Curvature Definition and 872 Threads

So that they become: ## g^{\sigma \rho} \nabla_\sigma \nabla_\rho R ~g_{\mu\nu} + R ~R_{\mu\nu} - \nabla_\mu \nabla_\nu R ##

I'm writing a paper on how to calculate the visible curvature of the horizon for any given altitude. I'm considering attempting to have it published. I wonder if anyone could suggest a publication that might be interested in such a paper. If this isn't the best forum for this request, please...

How do i sketch a psi function when there is no sinus or cosinus i see sometimes its curving but sometimes its directly going down when reaching the top whats the decisive factor here to sketch a graph.

Is space-time curvature a curvature of the Higgs field ? or just a mathematical curvature drawn on paper ?

In physics, velocity (speed) is the "change of position (spacial location) per change of time" and acceleration is the "change of velocity per change of time". Mathematically this is first and second derivative. Do further derivatives exist that correspont physical quantities like "change of...

All three questions concern general relativity. 1. Does the curvature of spacetime depend on the frame of reference? 2. Does kinetic energy exist? 3. If it exists, does it contribute to the curvature of spacetime?

Alan Macdonald claims in "4 Appendix: The Equivalence Principle" of his text "Special and General Relativity based on the Physical Meaning of the Spacetime Interval", that his calculation regarding a 2D-surface of a sphere proves, that the equivalence principle is violated. He defines the...

"Matter tells spacetime how to curve, and curved spacetime tells matter how to move" Taking the above into account, the space on the Earth's surface is slightly curved. What is the radius of curvature of space on the Earth's surface due to its mass?

The waterfall model is used in several places such as https://jila.colorado.edu/~ajsh/insidebh/waterfall.html how do we know so sure that the speed of the falling in arrows (wheatver that may be) can exceed the speed of light while on the other hand we assue the speed of light is the speed...

The formula for the radius of curvature. which fails, for example, at y=x^n (n>2) at x=0, which makes sense for odd n, but not for even n. That is, the curvature 1/R =0 for those examples, which seems wrong. What am I approaching incorrectly? Thanks. (formula copied from...

How is humanity technologically ready for testing below: Something analogous to Bell's theorem testing: for particle(s) in superposition of locations detecting spacetime curvature to test if spacetime can be in superposition. Does above test make sense theoretically even if far from...

Consider the electric and magnetic fields around a dipole antenna, Suppose these fields represent some type of curvature in space and time. Suppose where the fields are strong we have greater curvature. Also suppose these fields are really some very large but finite sum of "moving local...

I start with the 2D plane. Suppose y(x) is the curve that connects these two points. Its length is given by: $$S=\int_1^2 \, ds=\int_1^2 (1+y'^2)^{\frac {1}{2}} \, dx$$ Applying Euler's equation we get:$$\frac {\partial f} {\partial y'}=A$$$$\dfrac {y'}{(1+y'^2)^{\frac {1}{2}}}=A$$...

Thinking last night about what I should expect see of the Toronto skyline, which is 40 miles across the lake from me. Starting with an ideal scenario (nape-of-shore to nape-of-shore), a chord calculator tells me that for r=3900 mi. d=40 mi. the chord height will be 270 ft. I was surprised at...

Let us start with a simple two-dimensional space that is asymptotically flat, like e.g. sheet of paper on a desk. Geodesics in this space are straight lines in the traditional sense. Now add a bulge in the center of this space. without affecting the asymptotic flatness. Say, we simply distort...

Orodruin said in another thread that "the spacetime of a spatially flat universe generally has curvature". Is the expansion (or contraction) of a spatially flat universe the manifestation of the curvature of its spacetime? If so, does the expansion correspond to a positive or negative curvature?

For a flat universe, density parameter Ωuniverse=1. How does negative signage of a constituent density parameter, such as that of curvature index Ωk, which can be 0,1,-1 affect the signage of Ωuniverse? If Ωk were to be converted to its energy density, which is much less than the energy density...

Since space is curved within the Earth's gravitational field, every body that moves there will follow the curvature of space no matter what speed it has, so what will its trajectory be, how will it be straight, only if the launch is made absolutely vertically towards sea level? Only then can it...

The Riemann curvature tensor contains second derivatives of metric and squares of the first derivatives. The second derivatives have to be there because they are the ones that cannot be eliminated locally by a choice of coordinates. But other than being a mathematical artifact, is there a...

(a) $$\frac{ds}{dt}=|r'(t)|$$ $$=\sqrt{(x(t))^2+(y(t))^2+(z(t))^2}$$ $$=\frac{2}{9}+\frac{7}{6}t^4$$ $$s=\int_0^t |r'(a)|da=\frac{2}{9}t+\frac{7}{30}t^5$$ Then I think I need to rearrange the equation so ##t## is the subject, but how? Thanks Edit: wait, I realize my mistake. Let me redo

What would be a rough estimate for the Ricci scalar curvature of an astronomical object like the sun? Assuming the sun is a perfect fluid and you are calculating the rest frame of the sun, only the density component would be factored in. Assuming the sun is roughly 2*1030 kg. Please just make...

I'm studying Susskind's GR TTM book, in which he gives a nice explanation of why differential geometry is needed for GR. But there is one gap that I want to fill. The argument is: through a thought experiment, it seems that a uniform gravitation field can be seen as an artifact of going from an...

(I) Using the relevant equation I find this to be ## \frac{e^{x}}{2} ##. (II) Using the relation for the Ricci tensor, I find that the only non-zero components are...

Do force carriers follow the curvature of spacetime, or do they travel in perfectly straight lines? With black holes, gravity of course exists, so I'm thinking the force carriers (at least gravitons) don't follow spacetime curvature, since they would never escape the event horizon. Sounds like...

I believe that to find the curvature C2 is through the ABCD matrices and that C1 has only one phase inversion compared to C. In addition, that the pattern formed is like Newton's rings but I don't know how to find the sizes of the newton's rings depending on lens parameters

Based on the current understanding of general relativity, it is possible that curving spacetime in the back of a spacecraft would allow for faster-than-light travel. In general relativity, the curvature of spacetime is determined by the universe's distribution of matter and energy. If a...

Hi, I am working with leaf springs and studying the derivation of the formula for the deflection of such a structure. The derivation is shown here: My only doubt is how to obtain the following formula: $$\delta=\frac{L^{2}}{8R}$$ where: ##\delta## - deflection, ##L## - length of the beam...

After trying to kinda get a picture of the field of play in quantum physics according to the standard model, a question came up. I tried to formulate the known bosons each as a particle transferring some property. 1. Photons transfer electric charge: the electromagnetic force gives attraction...

Hello, I am attempting to work through problem 12.6 in MTW which involves formulating Newtonian Gravity using Curvature as opposed to the standard formulation. This is a precursor before standard GR. In it he states that the curvature tensor in this formulation is as follows...

For Ω=1, κ=0. Does the value of κ simply follow from the value of Ω, or can its value have an independent existence? So if Ω>1, does κ have to be 1?

According to the Planck 2018 results, the curvature component of the density parameter of the universe is Ωκ=0.001±0.002. From this data, would it be possible to determine the greatest possible positive curvature of the universe and the radius of the corresponding 3-sphere?

I'm reading Carroll's GR notes and I'm having trouble deciphering a particular expression for the Riemann curvature tensor. The coordinate-free definition is (eq. 3.71 in the notes): $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$$ An index-based expression is also given in (eq...

I've been trying to find a way to calculate Gaussian curvature from a 4D metric tensor. I found a program that does this in Mathematica using the Brioschi formula. However, this only seems to work for a 2D metric or formula (I would need to use something with more dimensions). I've found...

Hello : Have a question regarding the mathematical model of reflective curve where could i find information on it ? (pdf , webpages , ebooks ,...etc ) Other than Wikipedia Best Regards HB

Hi, The quote below has been taken from this article, https://math.ucr.edu/home/baez/einstein/node2.html, which I came across. The quote doesn't make any sense to me, especially the part in boldface. Could you please help me with it?

In the absence of a metric, we can not raise and lower indices at will. There are two sorts of Christoffel symbols, Christoffels of the first kind, ##\Gamma^a{}_{bc}## in component notation, and Christoffel symbols of the second kind, ##\Gamma_{abc}##. What's the relationship between the two...

I understand the mechanism of defining the curvature of a 2D manifold via triangle. But I don't understand how this works in 3D. Meanwhile, Lawrence Krauss mentioned in his book A Universe from Nothing it does. How does this work in 3D?

i am facing problems in the definition of dual oF some objects which has pair of anti symmetric indices e.g. Weyl curvature tensor. Double dual is there in the literature but given that how to find the anti self dual part of that. the problem is written in attached the file.

Hi there! I have a few related questions on Gaussian curvature (K) of surfaces and simply connected regions: Suppose that K approaches infinity in the neighborhood of a point (x1,x2) . Is there any relationship between the diverging points of K and (non) simply connected regions? If K diverges...

Can anyone out there give me a hint as to where to start with this problem? I've been looking at it for a while and can't see a way forward. What exactly is "the curvature itself" here?BTW I think the dynamic initial value equations 21.116 and 21.117 are incorrect. MTW should have inserted to...

I know some basic GR and encountered the Schwarzschild metric as well as the Riemann tensor. It is known that for maximally symmetric spaces there is a corresponding Riemann tensor and thus Ricci scalar. Question. How do you calculate the Ricci scalar ##R## and cosmological constant ##\Lambda##...

From the section[5.1] of 'Homogeneity and Isotropy' from General Relativity by Robert M. Wald (pages 91-92, edition 1984) whatever I have understood is that - ##\Sigma_t## is a spacelike hypersurface for some fixed time ##t##. The hypersurface is homogeneous. The metric of whole space is ##g##...

Physics novice here reading pop sci cosmology. Please bear with me. Premise 1: Whether or not expansion is slowing down or speeding up depends on a battle between two phenomena: the attractive gravitational pull of matter and the repulsive gravitational push of dark energy. What counts in this...

A question of sign. Is the curvature of Flamm's paraboloid positive or negative? If I've gotten the signs correct, it's a negative curvature. This is the opposite of the positive curvature of a sphere, and it implies that that geodesics drawn on Flamm's parabaloid should diverge. I think...

Suppose I measure the circumference of a circular orbit round a massive object and find it to be c. Suppose I then move to a slightly higher orbit an extra radial distance δr as measured locally. If space was flat I would expect the new circumference to be c + 2πδr. Will the actual measurement...

I know curvature (k) of a 2 dimensional graph y(x) is equal to y''/(1+(y')^2)^(3/2), were y' is the first derivative of y with respect to x, and y'' is the second derivative of y with respect to x. Is there a formula for the curvature at a point on a 3 dimensional graph z(x,y)? The curvature...

Hi, I know there is actually no way to set up a global coordinate chart on a 2-sphere (i.e. we cannot find a family of 2-parameter curves on a 2-sphere such that two nearby points on it have nearby coordinate values on ##\mathbb R^2## and the mapping is one-to-one). So, from a formal...

Is the curvature index κ necessarily zero in a flat universe with cosmological constant?

Here's what I read in a closed thread asking about the meaning of spacetime curvature: So does that mean if we were able to move to some spot "way above the earth/sun/distant galaxy" where we could watch light and trace it from a distant galaxy (which is otherwise hidden from us) pass around...

How did the author compute the highlighted term 2 from the highlighted term 1 in the following answer to the given question? If $\rho =\frac{d\psi}{ds}$, then the term 2 should be $\upsilon^2 \frac{d\hat{T}}{d\psi}\rho$, but instead, it was written...