What is Line element: Definition and 56 Discussions
In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc length, is a function of the metric tensor and is denoted by ds
Line elements are used in physics, especially in theories of gravitation (most notably general relativity) where spacetime is modelled as a curved Pseudo-Riemannian manifold with an appropriate metric tensor.
I would guess there’s some subtlety in the relationship between basis vectors and coordinates that I’m ignoring, but I really have no idea.
$$ ds^2 = -dt^2 + d\tilde{x}^2 $$
$$ d\tilde{t} = dt / \sqrt{\tilde{x}} $$
$$ \downarrow $$
$$ ds^2 = -\tilde{x} ~ d\tilde{t}^2 + d\tilde{x}^2 $$
$$ dx...
As explained here in Kruskal coordinates the line element for Schwarzschild spacetime is:
$$ds^2 = \frac{32 M^3}{r} \left( – dT^2 + dX^2 \right) + r^2 \left( d\theta^2 + \sin^2 \theta d\phi^2 \right)$$
My simple question is: why in the above line element are involved 5 coordinates and not just...
I know how to solve this problem by considering the Killing equation, namely ##\mathcal L_\xi g=0## that gives three differential equations involving the components of ##\xi=(\xi^\theta,\xi^\phi)## that can be integrated. The result I get, which I know to be true because this is a common result...
First I took the total derivative of these and arrived at
$$
dr=\frac{\partial r}{\partial x}dx+\frac{\partial r}{\partial y}dy \quad\rightarrow \quad r²dr=xdx+ydy
$$
$$
d\phi=\frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy \quad\rightarrow \quad r²dr
\phi=-ydx+xdy
$$...
I'm studying Eddington-Finkelstein coordinates for Schwarzschild metric. Adopting the coordinate set ##(t,r,\theta,\phi)##, the line element assumes the form:
$$
ds^2 = \left(1 - \frac{R_S}{r}\right)dt^2 - \left(1 - \frac{R_S}{r}\right)^{-1}dr^2 - r^2 [d\theta^2 + (\sin{\theta})^2d\phi^2],
$$...
We were taught that in cylindrical coodrinates, the position vector can be expressed as
And then we can write the line element by differentiating to get
.
We can then use this to do a line integral with a vector field along any path. And this seems to be what is done on all questions I've...
From Wikipedia article about Hyperbolic motion, I have the following coordinate equations of motion for Bob in his accelerated frame:
##t(T)=\frac{c}{g} \cdot \ln{(\sqrt{1+(\frac{g \cdot T}{c})^2}+\frac{g \cdot T}{c})} \quad (1)##
##x(T)=\frac{c^2}{g} \cdot (\sqrt{1+(\frac{g \cdot T}{c})^2}-1)...
Consider the line element
$$dl^2=d\theta^2 + \sin^2\theta\, d\varphi^2$$
where ##\theta\in [0,\pi]##. The standard interpretation of this line element is to take ##\varphi\in [0,2\pi)##, in which case the line element represents the standard metric of the sphere ##S^2##. However, from the line...
Could one derive a set of coordinate transformations that transforms events between different reference frames in the de Sitter metric using the invariant line element, similar to how the Lorentz Transformations leave the line element of the Minkowski metric invariant? Would these coordinate...
Assuming the line element ##ds^s=e^{2\alpha}dt^2-e^{2\beta}dr^2-r^2{d\Omega}^2 ##as usual into the form ##ds^s=e^{-2\alpha}dt^2-e^{-2\beta}dr^2-r^2{d\Omega}^2##, I found that the ##G_{tt}## tensor component of first expression do not reconcile with the second one though, it fits for ##G_{rr}...
i'm trying to find what sort of 2-d geometry this system is in, I've been given the line element
𝑑𝑠2=−sin𝜃cos𝜃sin𝜙cos𝜙[𝑑𝜃2+𝑑𝜙2]+(sin2𝜃sin2𝜙+cos2𝜃cos2𝜙)𝑑𝜃𝑑𝜙
where
0≤𝜙<2𝜋
and
0≤𝜃<𝜋/2
Im just not sure where to start. I've tried converting the coordinates to cartesian to see if it yields a...
Is there a general method to determine what geometry some line element is describing? I realize that you can tell whether a space is flat or not (by diagonalising the matrix, rescaling etc), but given some arbitrary line element, how does one determine the shape of the space?
Thanks
Hi! I have the following problem I don't really know where to start from:
A bowl with axial symmetry is built in flat Euclidean space ##R^3##, and has a radial profile giveb by ##z(r)##, where ##z## is the axis of symmetry and ##r## is the radial distance from the axis. What radial profile...
Hi, I'm the given the following line element:
ds^2=\Big(1-\frac{2m}{r}\large)d\tau ^2+\Big(1-\frac{2m}{r}\large)^{-1}dr^2+r^2(d\theta ^2+\sin ^2 (\theta)d\phi ^2)
And I'm asked to calculate the null geodesics.
I know that in order to do that I have to solve the Euler-Lagrange equations. For...
Hi! I'm asked to find all the non-zero Christoffel symbols given the following line element:
ds^2=2x^2dx^2+y^4dy^2+2xy^2dxdy
The result I have obtained is that the only non-zero component of the Christoffel symbols is:
\Gamma^x_{xx}=\frac{1}{x}
Is this correct?
MY PROCEDURE HAS BEEN:
the...
In a problem from Hartle's Gravity, we are asked to express the line element in non-Cartesian coordinates u, v which are defined with respect to x, y. I have no problem getting the new expression for the line element, but then we are asked if the new coordinate curves intersect at right angles...
Defining dS2 as gijdxidxj and
given dS2 = (dx1)2 + (dx2)2 + 4(dx1)(dx2). Find gijNow here is my take on the solution: Since the cross terms are present in the line element equation, we can say that it's a non-orthogonal system. So what I did was express the metric tensor in form of a 2x2...
Hello,
how can you imagine the geometrically meaning of the minus sign in ds2=-dx02+dx12+dx22+dx32, maybe similar to ds2=x12+dx22 is the length in a triangle with the Pythagoras theorem?
I've been looking all over the internet for the line element of the Misner solution to the Einstein Field equations, but I can't find it. Can someone please post the line element? Thank you.
Let us see how the line element transforms under conformal transformations. Consider the Minkovski metric gij, a line element ds2=dxigijdxj, and a conformal transformation
δk(x)=ak + λ xk + Λklxl + x2sk - 2xkx⋅s
We have δ(dxk)=dδ(x)k=λ dxk + Λkldxl + 2 x⋅dx sk - 2dxkx⋅s - 2xkdx⋅s
And so the...
Homework Statement
We've got a line element ds^2 = f(x) du^2 + dx^2 From that we should find the geodesic equation
Homework Equations
Line Element:
ds^2 = dq^j g_{jk} dq^k
Geodesic Equation:
\ddot{q}^j = -\Gamma_{km}^j \dot{q}^k \dot{q}^m
Christoffel Symbol:
\Gamma_{km}^j = \frac{g^{jl}}{2}...
With coordinates q en basis e ,textbooks use as line element :
ds=∑ ei*dqi But ei is a function of place, as one can see in deriving formulas for covariant derivative. Why don't they use as line element the correct:
ds=∑ (ei*dqi+dei*qi)
Same question in deriving covariant derivative,
Homework Statement
[/B]
Consider ##\mathbb{R}^3## in standard Cartesian co-ordinates, and the surface ##S^2## embedded within it defined by ##(x^2+y^2+z^2)|_{S^2}=1##. A particular set of co-ords on ##S^2## are defined by
##\zeta = \frac{x}{z-1}##,
##\eta = \frac{y}{z-1}##.
Express...
When we define line element of Minkowski space, we also define the metric tensor of the equation. What actually is the function of the tensor with the line element.
Homework Statement
Flat space-time in polar coordinate is considered. The line element is
ds2= -dt2+dr2+r2(dθ2+sin2θdΦ2)
The actual answers are given below, but I can't come up to them. Need urgent help.
Homework Equations
dA = √g11g22 dx1 dx2
dV = √g11g22g33 dx1 dx2dx3
The Attempt at a...
Can someone please type out the line element for the Godel metric (including any and all c terms and any other terms that one might omit if they were using natural units to set terms like c = 1)? I ask this because different sources on line have it written out in different ways which look...
ds2=guvdxudxv
Why is the invariance of ds2 as shown above?
Why can't it be something like
ds2=guudxudxu or
ds2=dxudxu ?
Isnt it based on Pythagoras theorem? Why must it have 1 u and 1 v instead of just 2 u or 2 v? Forgive me for such dumb question as I just started.
I've been struggling with this for the past hours and I can't find a good answer.
Using the integral for work W = \int_a^b \vec{F}\cdot d\vec{s} , when a > b , and the force is directed from a to b, i keep getting a negative result. I am considering d\vec{s} as the infinitesimal difference...
Homework Statement
Hello Guys, I am reading Hobson's General Relativity and I have come across an exercise problem, part of which frustrates me:
3.20 (P. 91)
In the 2-space with line element
ds^2=\frac{dr^{2}+r^{2}d\theta^{2}}{r^{2}-a^{2}}-\frac{r^{2}dr^{2}}{{(r^{2}-a^{2})}^{2}}
and...
What is the difference between the line element and length of a four vector? They both seem to have the same definition just with slightly different notation, so is the line element just the length of a specfic vector.
Also, if the magnitude of a four-vector is calculated to b -1 is this...
Hi, I have to write a spacetime line element for the shape of a cube of cosmological dimensions. This cube is expanding like this:
i)With time, the cube becomes elongated along the z-axis, and the square x-y shape doesn't change.
ii)The line element must be spatially homogeneus. (I don't...
Homework Statement
Transform the line element of special relativity from the usual (t, x, y, z) coordinates rectangular coordinates to new coordinates (t', x', y', z') related by
t =\left (\frac{c}{g} + \frac{x'}{c} \right )sinh\left (\frac{gt'}{c} \right)
x =\left (\frac{c}{g} +...
Dear All
I want to know the formula to extract the longitude of the satellite from the TLE below.
I hope someone can help.
Thanks
1 24652U 96063A 97251.85429118 .00000144 00000-0 00000-0 0 568
2 24652 0.0850 104.0202 0008355 303.2059 279.6585 1.00288964 3492
Homework Statement
Say we have a function such that
x = uv , y = (u^2 - v^2) /2
Hence our line element in Cartesian coordinates is.
ds^2 = dx^2 + dy^2
Now I have two questions. I like to work on math problems algebraically if possible so I thought to convert our line element I could take...
Hi all,
Papapetrou line element describes an axial symmetric stationary spacetimes and the coordinates that appear in this metric are just similar to cylindrical coordinates; I mean they are labeled ρ, z and phi. I want to know if they are really the cylindrical coordinate or not; In other...
In one of the early chapters of Gravity by Hartle, he is developing the line element on a sphere in preparation for developing the concept of a spacetime interval. Whilst finishing up the proof Hartle sort of implicitly says that if two lines are orthogonal the line element connecting two points...
Homework Statement
Show that a line element of form ds2 = gabdXadXb transforms like a scalar under any general coordinate transform
Homework Equations
The Attempt at a Solution
I think I've actually found the solution here, but I can't make sense of it...
Homework Statement
Consider the two-dimensional spacetime spanned by coordinates (v,x) with the line element
ds^2=-xdv^2 +2dvdx
Calculate the light cone at a point (vx)
The Attempt at a Solution
I don't even know how the light cone for flat spacetime is calculated. So if that one's...
Hello! :-)
I've been thinking about the RW line element (c^2 d tau^2), and I understand that there is a term which describes the geometry of the Universe (d sigma^2) and then a term corresponding to the cosmic time (c^2 dt^2). The resulting (c^2 d tau^2 ) term would be the line measured by...
I would like some help in calculating the basic Friedman equations starting from the flat FRW universe line element, Once I have calculated the Christoffel symbols for this metric how do i get to the Friedman equation, any link to a good book will be really helpful, thanks, seetesh.
I noticed somewhere the line element of a two-dimensional torus is written in the form
ds^2=r^2(d\theta^2_1+d\theta^2_2)
The author only states that he assumes same radius parameter for simplicity and no further explanation is given. But I do not understand how that form is possible. I...
I've read Schwarzschild paper and I don't understand his conditions
"The solution is spatially symmetric with respect to the origin of the co-ordinate system in the sense that one finds again the same solution when x,y,z are subjected to an orthogonal transformation(rotation)"
Could...
If the Ricci-scalar R is constant for a given spatial hypersurface, then the curvature of that region should be homogenous and isotropic, right?
A homogenous and isotropic hypersurface (disregarding time) has by definition the following line element (due to spherical symmetry):
d\sigma^2 =...