Curvature Definition and 920 Threads

Hello all. I am working on a research project involving the Stark effect and its application in molecular guides and came across a bit of math in a paper that I don't understand. In this paper http://arxiv.org/abs/physics/0310046 there is an equation in the introduction concerning the electric...

Here's what I'm watching: At about 1:35:00 he leaves it to us to look at a parallel transport issue. Explicitly to caclculate ##D_s D_r T_m - D_r D_s T_m## And on the last term I'm having some difficulties, the second christoffel symbol. So we have ##D_s [ \partial_r T_m - \Gamma_{rm}^t T_t]##...

hi, I am just curious about, and really wonder if there is a proof which demonstrates that curvature tensor is 0 in all flat space coordinates. Nevertheless, I have seen the proofs related to curvature tensor in Cartesian coordinates and polar coordinates, but have not been able to see that zero...

According to this paper, eternal inflation would be falsified by positive curvature: http://arxiv.org/pdf/1203.6876v2.pdf However the proposer of eternal inflation, Alex Vilenkin, has suggested spontaneous creation of the universe from"nothing". Apparently this doesn't violate the conservation...

I was reading a paper by Starobinsky on spectrum of relict gravitational radiation. He uses the term Planck curvature as follows "We construct a model in which the universe was perpetually in the quantum state with the radius of curvature of the order of Planck curvature, and later left this...

If I know the density curvature parameter for today $$ \Omega_{k,0} = \frac{-c^{2}k}{R_{0}H_{0}} $$ then is it possible to surmise what it would be, say, during the matter dominated era $$ \Omega_{k,t} = \frac{-c^{2}k}{R(t)H(t)}$$ ?

I want to program space curvature vizualizaion. I want to have an observer as a player that moves in 3d curved space and surrounding objects that will show curvature by distortion when player passes near them. I am concerned about some points: 1. What curvature to choose in order to experience...

Homework Statement A circular rod has a radius of curvature R = 8.11 cm, and a uniformly distributed positive charge Q = 6.25 pC and subtends an angle theta = 2.40 rad. What is the magnitude of the electric field that Q produces at the center of curvature? Homework Equations E = kQ/r^2 6.25 pC...

Dear PF Forum, I have a confusion about gravity. And frankly I don't know if this question belongs to this sub forum (cosmology, general physic?). Gravity attracts object - Newton Gravity curves space time - Einstein. Why we revolve around a massive object? Because that massive object curves...

Can someone explain mathematically why do we say Riemann Curvature Tensor has all the information about curvature of Space Thank You

I can understand that a free falling frame in a curved space with a non-zero Riemann tensor has a zero Ricci tensor but I have a doubt about the opposite; does an accelerating frame in a vacuum space have a non-zero Ricci or Riemann tensor? if so, where do components of energy momentum tensor...

From what I read attempts to measure the curvature of space have not succeeded. It would seem there may not be a curvature of space time. If this is true then what may be implied is that space goes on forever. If this is true how could the big bang theory, if it could, give a reasonable answer...

Homework Statement Page 16 (attached file) \frac{dH}{dt}|_{t=0} = Δ_{Σ}φ + Ric (ν,ν)φ+|A|^{2}φ \frac{d}{dt}(dσ_{t})|_{t=0} = - φHdσ H = mean curvature of surface Σ A = the second fundamental of Σ ν = the unit normal vector field along Σ φ = the scalar field on three manifold M φ∈C^{∞}(Σ)...

Hey, I've been looking into some civil/structural engineering for a school project, and came across bridge design. I've decided to try some integrating and optimising to do with an arch bridge (optimal cost/strength proportions). The math isn't too hard, but what I'm struggling with so far is...

I understand(or assume understand) that geodesic deviation describes how much parallel geodesics diverge/converge on manifolds while moving along these geodesic. But is not it a definition for intrinsic curvature? If it is same as Riemann curvature tensor in terms of describing curvature, why...

Hello Forum, I have read Einstein famous thought experiments about the elevator. 1) Being inside an elevator accelerating upward in absence of a gravitational field is equivalent to being inside the same elevator at rest inside a homogeneous gravitational field. 2) An elevator in free fall...

Please bear with me because I'm only in Pre-calculus and am taking basic high school physics. This is completely outside of my realm but curiosity has taken the better of me. I just learned last week about the difference between Euclidean Geometry and Riemmanian Geometry (from another thread...

G-Waves is a buzzword recently :) At the beginning I thought G-waves as the propagation of the changes of the curvature caused by a mass when the status of the mass (e.g. value or location) changes...But moment ago, I was told that G-waves are different from the waves that transmitting the...

Homework Statement why when ℓ / R ≪ 1 , the curvature effect can be neglected ?? Homework EquationsThe Attempt at a Solution no matter how small or how big is R , it is strill considered as curvature , right ??

I know of no scientific reason to suppose that "dark energy" is anything more than the cosmological curvature constant identified by Einstein in 1917 as occurring naturally in the GR equation for spacetime curvature. It might eventually turn out to be related to some type of energy. That's...

Hi, The following textbook Heisenberg's Quantum Mechanics shows an example of calculating Berry's curvature (top page on pg 518). It led to a following equation Vm= (- 1/B2 ) * i *∑ ( <m,B|S|n,B> ∧ <n,B|S|m,B> ) / A2 ...[1] the textbook claims that we add the term m = n since <m|S|m> ∧ <m|S|m>...

Should I post in Diff Geometry? I searched that forum, and did not see what I was looking for. I want to compute all 256 components of the curvature tensor. Do I start with the equation of geodesic deviation in component form, or can I go straight to the definition of the components in terms of...

If space is warped around heavy objects in space, i feel that space would be FUBAR around black holes. So, my question is, Does light get sucked in by gravity, or does it just get caught in the warped space around a black hole?

So I'm working with some group manifolds. The part that's getting to me is the Ricci scalar I'm using to describe the curvature. I have identified the groups that I'm using but that's not really relevant at the moment. We're using a left-invariant metric ##\mathcal{M}_{ab}##. Now I've got the...

Homework Statement This is a problem from A. Zee's book EInstein Gravity in a Nutshell, problem I.5.5 Consider the metric ##ds^2 = dr^2 + (rh(r))^2dθ^2## with θ and θ + 2π identified. For h(r) = 1, this is flat space. Let h(0) = 1. Show that the curvature at the origin is positive or negative...

Homework Statement Homework Equations I know that the tangential accel is v = wr and that Centripetal = v^2/r The Attempt at a Solution For A, I thought it would be straight forward if I had the radius as well as omega. I know that the distance between A and B is 60in, but I don't think it...

Spacetime shrinks in a gravitational field. As I understand it, objects falling into a black hole will appear to contract in size and run slower as they approach the horizon. This is similar to how things contract and slow down when traveling close to the speed of light because they are...

Since the very rapid expansion of the Universe, ie Inflation, caused curvature to be smoothed out, the assumption must be that curvature existed between the time of the Big Bang and the start of Inflation. If that is correct, is this borne out by the Planck or any other theory?

I've been playing around with the Carminati-McLenaghan invariants https://en.wikipedia.org/wiki/Carminati–McLenaghan_invariants , which are a set of curvature scalars based on the Riemann tensor (not depending on its derivatives). In general, we want curvature scalars to be scalars that are...

Hi! I would like to calculate the curvature for a surface S:R^2->R'3 numerically. The problem: I simply have the surface as a mesh like you see in the image attached. I calculated the linearly interpolated face normals in the nodes, too. You see the vectors. Question: How would you calculate...

Geroch 1968 touches on the Kundt type I and II curvature invariants. If I'm understanding correctly, then type I means curvature polynomials. Type II appears to be something else that I confess I don't understand very well. (I happen to own a copy of the book in which the Kundt paper appeared. I...

Wikipedia states that: "If the measurement is close enough to the surface, light rays can curve downward at a rate equal to the mean curvature of the Earth's surface. In this case, the two effects of assumed curvature and refraction could cancel each other out and the Earth will appear flat in...

Homework Statement For the first problem I am asked to find the curvature for y=cosx We are studying vector value functions so I tried to rewrite this as a vector valued function so I can find the curvature. I just chose r(t)= <t,cost,0>. I found rI(t)=<1,-sint,0> and rII(t)=<0,-cost,0> and...

Reading Geroch's "What is a Singularity in General Relativity?", it seems that polynomial scalar invariants constructed from the Riemann tensor can diverge if we are at infinite distance, and not in a true singularity. Can someone give an example of space-time whose scalar invariant diverges...

Does anyone know of an example, preferably a simple one, that can be used to demonstrate that we can have a curvature singularity without a singularity in the Kretschmann scalar?

Going from the Newtonian to relativistic version of Friedmann's equation we use the substitution kc^{2} = -\frac{2U}{x^{2}} The derivation considers the equation of motion of a particle with classic Newtonian dynamics. I can sought of see that if space is flat the radius of curvature will be...

Suppose we are in a Minkowskian space, away from all the source of gravity, and observe an accelerated frame from this frame. Acoording to Equivalence principle, we can consider the accelerated frame to be at rest and assume we have gravity in the accelerated frame. Thus, observer in the...

I don't know much about differential geometry but I hope this is a good place to ask and that my question "makes sense" I have heard that an ornamental cabbage leaf is an example of a surface with an intrinsic curvature. If one wanted to make such a surface from scratch(and to detailed...

Homework Statement For a generic function y=f(x) which is twice-differentiaable, is it possible for there to be a curvature on the curve of that function that is greater than the curvature at its relative extremum?Homework Equations The Attempt at a Solution From visualization and a sketch...

Homework Statement Given the polar curve r=e^(a*theta), a>0, find the curvature K and determine the limit of K as (a) theta approaches infinity and (b) as a approaches infinity. Homework Equations x=r*cos(theta) y=r*sin(theta) K=|x'y''-y'x''|/[(x')^2 + (y')^2]^(3/2) The Attempt at a Solution...

Homework Statement Let T be the tangent line at the point P(x,y) to the graph of the curve ##x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}, a>0##. Show that the radius of curvature at P is three times the distance from the origin to the tangent line T.Homework Equations R=1/K ##R=\frac{\left...

Homework Statement A highway has an exit ramp that beings at the origin of a coordinate system and follows the curve ##y=\frac{1}{32}x^{\frac{5}{2}}## to the point (4,1). Then it take on a circular path whose curvature is that given bt the curve ##y=\frac{1}{32}x^{\frac{5}{2}}## at the point...

Find the curvature. r(t) = 9t i + 5 sin(t) j + 5 cos(t) k

Find the curvature at the point (2, 4, −1). x = 2t, y = 4t3/2, z = −t2

Find the curvature. r(t) = 3t i + 5t j + (6 + t2) k κ(t) = This is what i have so far: derivative of r(t) = 3i + 5j + 2k I know the next step is to find the cross product of r(t) and r'(t) but I'm not sure how to go about it, especially how to make use of the k vector of r(t)

Hi, I am taking Calc III, but I am having a hard time understanding some of the concepts. Right now I am struggling with understanding the curvature of a line. What I have in my notes is this: Curvature second derivative (rate of change of tangent line)(rate of change w/ respect to arc length)...

How does one derive the general form of the Riemann tensor components when it is defined with respect to the Levi-Civita connection? I assumed it was just a "plug-in and play" situation, however I end up with extra terms that don't agree with the form I've looked up in a book. In a general...

Do gravitational forces have to follow spacetime in the same way as light? Or does gravity act in a higher dimension? Thanks, T

Hello, I just joined. I have no formal background in physics, just curiosity. So, my questions may well be simplistic to most of you. Hopefully, that is permissible, now and then! First question, if space, as it's usually defined, is empty, nothing, how can it be curved? Hankb

What is a nonscalar curvature singularity, in the context of "the https://en.wikipedia.org/w/index.php?title=Wave_of_death&action=edit&redlink=1 is a gravitational plane wave exhibiting a strong nonscalar null...