Curvature Definition and 872 Threads

When spacetime is not bent the two objects, red ball and blue ball, will move strait up the y-axis as they move through time. (Space is x and time is y). Now I've made the assumption that either a) All things want to move the smallest possible distance to the next point in time or b) all...

Gravity causes spacetime to warp. Relative speed also causes an apparent warp from the point of view of the stationary observer. But warp due to relative speed will cause rods to contract, rods will effectively measure shorter for the stationary observer. Accordingly we should also infer that...

As far as I understand it, Einstein theorized that gravity was the result of the curvature of space created by the presence of mater/energy, but that idea seems like it does not meld well with the idea that gravity is the result of a specific force carrying particle, as with the other...

Hi, So according to GR, energy bends spacetime, right? So that would include both light and mass. If I am understanding this right, light bends spacetime, and is also affected by the spacetime curvature. Could someone explain exactly what the spacetime curvature does to light (I mean like how...

Dear PF Forum, I realize that there are 4 basic forces in our universe. Two of them are gravity and electromagnetic force. Electromagnetic travels at the speed of light. And it seems that Gravity travels/propagates at the speed of light also. Light is bent by gravity. What about gravity? Is...

In the Einstein Field Equations: Rμν - 1/2gμνR + Λgμν = 8πG/c^4 × Tμν, which tensor will describe the coordinates for the curvature of spacetime? The equations above describe the curvature of spacetime as it relates to mass and energy, but if I were to want to graph the curvature of spacetime...

What is the value of curvature which caused by earth(geodetic effect) on space-time?

Hi, ##x(s)=\cos\frac{s}{\sqrt{2}}## ##y(s)=\sin\frac{s}{\sqrt{2}}## ##z(s)=\frac{s}{\sqrt{2}}##, it is a unit-speed helix. Its curvature is ##\kappa=||\ddot{r}||=\frac{1}{2}##. Principal unit normal is ##{\mathbf n}=(\cos\frac{s}{\sqrt{2}},\sin\frac{s}{\sqrt{2}},0)##. So far so good... But the...

Homework Statement Solve the following equation: v is the dependent variable, x is the independent variable Homework Equations \frac{d^2v/dx^2}{(1+\frac{dv}{dx}^2)^{3/2}}=1 The Attempt at a Solution Hi, I am trying to solve the following equation...

could the bend be described mathematically? Not the vector the curvature.

[Moderator note: This has been spun off into a separate thread from here.] I've never been on a forum so I apologize if I'm asking a question in the wrong place. If so, would someone kindly directing me where to post. If gravity warps space around mass, wouldn't the curving of space create...

As per ww.physicsclassroom.com/class/refln/Lesson-3/The-Anatomy-of-a-Curved-Mirror The point in the center of the sphere from which the mirror was sliced is known as the center of curvature,I am not able to understand this.Please help.

A 2013 paper ( http://arxiv.org/pdf/1312.4854v1.pdf ) gives the following: Ωk = 0.0010 ± 0.0029. Can a value for the radius of curvature be determined from Ωk = 0.0010? If so, what is it, and what is the formula? If not, why not?

If the fabric of the universe is made of both space and time, and curving spacetime affects time, then I'm guessing it also affects space. I'm aware that an object shortens in length as it approaches the speed of light. But in the case of gravity, is space relative like time? Does an object on...

Hello friends, I am working on a design project for my capstone course in my engineering curriculum. Part of the design involves a cord consisting of wires tightly wrapped helically (à la a spring) around a nylon rope core. An important specification of this design is the radius of curvature...

Homework Statement Find the curvature at the point (x, y) on the ellipse x^2/9+y^2/4=1. Homework Equations None. The Attempt at a Solution x^2/a^2+y^2/b^2=1 so I know that a=3 and b=2 for this problem. x(t)=acos(t) and y(t)=bsin(t) so x(t)=3cos(t) and y(t)=2sin(t) now what? What's the...

Hi, I am a first time poster and I am completely lost with this question. Any help would be greatly appreciated Filling the space between a contact lens and the cornea is a small quantity of liquid of refractive index of 1.336. Assuming the refractive index of the lens material is 1.490 find...

Hi, I am a first time poster and I am completely lost with this question. Any help would be greatly appreciated Filling the space between a contact lens and the cornea is a small quantity of liquid of refractive index of 1.336. Assuming the refractive index of the lens material is 1.490 find...

Currently reading Peter Coles, Cosmology a very short introduction. There is a bit I don't understand. In a section discussing Friedmann Models, and how going on the cosmological principle density of the universe is the same in every place, and therefore space must be warped in the same way at...

For a physical singularity I think it is sufficient that anyone scalar quantity blows up, Why is it not a necessary condition that all blow up? For a curvature singularity am I correct in thinking that it is a sufficient condition to find a coordinate system in which the metric coefficient no...

I am trying to get my head around curvature, geodesics and acceleration in GR. I've put together the following paragraph that attempts to describe qualitatively how I think these things play together. In Newtonian mechanics, a freely falling object accelerating towards the Earth implies a force...

My question is related to M. Alcubierres paper on the warp drive within general relativity. I was wondering about the realizability of this, setting the three energy conditions aside for the moment. Assuming the highest energy density known to me as an energy source, namely something like...

The ratio of the circumference of a circle to the radius is Pi. Our value of Pi is an irrational number and is calculated assuming a flat curvature of space-time. But our universe as a whole, while very flat, probably has some small amount of curvature -- in addition to the greater local...

Assuming a mass warp spacetime such that the curvature of spacetime extend one light year away from that object. If I am standing at 1 light year away from the object and the object start losing mass by emitting light, will you feel the change in the curvature first before the radiation reach...

Homework Statement Find the curvature of y=sec x. Homework Equations None. The Attempt at a Solution k(x)=abs(y")/[1+(y')^2]^(3/2) y'=secx*tanx y"=secx(sec^2 x+tan^2 x) k(x)=abs(secx(sec^2 x+tan^2 x))/[1+(secx*tanx)^2]^(3/2) how do I simplify this?

Hello, given a stationary pointlike particle with mass m at some position, I'm trying to compute just how much space is curved/deformed at a distance r from that particle due to its gravitational field. I'm not really into all that tensor calculus, so I really struggle with the equations given...

Homework Statement Find the curvature of r(t)=t*i+(1/2)t^2*j+t^2*k. Homework Equations None. The Attempt at a Solution r'(t)=<1, t, 2t> r"(t)=<0, 1, 2> r'(t)xr''(t)=<0, t, 4t>

I am studying differential geometry of surfaces. I am trying to understand some features of the first fundamental form. The first fundamental form is given by ds2 = αijdxidxj Now if the αijs are all constants (not functions of your variables) then I think (correct me if I'm wrong) that the...

How can I mathematically determine curvature for a satellite dish for a specific radio wavelength?

How is 3-sphere curvature measured? If a 2-D being living "in" the surface of a sphere tried to measure the 3-D curvature of the sphere, how would they go about it? They couldn't detect the curvature by looking for curvature in the paths of signals, because if the surface of their sphere was as...

which formula is correct? or

I just watched susskind video on EFE but he didnt show us how to convert curvature tensor(the one with 4 indices) to that of Ricci tensor. Can anyone help me with this? Try to simplify it as I just started this.

I recently watched Susskind video on general relativity. I am unsure why the commutator of the covariant derivative of the vectors is nonzero when there is curvature. E.g. DrDsVm-DsDrVm In flat space, that difference is zero. But why is it non zero in curved space? Someone please enlightened...

I'm looking at lecture notes on General Relativity by Sean M. Carroll, and after defining the Riemannanian tensor in the usual theorem, the extent to which the partial derivatives of a vector field fail to commute, it says ' having defined curvature tensor as something which characterizes the...

Homework Statement Find the curvature of r(t)=2ti+2tj+k. Homework Equations None. The Attempt at a Solution The answer is 0 in the book. I know the formula for curvature is k(t)=abs(r'(t)xr"(t))/abs(r'(t))^3. I know that r'(t)=2i+2j and r"(t)=0, so r'(t)xr"(t)=0? How to find r'(t)xr"(t) and...

I am having a hard time grasping contact curvature sums. Can someone give me a link to where there is a guide or a youtube video? or can someone help me here please. Here are the equations: 1/Rx = 1/rax + 1/rbx 1/Ry = 1/ray + 1/rby 1/R = 1/Rx + 1/Ry The question is this: The ball-outer...

Hi, as we all know, the FLRW metric has 3 types of spatial curvatures, spherical, flat, and hyperbolic. I understand that of course due to homogeneity, this curvature must be spatially everywhere the same and so can not depend on the spatial coordinates. However, I can't recall what is the...

Hello all, I have a quick question regarding the relation of the space-time metric and the curvature. I have determined the space-time metric, g_(alpha beta), but I am unsure as how to go from the line element ds^2 = [ 1 + (dz/dr)^2] dr^2 + r^2 dtheta^2 and the space-time metric g to the...

Assuming that my understanding is correct, I believe it was Einstein who proposed that gravity is the result of the warping or curving of space-time. My question is this: if gravity, which is solely attractive in nature, is the result of warped or curved space time, then is it possible for the...

A ball thrown up comes down again because the ball follows the curvature of spacetime. So is it possible that the ball comes down and before hitting the surface, goes up again due to curvature of the spacetime,which now point upward.

Hello guys. I was thinking about alternatives to inflation, especially old ones (such as the hawking-hartle state and imaginary time) and I remebered a theory put foward by Penrose, in which his relatively new CCC is based. Called the Weyl Curvature Hypothesis. No idea of what it is. Could you...

I teach a class on astronomy and recently tried to explain the curving of space time by massive objects like neutron stars and black holes. I even used a sheet of spandex to represent space-time which we bent using different weights. However my students were very confused how space, which they...

Homework Statement (Self study.) Several sources give the following for the Riemann Curvature Tensor: The above is from Wikipedia. My question is what is \nabla_{[u,v]} ? Homework Equations [A,B] as general purpose commutator: AB-BA (where A & B are, possibly, non-commutative operators)...

So I am working on the next quadratic Lagrangian: $$ L = \alpha R_{\mu\nu}R^{\mu\nu} + \beta R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} + \gamma R² $$ I have already derived $$ \delta (R_{\mu\nu}R^{\mu\nu}) = [- \frac{1}{2}g_{\mu\nu}R_{\alpha\beta}R^{\alpha\beta} +...

I have come about few mathematical problems related to Riemann Tensor analysis while learning General Relativity. Should I ask these questions in this section or in the homework section. They are pretty hard!

Homework Statement Assume that the surface has the first fundamental form as E = G = 4(1+u2+v2)-2 F = 0[/B] Homework Equations K = \frac{-1}{2\sqrt{EG}}[(\frac{E_v}{\sqrt{EG}})_v + (\frac{G_u}{\sqrt{EG}})_u][/B]The Attempt at a Solution Ev = -16v*(1+u2+v2)-3 Gu = -16u*(1+u2+v2)-3 When...

We know that a charged particle will have a drift velocity in both a curved magnetic field and when there is a transverse spatial gradient in the magnitude of the magnetic field. This drift velocity is added to the rotation velocity around the the field line. In both cases the force vector on...

Homework Statement Consider a simplified model of the human eye, in which all internal elements have the same refractive index of n = 1.40. Furthermore, assume that all refraction occurs at the cornea, whose vertex is 2.50 cm from the retina. Calculate the radius of curvature of the cornea such...

Good evening. I am a current astrophysics undergraduate who is currently having a $10 bet with my classical mechanics professor (chemistry/mathematics background) over what the official curvature of the universe is. While my academic level of understanding is still not quite high enough to...

Hello all I just joined this forum so forgive me for jumping right in but I have a question about Gravity and the curvature of space time that I can't get answer with a Google search. My question: though I understand that an object remains in orbit because of the curvature of space time and it...