Metric Definition and 1000 Threads

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.

hi why only scalar cam build with second order of derivative of metric is Ricci scalar? thanks

Let $f$ be a continuous mapping of a compact metric space $X$ into a metric space $Y$ then $f$ is uniformly continuous on $X$. I have seen a proof in the Rudin's book but I don't quite get it , can anybody establish another proof but with more details ?

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

Homework Statement It is clear that a countable complete metric space must have an isolated point,moreover,the set of isolated points is dense.what example is there of a countable complete metric space with points that are not isolated? Homework Equations The Attempt at a Solution

The textbooks claim that the weak field (Newtonian) metric is more intuitive than the Schwarzschild metric, but I don’t agree.The time correction factor for the weak field metric is the same as that for the Schwarzschild metric. But for the length correction factor for the weak field metric is...

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...

Homework Statement Consider metric ds2 = dx2 + x3 dy2 for 2D space. Calculate all non-zero christoffel symbols of metric. Homework Equations \Gammajik = \partialei / \partial xk \times ej The Attempt at a Solution Christoffel symbols, by definition, takes the partial of each...

Hi all, Perhaps I'm asking the wrong question but I am wondering about the relationship between different definitions of, for the sake of argument, the torus. We can define it parametrically (or as a single constraint) and from there work out the induced metric as with any surface. But...

Homework Statement Let (X,d) be any metric space. Fix a in X and for each x in X define fx:X→ℝ by: fx(z)=d(z,x)-d(z,a) for all z in X. Show that fx(z) is bounded and continuous. The Attempt at a Solution I can't figure out how to tell if it is bounded. Any hints? I'm sure...

Hi, I am working through GR by myself and decided to derive the Friedmann equations from the RW metric w. ( +,-,-,-) signature. I succeeded except that I get right value but the opposite sign for each of the Ricci tensor components and the Ricci scalar e.g. For R00 I get +3R../R not -3R../R . I...

Hello! I'm trying to prove that ##d_1\left({x,y}\right)=\max\{\left|{x_j-y_j}\right|:j=1,2,...,k\}## is a metric. I know that since ##d_1\left({x,y}\right) = \left|{x_j-y_j}\right|## for some ##j## that ##d_1\left({x,y}\right) \geq 0## and since ##\left|{x_j-y_j}\right| =...

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

I'm interested in part iv) on the attachment. This is my work so far: e=(1,0) and e'=(0,1) form a basis of the tangent space at any point z=(x,y). Making the identification (x,y)->x+iy, we get g(e,e')=0 and g(e,e)=g(e',e')=$\frac{1}{im(z)^2}$. a(t)=z+t and b(t)=z+it are generating curves for...

In this documentation from Nasa a procedure to get to what I guess is the gravitational acceleration according to the post-Newtonian expansion at the 1PN-level for the spherically symmetric case is found: http://descanso.jpl.nasa.gov/Monograph/series2/Descanso2_all.pdf The procedure is...

Homework Statement Show that the sequence (xn)n\inN \inZ given by xn = Ʃ from k=0 to n (7n) for all n \in N is a cauchy sequence for the 7 adic metric. Homework Equations In a metric space (X,dx) a sequence (xn)n\inN in X is a cauchy sequence if for all ε> 0 there exists some M\inN such...

Hello, I don't know whether I have mentioned the subject line properly. Many times while reading over General Relativity I come across the following equation: ds^2=dx1^2+dx2^2+dx3^2+dx4^2 =dx^2+dy^2+dz^2-c^2dt^2. Now, my question from the above equation is: (a) Are we putting...

What is the most general form of the metric for a homogeneous, isotropic and static space-time? For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) signature) ds^2=dt^2+a^2(t)g_{ij}(\vec x)dx^idx^j Now the static condition. If I'm not mistaken...

I can't see how to get the following result. Help would be appreciated! This question has to do with the Riemann curvature tensor in inertial coordinates. Such that, if I'm not wrong, (in inertial coordinates) R_{abcd}=\frac{1}{2} (g_{ad,bc}+g_{bc,ad}-g_{bd,ac}-g_{ac,bd}) where ",_i"...

I'm confused about the the definition of a function not being continuous. Is it correct to say f(x) is not continuous at x in the metric space (X,d) if \existsε>0 such that \forall\delta there exists a y in X such that d(x,y)<\delta implies d(f(x),f(y))>ε Is y dependant on \delta? It...

There seems to be an emphasis in several books on general relativity that the metric (components) in itself does not reflect anything physical, only our choice of coordinates. On the other hand it can seem like the authors, instead of being true to this, treat the metric (components) as...

I must be overlooking something! Given a metric space (E,d), the improper subset E is open in E. How? Here is my understanding: 1) We call a set S(subset of E) open iff for all x(element of S) there exist epsilon such that an open ball of radi epsilon centered about s is wholly contained in...

Hi guys. This question is related to Problem 6.3 in Wald which involves deriving the Reissner-Nordstrom (RN) metric. We start with the source free Maxwell's equations ##\nabla^{a}F_{ab} = 0,\nabla_{[a}F_{bc]} = 0## in a static spherically symmetric space-time which, in the coordinates adapted to...

Homework Statement Show that observers whose world-lines are the curves along which only ξ varies undergo constant proper acceleration with invariant magnitude ημν(duμ/dτ )(duν/dτ ) = 1/χ2. Homework Equations ds2 = −a2χ2dξ2 + dχ2 + dy2 + dz2 xμ = {ξ, χ, y, z} t = χsinh(aξ) x...

Thanks in advance - this problem has been bothering me for a while! I'm working with an unpowered spaceship orbiting a large mass M. The orbit is circular and it is following the geodesic freely. It has an orbit radies of r = R. My question is this. The metric of the space-time curvature...

Homework Statement "Evaluate: g^{\mu \nu} g_{\nu \rho} where ds^2 = g_{\mu \nu} dx^\mu dx^\nu , ds^2 = c^2 dt^2 - dx^2 - dy^2 - dz^2 " Homework Equations None necessary, just a notation issue The Attempt at a Solution Just using raising and lowering rules, I imagine each raising and...

Looking for clarification on something. Take the metric in this form ds^2 = dt^2 - dx^2 Brian Green likes to say a null path is one where motion through space is shared equally with motion through time, or (ds=0). Motion through space is not allowed that is "faster" than motion...

hi , how i can demonstrate the expression of metric kerr?

How does one go about finding what the gravitational time dilation is from the metric? Is it simply t'/t_0=1/\sqrt{g_{tt}}? It seems that could be true for static metrics, but perhaps not more dynamic ones like the Kerr metric. My confusion on this arises on how to treat the time cross terms...

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,

Homework Statement If S is a subspace of the metric space X prove (intxA)\capS\subsetints(A\capS) where A is an element of ΩX(Open subsets of X) The Attempt at a Solution So intxA=\bigcupBd(a,r) where d is the metric on X and the a's are elements of A and I think...

Homework Statement A dissimilarity measure d(x, y) for two data points x and y typically satisfy the following three properties: 1. d(x, y) ≥ 0 and d(x, y) = 0 if and only if x = y 2. d(x, y) = d(y, x) 3. d(x,z) ≤ d(x, y) + d(y,z)The following method has been proposed for measuring the...

Hi everyone, I posted this a couple days ago and didn't get a response, so I thought I'd try again. Let me know if something about this is confusing. Thanks! Homework Statement Let X be a metric space and let x\in{X} be any point. Prove that the set \left\{x\right\} is closed in X...

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.

Hello all, I've been puzzled by this problem for some time now and was wondering if anyone here could help me out. Textbooks on GR (specifically when going into gravitational waves) tend not to elucidate this. It's often taken for granted that through the gauge diffeomorphism invariance (or...

Given the proposition: A subset U of a metric space is an open set iff U is a union of open balls. Then we are told that it follows from the above proposition that two metrics are equivalent if each open ball of either of the metric topolgies is an open set of the other topology. This is...

a metric on the plane determines an action of SO(2) on is unit circle by rotation. Suppose one starts with a free transitive action of SO(2) on a circle. When does this come from a metric? Always?

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.

So I ran into a question; Show that there is no metric on S^2 having curvature bounded above by 0 and no metric on surface of genus g which is bounded below by 0. honestly I have no idea what is going on here. I know that a Genus is the number of holes in some manifold or the number of...

Hi can anyone explain how to find \delta \sqrt{-g} when varying with respect to the metric tensor g^{\mu\nu}. i.e why is it equal to \delta \sqrt{-g} = -\frac{1}{2} \sqrt{-g}g_{\mu\nu} \delta g^{\mu \nu}

If you are given a metric gαβ and you are asked to find if it describes a flat space, is there any way to answer it without calculating the Riemman Tensor Rλμνσ? and how can I find for that given metric the coordinate transformation which brings it in conformal form? For example I'll give you...

Metric Space and Topology HW help! Let X be a metric space and let (sn )n be a sequence whose terms are in X. We say that (sn )n converges to s \ni X if \forall \epsilon > 0 \exists N \forall n ≥ N : d(sn,s) < \epsilon For n ≥ 1, let jn = 2[(5^(n) - 5^(n-1))/4]. (Convince yourself...

Homework Statement X is a metric and E is a subspace of X (E\subsetX) The boundary ∂E of a set E is defined to be the set of points adherent to both E and the complement of E, ∂E=\overline{E}\cap(\overline{X\E}) (ignore the red color, i can't get it out) Show that E is open if and only...

So I was flipping through Lang's Undergraduate Analysis and noticed the absence of the important concept of metric spaces. I checked the index and was referred to problem 2, Chapter 6 Section 2. There he defines a metric space, what a bounded metric is, and gives a few straightforward problems...

Are the axioms of a Norm different from those of a Metric? For instance Wikipedia says: a NORM is a function p: V → R s.t. V is a Vector Space, with the following properties: For all a ∈ F and all u, v ∈ V, p(av) = |a| p(v), (positive homogeneity or positive scalability). p(u + v) ≤ p(u)...

could someone specify a metric on (0,1) that defines (the same topology) as the abs. value (i.e. usual) metric and makes this open interval into a complete set? Thanks

Am i correct in thinking that in a simple ds^2 =dt^2 - a^{2}d\underline{x}^{2} metric that: g_{\mu\nu}=diag(1,-a^2,-a^2,-a^2) and g^{\mu\nu}=diag(1,-\frac{1}{a^2},-\frac{1}{a^2},-\frac{1}{a^2}) thx

Homework Statement Let X be an infinite set. For p\in X and q\in X, d(p,q)=1 for p\neq q and d(p,q)=0 for p=q Prove that this is a metric. Find all open subsets of X with this metric. Find all closed subsets of X with this metric. Homework Equations NA The Attempt at a...

Hi does anyone know how to calculate: \delta (det|g_{\mu\nu}|) or simply \delta g