What is Curved space: Definition and 80 Discussions
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Curved spaces play an essential role in general relativity, where gravity is often visualized as curved space. The Friedmann-Lemaître-Robertson-Walker metric is a curved metric which forms the current foundation for the description of the expansion of space and shape of the universe.
Let's say we have some observer in some curved spacetime, and we have another observer moving relative to them with some velocity ##v## that is a significant fraction of ##c##. How would coordinates in this curved spacetime change between the two reference frames?
For example, imagine a...
In special relativity we've the invariant ##\begin{aligned} d s^2=&-d t^2 \\ &+d x^2 \\+d y^2+d z \end{aligned}##.
For a clock moving along a worldline the above equation reduces to ##\begin{aligned} d s^2=&-d t^2\end{aligned}## , hence we can say that the time measured by the clock moving...
I don't want to post this in a math forum because it's very basic and I just want a straightforward answer, not something math heavy . What's the definition of angle in a cuved space embedded in a higher eucledian space? Like when I have a spherical surface in 3d eucledian space and want to work...
By definition of the vector potential we may write
\nabla \times A =B
at least in flat space. Does this relation hold in curved space? I am particularly interested if we can still write this in a spatially flat Friedmann-Robertson-Walker background with metric ds^2=dt^2-a^2(dx^2+dy^2+dz^2) and...
It is said that: It is not possible to write a position vector in a curved space time. What is the reason?
How can one describe a general vector in a curved space time?
Can you please suggest a good textbook or an article which explains this aspect?
The metric of a 3-D positively curved space is dr2+ Sk(r)2(dθ2+sin2θdΦ2).
Now if this space expands with a scale factor a(t) from r to r'.
Whether the change in the radial component be a(t)dr and angular component be Sk(r')dθ and Sk(r')sinθdΦ since the change due to expansion is already...
To explain the concept of curved space time, we often use analogy of rubber sheet. If we put a heavy ball at the centre of sheet then it creates a depression and now a smaller ball will fall towards that heavy ball because of depression. But in this analogy smaller ball is falling down the slope...
Let us assume a "toy-metric" of the form
$$ g=-c^2 \mathrm{d}t^2+\mathrm{d}x^2+\mathrm{d}y^2+\mathrm{d}z^2-\frac{4GJ}{c^3 r^3} (c \mathrm{d}t) \left( \frac{x\mathrm{d}y-y\mathrm{d}x}{r} \right)$$
where ##J## is the angular-momentum vector of the source.
Consider the curve
$$ \gamma(\tau)=(x^\mu...
Homework Statement
I think I have managed to do the first three parts of this problem ok, but I am struggling with part 4.
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A 2D negatively curved surface can be described in 3D Euclidean Cartesian coordinates by the equation:
##x^2 + y^2 + z^2 = −a^2##.
1) Find the 2D line element for...
Hi all,
I'm doing some reading about special/general relativity, and have come across the ideas of curved space etc. I've very much a novice in physics, so please excuse my (possibly) stupid questions. For background, I'm interested in writing a sci-fi story, and would like to have at least...
I am trying to understand Wen and Zee's article on topological quantum numbers of Hall fluid on curved space: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.69.953
They passingly mentiond the fact that a spinning particle with orbital angular momentum $s$ moving on a manifold with...
We are told that planets and comets orbit the sun in an ellipse (Kepler's 3 laws) as shown below:
We are also told that according to Einstein's theory of gravity, there is no force applied. Implied is that the planets move in straight lines through curved space. We know that the effect of...
hi, I really wonder what the difference between curvilinear coordinates in a Euclidean space and embedding a curved space into Euclidean space is ? They resemble to each other for me, so Could you explain or spell out the difference? Thanks in advance...
If we study the high temperature limit (near Hagedorn) of a string gas, most of the energy is concentrated in a single long string. If we model the string by a fixed number of rigid links of length ls and calculate the number of possible configurations, we get the density of states:
$$\omega(E)...
In this thread, ramparts asked how integration by parts could be used in general relativity.
suppose you have
##\int_M (\nabla^a \nabla_a f) g .Vol##
Can it be written like
##\int_M (\nabla^a \nabla_a g) f .Vol## plus a boundary integration term (by integrating twice by parts)?
I think thay it...
I was going to ask a question about whether or not pi was constant or changed with curved space. I found the answer on here that it does indeed change. Then I started thinking about the ramifications of that. sine waves are dependent on pi, so they should change too. Does sin(theta) =...
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
Follow along at http://star-www.st-and.ac.uk/~hz4/gr/GRlec4+5+6.pdf and go to PDF page 9 or page 44 of the "slides." I'm trying to see how to go from the first to the third line. If we write the free particle Lagrangian and use q^i-dot and q^j-dot as the velocities and metric g_ij, how is it we...
Given an object spinning on its own axis, in orbit around a planet with mass and that the object is traveling in the direction of its axis, does the axis continue to point at the same point in infinity as it rotates around the planet or does the axis follow the curvature of space around the...
In Euclidean space, we may define covariant basis by the partial derivative of position vector with respect to each coordinates, i.e.
##∂R/(∂z^i )=z_i##
But in curved space (such as, the two dimensional space on a sphere) how can we define covariant basis 'intrinsicly'?(as we have no position...
I am trying to get an understanding of general relativity one tidbit at a time. I have a vague concept of why curved spacetime causes the effect we call gravity. However, there's an aspect of it (ok, there' are quite many aspects of it, but I'm concentrating on this one right now) that I can't...
In flat space the atoms in a metal have regular packing structures.
A slight curvature of space would mean this wasn't geometrically possible. As a consequence do we expect metals to have a significantly lower density with a slight curvature of space?
Obviously, this doesn't just apply to...
If the space around the Earth is curved according to general theory of relativity no lateral force is required to put the satellite in orbit because when the rocket carrying satellite has reached the certain height the satellite should spontaneously start sliding along the curved path traced...
NOT including the prediction capabilities of the particular math equations of GR or SR.
In particular, hard evidence such as, or close to; here's an electron, because we measured it directly, or saw it in an electron microscope.
Or here's a cell under a microscope.
Or this is a brain scan/MRI...
Hi when trying to derive this equation, i am stuck on:
[\Gamma_{\mu}(x),\gamma^{\nu}(x)]=\frac{\partial \gamma^{\nu}(x)}{\partial x^{\mu}} + \Gamma^{\nu}_{\mu p}\gamma^{p} .
This [\Gamma_{\mu}(x) term is the spin connection, if this is an ordinary commutator:
a) is it a fermionic so +...
I have to compute the square of the Dirac operator, D=γaeμaDμ , in curved space time (DμΨ=∂μΨ+AabμΣab is the covariant derivative of the spinor field and Σab the Lorentz generators involving gamma matrices). Dirac equation for the massless fermion is γaeμaDμΨ=0. In particular I have to show that...
Am I correct in saying that the angular deficit (change in angle) of a vector transported around a closed surface on a curved surface can only be observed by flattening the surface?
Actually a further problem- I understand it from the flat sheet to a cone: Cut out a pie from a sheet, Draw a...
Hi,
I've been looking at the Klein Gordon equation, Maxwell's equation, and the Dirac equation in curved space and I was wondering if there is an underlying formalism regarding how to derive them from their flat space counterparts.
What I mean is, at the heart of the whole process for all...
The Dirac delta function is defined as:
\int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = 1
Or more generally the integral is,
\int_{ - \infty }^{ + \infty } {\delta (\int_{{x_0}}^x {dx'} )dx}
But if the metric varies with x, then the integral becomes,
\int_{ - \infty }^{ + \infty }...
Considering two equally massive stars stationed at some small distance from each other, would a clock stationed between the two stars equadistance from both tick more slowly due to the proximity to the massive stars, or would the effective cancellation of the gravitational attraction also cancel...
When I write down a quantum field (for instance to compute T^00 or some expectation value)
I write it as an integral over momentum space.
If I am working in curved space
should this be divided by sqrt [g]?
(and why or why not?)
Suppose I construct a metal triangle in flat space the sum of the interior angles will be 180°. If I then move the structure into a curved space in which the sum of the interior angles of the triangle will be greater then 180° do the corners of my triangle experience stress? Or, do I simply...
Recall the gravitational effect of a hollow sphere upon a test mass inside ... no net attraction of the test mass anywhere within the sphere.
Let this sphere be large and dense enough to have 1 G of attraction on a test mass resting on its outer surface.
Plotting the measure of effect on...
Greetings,
I read a lot. It is often repetitious. Occasionally I read an idea or way of looking at something that I had not read before.
In one book, I read the matter did not curve space, but rather, matter *is* curved space. This is a paradigm shift in the way of thinking about matter...
I've always been confused by the typical analogies I see when gravity as a space-time curvature is explained.
In 2-D it is usually a plane with field lines, and the surface of the plane is curved around an object. And so we are told a mass placed in this curvature will "fall" down the curve...
There is a new theory being put forth that gravity may amplify vacuum energy to the point that the amplified vacuum energy may predominate over classical vacuum energy, which would cause it to influence astrophysical processes:
http://www.physorg.com/news193330592.html
It's just a...
Example: take curved 2D space with positive constant curvature everywhere. You say, sphere with radius R? no, there are 2 different solutions in topology: sphere and half-sphere. Half sphere (1/2 of sphere where points across the 'equator' are connected to the opposite sides) can’t be 'embedded'...
If you think of a bow (as in bow and arrow) placed in space in the proximity of a massive object. This bow for arguments sake is 100 km from tip to tip.
Now replace the bow (wooden part and string) with what might be considered waypoints. The curved wooden part represents a straight line in...
I have been contemplating my confusion about my intuition regarding GR and believe I have tracked down the primary source of confusion.
The classical theories I have been taught assumed flat space with independent time and used the divergence theorem to derive inverse squared laws for fields...
Relativity, Gravity, "Attraction", Curved Space
Okay, I am by no means knowledgeable in this field, probably more dangerous than anything. But I have a question about gravity. If two masses are identical (in an idea situation) and have no other forces acting on them, and start from a...
We've all seen the "ball on a rubber sheet" analogy, showing how warped space near a planet can cause a light beam to alter its path. We are told that the light is actually following the shortest path in curved space.
When it comes to a *stationary* object near a planet, however, I have a...
In a flat space, the momentum of a photon gas distributes isotropically. Every direction is equivalent. If the space is curved,like the space outside a black hole, what will happen to the photon gas? Will the momentum distribution be not isotropic any more?
From another thread:
(Bob for short's reply)
I thought acceleration DID curve space...
I'm coming from this perspective:
say in the rotating "rigid" disc...and via Einstein's equivalence principle...
for example, Brian Greene in THE ELEGANT UNIVERSE says:
any clarifications...
How do we distinguish (mathematically) between curved space and the choice of coordinates? For example, the flat space metric in spherical polar coordinates looks as if it is curved space. I can ask the same for gravitational waves - how do we know that it isn't the TT gauge which is wavelike...