfect fluid), without getting into an expression with multiple derivatives of the metric, are there any cases where it would be possible to deduce the form of the metric?

Homework Statement I've searched everywhere, and I cannot find an example of calculation of Lie derivation of a metric. If I have some vector field \alpha, and a metric g, a lie derivative is (by definition, if I understood it): \mathcal{L}_\alpha g=\nabla_\mu \alpha_\nu+\nabla_\nu...

In a paper about field theory in curved spacetime, an author says that the Lagrangian density for a free scalar particle is L = \sqrt{-g} ((\nabla_\mu \Phi)(\nabla^\mu \Phi) - m^2 \Phi^2) Is there a simple explanation for why this is scaled by \sqrt{-g}?

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...

Hi, If X \LARGE is a metric space and E \subset X is a discrete set then is E \LARGE open or closed or both? Here's my understanding: E \LARGE is closed relative to X \LARGE. proof: If p \subset E then by definition p \LARGE is an isolated point of E \LARGE, which implies that p...

This may be a stupid question, but why can't the expansion/contraction of spacetime from a gravitational wave be used to create the areas of expansion/contraction required in the Alcubierre metric, instead of using regions of positive/negative energy density? I saw on the forums about the...

Hi, In Baby Rudin, Thm 3.6 states that If p(n) is a sequence in a compact metric space X, then some subsequence of p(n) converges to a point in X. Why is it not the case that every subsequence of p(n) converges to a point in X? I would think a compact set would contain every sequence...

Hi, I'm reading Baby Rudin and have a quick question regarding topology. Given a nonempty subset E of a metric space X, is it true that the only points in E are either isolated points or limit points? (b/c all interior points are by definition limit points, but not all limit points are...

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}}}

Homework Statement show the set {f: ∫f(t)dt>1(integration from 0 to 1) } is an open set in the metric space ( C[0,1],||.||∞） and if A is the subset of C[0,1] defined by A={f:0<=f<=1} is closed in the norm ||.||∞ norm. Homework Equations C[0,1] is f is continuous from 0 to 1.and ||.||∞...

The usual proof of this theorem seems to assume that the topology of the metric space is the one generated by the metric. But if I use another topology, for example the trivial, the space need not be Hausdorff but the metric stays the same. Am I missing something or is the statement of the...

I need to build a tensor from the product of the metric components, like this (using three factors, not less, not more) : H^{\mu \nu \lambda \kappa \rho \sigma} = g^{\mu \nu} \, g^{\lambda \kappa} \, g^{\rho \sigma} + g^{\mu \lambda} \, g^{\nu \kappa} \, g^{\rho \sigma} + ..., however, that...

hi we know that our universe is homogenous and isotropic in large scale. the metric describe these conditions is FRW metric. In FRW, we have constant,k, that represent the surveture of space. it can be 1,0,-1. but the the Einstan Eq, Ricci scalar is obtained as function of time! and this...

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}

In my general relativity course we recently covered the definition of a killing vector and their importance. However, I am not completely comfortable calculating the killing vectors for a given metric (in a particular case, the 2-sphere), and would like to know if anyone knows of a good...

hello Whic one of these to metric are Minkowski metric ds^2 =-(cdt)^2+(dX)^2 ds^2 =(cdt)^2-(dX)^2 and what about timelike (ds^2<0) and spacelike (ds^2>0) for each metric? With my appreciation to those who answer

Homework Statement Let X and Y be metric spaces such that X is complete. Show that if {fα(x) : α ∈ A} is a bounded subset of Y for each x ∈ X, then there exists a nonempty open subset U of X such that {fα(x) : α ∈ A, x ∈ U} is a bounded subset of Y. Homework Equations Definition of...

Hello everybody! I have some questions concerning the structure of the Schwarzschild metric, which is given by $$ ds^2=-(1- \frac{2GM}{r})dt^2+(1-\frac{2GM}{r})^{-1}dr^2+ r^2(d\theta^2+ \sin^2(\theta) d\phi^2) $$ where we set $c=1$. I would like to know the following: \\ \\ 1. Why is it...

In the Wiki article http://en.wikipedia.org/wiki/Vaidya_metric it states that but it looks to me as if the 'particles' are traveling at the speed of light ( null propagation vector field ) and so must have zero rest mass. Is this a typo or have I misunderstood something ?

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.

Double contraction of curvature tensor --> Ricci scalar times metric I'm trying to follow the derivation of the Einstein tensor through double contraction of the covariant derivative of the Bianchi identity. (Carroll presentation.) Only one step in this derivation still puzzles me. What I...

Hello guys , beside Einstein's General Theory of Relativity by hervik , is there any lecture or book about this case ? Thx

Hey guys! I am considering a space with a diagonal metric, which is maximally symmetric. It can be proven that in that case of a diagonal metric the following equations for the Christoffel symbols hold: \Gamma^{\gamma}_{\alpha \beta} = 0 \Gamma^{\beta}_{\alpha \alpha} =...

Hi all, S. Martin's Supersymmetry primer (http://arxiv.org/abs/hep-ph/9709356) is a wonderful source from which to learn SUSY. But, what really causes me (and others around me) huge consternation is Martin's use of mostly plus metric, when particle physicists use the mostly minus metric...

Let $$f:U \to \mathbb{R}^3$$ be a surface, where $$U=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|<3, |u^2|<3\}.$$ Consider the two closed square regions $$4F_1=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq1, |u^2|\leq1\}$$ and $$F_2=\{(u^1,u^2)\in \mathbb{R}^2:|u^1|\leq2, |u^2|\leq2\}.$$ While the first...

Kerr–Newman metric: c^{2} d\tau^{2} = - \left(\frac{dr^2}{\Delta} + d\theta^2 \right) \rho^2 + (c \; dt - \alpha \sin^2 \theta \; d\phi)^2 \frac{\Delta}{\rho^2} - ((r^2 + \alpha^2) d\phi - \alpha c \; dt)^2 \frac{\sin^2 \theta}{\rho^2} I used the Kerr–Newman metric equation form listed on...

Given a Finsler geometry (M,L,F) and $$g_{ab}^L=\frac{1}{2} \frac{\partial^2 L}{\partial y^a \partial y^b}$$ $$g_{ab}^F=\frac{1}{2} \frac{\partial^2 F^2}{\partial y^a \partial y^b}$$ $$F(x,y)=|L(x,y)|^{1/r}$$ I manage to get the following form $$g_{ab}^F=\frac{2|L|^{2/r}}{rL}(...

yeh well, I once understood this, but now looking at it today I can't get it to make sense intuitively. The quantity: dx2+dy2+dy2-c2dt2 is the same for every intertial frame in SR - just like length is the same in all inertial frames in classical mechanics. Now I am not sure that I...

Hi all, I am trying to understand geometric flows, and in particular the Ricci flow. I understand how to calculate the metric tensor from the parametrization of a surface, but I am facing a problems in the concept phase. A metric tensor's purpose is to provide a coordinate invariant...

Y ⊂ X where X is a metric space with the function d. Prove that (Y,d) is a metric space with the same function d. The metric function d: X x X -> R. I know that the function for Y is: d* : Y x Y -> R How do I show that d is the same as d*.

Like any mathematical relativity between them as per General Relativity?

http://imageshack.us/a/img12/8381/37753570.jpg I am having trouble with this question, like I do with most analysis questions haha. It seems like I must show that the maximum exists. So E is compact -> E is closed To me having E closed seems like it is clear that a maximum distance...