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.

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  1. Pencilvester

    I Minkowski to Euclidean line element from coordinate changes?

    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...
  2. cianfa72

    I Schwarzschild spacetime in Kruskal coordinates

    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...
  3. alexriemann

    How Do Killing Vector Fields Form a Basis on S2?

    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...
  4. C

    Line element in cylindrical coordinates

    First I took the total derivative of these and arrived at $$ dr=\frac{\partial r}{\partial x}dx+\frac{\partial r}{\partial y}dy \quad\rightarrow \quad r²dr=xdx+ydy $$ $$ d\phi=\frac{\partial \phi}{\partial x}dx+\frac{\partial \phi}{\partial y}dy \quad\rightarrow \quad r²dr \phi=-ydx+xdy $$...
  5. A

    I Eddington-Finkelstein coordinates for Schwarzschild metric

    I'm studying Eddington-Finkelstein coordinates for Schwarzschild metric. Adopting the coordinate set ##(t,r,\theta,\phi)##, the line element assumes the form: $$ ds^2 = \left(1 - \frac{R_S}{r}\right)dt^2 - \left(1 - \frac{R_S}{r}\right)^{-1}dr^2 - r^2 [d\theta^2 + (\sin{\theta})^2d\phi^2], $$...
  6. chiyu

    I Vector calculus: line element dr in cylindrical coordinates

    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...
  7. Lluis Olle

    B Metric Line Element Use: Do's & Don'ts for Accelerated Dummies?

    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)...
  8. Demystifier

    I Geometries from Line Element $$dl^2=d\theta^2 + \sin^2\theta\, d\varphi^2$$

    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...
  9. Sciencemaster

    I Coord Transform in de Sitter Space: Phys Significance &Linearity?

    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...
  10. Bishal Banjara

    I Inverting the metric coefficients in the Schwarzschild line element

    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}...
  11. steve1763

    A Find 2D Geometry of Line Element in Coordinates

    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...
  12. steve1763

    A Determining Geometry of Line Element: A General Method?

    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
  13. C

    Symmetric bowl associated with a line element

    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...
  14. C

    Null geodesics given line element

    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...
  15. C

    I Christoffel symbols knowing Line Element (check my result)

    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...
  16. beefbrisket

    I Meaning of cross terms in line element

    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...
  17. Destroxia

    Calculating Christoffel Symbols from a given line element

    Homework Statement Given some 2D line element, ## ds^2 = -dt^2 +x^2 dx^2 ##, find the Christoffel Symbols, ## \Gamma_{\beta \gamma}^{\alpha} ##. Homework Equations ## \Gamma_{\beta \gamma}^{\alpha} = \frac {1}{2} g^{\delta \alpha} (\frac{\partial g_{\alpha \beta}}{\partial x^\gamma} +...
  18. Sayak Das

    Finding the inverse metric tensor from a given line element

    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...
  19. J

    I Line Element in Minkowski Space: Geometric Meaning

    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?
  20. A

    I Where does the line element of Minkowiski space come from?

    How is it derived? ds^2 =-cdt^2+dx^2+dy^2+dz^2
  21. Spinnor

    I Line element from Kaluza-Klein for Kids

    In http://vixra.org/abs/1406.0172, the five-dimensional Kaluza-Klein line element d˜s^2 is given by, Does this look correct? Thanks!
  22. C

    I What is the hyper-cylindrical line element?

    What is the hyper-cylindrical line element?
  23. S

    I Misner Solution Line Element | Einstein Field Equations

    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.
  24. physicality

    I Conformal transformation of the line element

    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...
  25. C

    Finding the geodesic equation from a given line element

    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}...
  26. Alain De Vos

    Line element and derivation of lagrange equation

    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,
  27. H

    Calculating the line element after a change of coordinates

    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...
  28. Tony Stark

    Metric Tensor of a line element

    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.
  29. Tony Stark

    Line element in Euclidean Space

    The line element is defined as How is dx2+dy2+dz2 be written as gijdqidqj. Is some sort of notation used??
  30. Tony Stark

    Calculating Area and Volume of a sphere through 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...
  31. S

    Type Godel Metric Line Element - Get Help Here!

    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...
  32. T

    Expression for the line element ds^2

    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.
  33. B

    Bondi's Radiating Line Element: Solve the Mystery!

    Hello, Some knows how to Bondi deduce his radiating line element? I read some papers and there isn't any hint about hit. Best regards
  34. B

    Cylindrically symmetric line element canonical form

    Hello, What is the most general cylindrically symmetric line element in the canonical form? Best regards.
  35. P

    Meaning of line element vector on work formula

    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...
  36. C

    Line element under coordinate transformation to get polar form

    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...
  37. A

    Understanding the Relationship between Line Element and Length in Four Vectors

    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...
  38. L

    Spacetime line element to describe an expanding cube

    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...
  39. soothsayer

    Transform of Line Element in Special Relativity

    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} +...
  40. H

    How to get longitude from Two Line Element (TLE)

    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
  41. O

    Question about changing line element.

    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...
  42. S

    Is Papapetrou line element the same as cylindrical coordinates?

    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...
  43. Vorde

    Orthogonal Lines and their line element

    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...
  44. C

    Show that a line element transforms like a scalar

    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...
  45. R

    How do you calculate the light cone for the following line element?

    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...
  46. R

    Understanding the Relationship Between the RW Line Element and Universe Density

    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...
  47. S

    Friedman equations from flat FRW universe line element

    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.
  48. A

    Form of line element of a torus.

    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...
  49. O

    Why Is the Schwarzschild Solution Considered Spatially Symmetric?

    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...
  50. T

    Deriving the line element in homogenous isotropic space

    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 =...