Metric tensor Definition and 206 Threads

  1. snoopies622

    I How to keep the components of a metric tensor constant?

    I've noticed that a very easy way to generate the Lorentz transformation is to draw Cartesian coordinate axes in a plane, label then ix and ct, rotate them clockwise some angle \theta producing axes ix' and ct', use the simple rotation transformation to produce ix' and ct', then just divide...
  2. D.S.Beyer

    B Density of the early Universe contributing to the red-shift?

    Does the relative density of the early universe contribute to the red-shift of distant galaxies? If so, by how much? How would this be calculated? Asked another way : Assuming both the early universe and the current universe are flat, could the relative difference of their space time metric...
  3. shahbaznihal

    A On metric and connection independence

    Some models of gravity, inspired by the main theme of spacetime fabric of Classical GR, treat the metric of the manifold and the connection as independent entities. I want to study this theory further but I am unable to find any paper on this, on ariXiv atleast. I will be very thankful if...
  4. Abhishek11235

    A Finding the unit Normal to a surface using the metric tensor.

    Let $$\phi(x^1,x^2...,x^n) =c$$ be a surface. What is unit Normal to the surface? I know how to find equation of normal to a surface. It is given by: $$\hat{e_{n}}=\frac{\nabla\phi}{|\nabla\phi|}$$However the answer is given using metric tensor which I am not able to derive. Here is the answer...
  5. M

    B G11 Metric Tensor: What is it & How Does it Work?

    What is g11? I am very curious, can someone briefly describe what the metric tensor is, please?
  6. S

    A Causal Structure of Metric Prop.: Matrix Size Differs

    Proposition: Consider an ##n + 1##-dimensional metric with the following product structure: $$ g=\underbrace{g_{rr}(t,r)\mathrm{d}r^2+2g_{rt}(t,r)\mathrm{d}t\mathrm{d}r+g_{tt}(t,r)\mathrm{d}t^2}_{:=^2g}+\underbrace{h_{AB}(t,r,x^A)\mathrm{d}x^A\mathrm{d}x^B}_{:=h} $$ where ##h## is a Riemannian...
  7. 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...
  8. F

    I Lie derivative of a metric determinant

    I’m hoping to clear up some confusion I have over what the Lie derivative of a metric determinant is. Consider a 4-dimensional (pseudo-) Riemannian manifold, with metric ##g_{\mu\nu}##. The determinant of this metric is given by ##g:=\text{det}(g_{\mu\nu})##. Given this, now consider the...
  9. S

    B Metric Tensor and The Minkowski metric

    Hi, I have seen the general form for the metric tensor in general relativity, but I don't understand how that math would create a Minkowski metric with the diagonal matrix {-1 +1 +1 +1}. I assume that using the kronecker delta to create the metric would produce a matrix that has all positive 1s...
  10. vibhuav

    I Requesting clarification about metric tensor

    I am a little bit confused about the metric tensor and would like some feedback before I proceed with my learning of GR. So I understand that metric tensor describes the geometry of the space itself. I also understand that the components of the metric tensor (any tensor for that matter) come...
  11. vibhuav

    I Evaluating metric tensor in a primed coordinate system

    I am trying to learn GR. In two of the books on tensors, there is an example of evaluating the inertia tensor in a primed coordinate system (for example, a rotated one) from that in an unprimed coordinate system using the eqn. ##I’ = R I R^{-1}## where R is the transformation matrix and...
  12. DeathbyGreen

    Trouble with Peskin QFT textbook

    I'm trying to work through a scattering calculation in the Peskin QFT textbook in chapter 5, specifically getting equation 5.10. They take two bracketed terms 4[p'^{\mu}p^{\nu}+p'^{\nu}p^{\mu}-g^{\mu\nu}(p \cdot p'+m_e^2)] and 4[k_{\mu}k'_{\nu}+k_{\nu}k'_{\mu}-g_{\mu\nu}(k \cdot...
  13. T

    I Finding distance in polar coordinates with metric tensor

    Hi, I'm getting into general relativity and am learning about tensors and coordinate transformations. My question is, how do you use the metric tensor in polar coordinates to find the distance between two points? Example I want to try is: Point A (1,1) or (sq root(2), 45) Point B (1,0) or...
  14. B

    How to Derive the Relation Using Inner Products of Vectors?

    Homework Statement I am trying to derive the following relation using inner products of vectors: Homework Equations g_{\mu\nu} g^{\mu\sigma} = \delta_{\nu}^{\hspace{2mm}\sigma} The Attempt at a Solution What I have done is take two vectors and find the inner products in different ways with...
  15. JTC

    A Understanding Metric Tensor Calculations for Different Coordinate Systems

    Good Day, Another fundamentally simple question... if I go here; http://www-hep.physics.uiowa.edu/~vincent/courses/29273/metric.pdf I see how to calculate the metric tensor. The process is totally clear to me. My question involves LANGUAGE and the ORIGIN LANGUAGE: Does one say "one...
  16. T

    I What constrains the metric tensor field in GR?

    Do the field equations themselves constrain the metric tensor? or do they just translate external constraints on the stress-energy tensor into constraints on the metric tensor? another way to ask the question is, if I generated an arbitrary differentiable metric tensor field, would it translate...
  17. F

    I Calculating Perturbative Expansion of Metric Inverse in Cosmology

    As I understand it, in the context of cosmological perturbation theory, one expands the metric tensor around a background metric (in this case Minkowski spacetime) as $$g_{\mu\nu}=\eta_{\mu\nu}+\kappa h_{\mu\nu}$$ where ##h_{\mu\nu}## is a metric tensor and ##\kappa <<1##. My question is, how...
  18. N

    I Metric Tensor as Simplest Math Object for Describing Space

    I've been reading Fleisch's "A Student's Guide to Vectors and Tensors" as a self-study, and watched this helpful video also by Fleisch: Suddenly co-vectors and one-forms make more sense than they did when I tried to learn the from Schutz's GR book many years ago. Especially in the video...
  19. M

    A How to obtain components of the metric tensor?

    In coordinates given by x^\mu = (ct,x,y,z) the line element is given (ds)^2 = g_{00} (cdt)^2 + 2g_{oi}(cdt\;dx^i) + g_{ij}dx^idx^j, where the g_{\mu\nu} are the components of the metric tensor and latin indices run from 1-3. In the first post-Newtonian approximation the space time metric is...
  20. DaTario

    A Simple 1D kinematic exercises with metric tensor

    Hi All I would like to know if there is a way to produce simple one dimensional kinematic exercises with space-time metric tensor different from the Euclidean metric. Examples, if possible, are welcome. Best wishes, DaTario
  21. T

    I Metric tensor : raising/lowering indices

    Hi everyone, I'm currently studying Griffith's Intro to Elementary Particles and in chapter 7 about QED, there's one part of an operation on tensors I don't follow in applying Feynman's rules to electron-muon scattering : ## \gamma^\mu g_{\mu\nu} \gamma^\nu = \gamma^\mu \gamma_\mu## My...
  22. K

    I Understanding the Derivation of the Metric Tensor

    Hello, I have a question regarding the first equation above. it says dui=ai*dr=ai*aj*duj but I wonder how. (sorry I omitted vector notation because I don't know how to put them on) if dui=ai*dr=ai*aj*duj is true, then dr=aj*duj |dr|*rhat=|aj|*duj*ajhat where lim |dr|,|duj|->0 which means...
  23. V

    I Metric tensor derived from a geodesic

    Let we have a 2D manifold. We choose a coordinate system where we can construct all geodesics through any point. Is it enough to derive a metric from geodesic equation? Or do we need to define something else for the manifold?
  24. R

    B Metric tensor of a perfect fluid in its rest frame

    The stress-energy tensor of a perfect fluid in its rest frame is: (1) Tij= diag [ρc2, P, P, P] where ρc2 is the energy density and P the pressure of the fluid. If Tij is as stated in eq.(1), the metric tensor gij of the system composed by an indefinitely extended perfect fluid in...
  25. F

    A Is the Metric Tensor Invariant under Lorenz Transformations in M4?

    I'm stuck on an apparently obvious statement in special relativity, so I hope you can help me. Can I define Lorenz transformations as transformations that don't change the spacetime interval in M4 and from this deduct that the metric tensor is invariant under LT? I've always read that the...
  26. redtree

    I Gravitational Redshift: Derivation from Static Metric

    I am trying to find a derivation of gravitational redshift from a static metric that does not depend on the equivalence principle and is not a heuristic Newtonian derivation. Any suggestions?
  27. arpon

    I Two metric tensors describing same geometry

    Consider two coordinate systems on a sphere. The metric tensors of the two coordinate systems are given. Now how can I check that both coordinate systems describe the same geometry (in this case spherical geometry)? (I used spherical geometry as an example. I would like to know the process in...
  28. K

    I Writing Components of a Metric Tensor

    I wonder if it is possible to write the components of the metric tensor (or any other tensor) as a summ of functions of the coordinates? Like this: g^{\mu\nu} = \sum_{\mu}^{D}\sum_{\nu}^{D} g_{_1}(x^{\mu}) g_{_2}(x^{\nu}) where g1 and g2 are functions of one variable alone and D is the...
  29. K

    I Is Symmetry on μ and α Valid for the Derivative of the Metric Tensor?

    I was thinking about the metric tensor. Given a metric gμν we know that it is symmetric on its two indices. If we have gμν,α (the derivative of the metric with respect to xα), is it also valid to consider symmetry on μ and α? i.e. is the identity gμν,α = gαν,μ valid?
  30. S

    Metric tensor and gradient in spherical polar coordinates

    Homework Statement Let ##x##, ##y##, and ##z## be the usual cartesian coordinates in ##\mathbb{R}^{3}## and let ##u^{1} = r##, ##u^{2} = \theta## (colatitude), and ##u^{3} = \phi## be spherical coordinates. Compute the metric tensor components for the spherical coordinates...
  31. tomdodd4598

    I Special Relativity Approximation of Gravitation

    Hey there, I have two questions - the first is about an approximation of a central gravitational force on a particle (of small mass) based on special relativity, and the second is about the legitimacy of a Lagrangian I'm using to calculate the motion of a particle in the Schwarzschild metric...
  32. redtree

    A Riemann Tensor Equation: Simplifying the Riemann-Christoffel Tensor

    The Riemann-Christoffel Tensor (##R^{k}_{\cdot n i j}##) is defined as: $$ R^{k}_{\cdot n i j}= \frac{\delta \Gamma^{k}_{j n}}{\delta Z^{i}} - \frac{\delta \Gamma^{k}_{i n}}{\delta Z^{j}}+ \Gamma^{k}_{i l} \Gamma^{l}_{j n}- \Gamma^{k}_{j l} \Gamma^{l}_{i n} $$ My question is that it seems that...
  33. redtree

    A Relationship between metric tensor and position vector

    Given the definition of the covariant basis (##Z_{i}##) as follows: $$Z_{i} = \frac{\delta \textbf{R}}{\delta Z^{i}}$$ Then, the derivative of the covariant basis is as follows: $$\frac{\delta Z_{i}}{\delta Z^{j}} = \frac{\delta^2 \textbf{R}}{\delta Z^{i} \delta Z^{j}}$$ Which is also equal...
  34. mertcan

    A How Is the Taylor Expansion Applied to Metric Tensors?

    hi, when I dug up something about metric tensors, I found a equation in my attached file. Could you provide me with how the derivation of this ensured? What is the logic of that expansion in terms of metric tensor? I really need your valuable responses. I really wonder it. Thanks in advance...
  35. A

    A Metric with Harmonic Coefficient and General Relativity

    Goodmorning everyone, is there any implies to use in general relativity a metric whose coefficients are harmonic functions? For example in (1+1)-dimensions, is there any implies for using a metric ds2=E(du2+dv2) with E a harmonic function? In (1+1)-dimensions is well-know that the Einstein...
  36. arpon

    I Is there any 2D surface whose metric tensor is eta?

    Does there exist any 2D surface whose metric tensor is, ##\eta_{\mu\nu}= \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}##
  37. W

    I The Order and Valence of Tensors

    I'm having a bit of trouble understanding the nature of tensors (which is pretty central to the gen rel course I'm currently taking). I understand that the order (or rank) of a tensor is the dimensionality of the array required to describe it's components, i.e. a 0 rank tensor is a scalar, a 1...
  38. A

    Diagnosing an Equation in General Relativity

    Hello, since gμν gμν = 4 where g = diag[1,-1,-1,-1], see: https://www.physicsforums.com/threads/questions-about-tensors-in-gr.39158/ Is the following equation correct? xμ xμ = gμνxν gμνxν = gμν gμνxν xν= 4 xμ xμ If not, where is the problem? Cheers, Adam
  39. DiracPool

    GR Metric Tensor Rank 2: Quadratic vs Shear Forces

    Is the metric tensor a tensor of rank two simply because the line element (or equivalent Pythagorean relation between differential distances) is "quadratic" in nature? This would be in opposition to say, the stress tensor being a tensor of rank two because it has to deal with "shear" forces. I...
  40. D

    Non-Euclidean geometry and the equivalence principle

    As I understand it, a Cartesian coordinate map (a coordinate map for which the line element takes the simple form ##ds^{2}=(dx^{1})^{2}+ (dx^{2})^{2}+\cdots +(dx^{n})^{2}##, and for which the coordinate basis ##\lbrace\frac{\partial}{\partial x^{\mu}}\rbrace## is orthonormal) can only be...
  41. A

    Varying The Gibbons-Hawking Term

    The Gibbons Hawking boundary term is given as ##S_{GHY} = -\frac{1}{8 \pi G} \int_{\partial M} d^dx \sqrt{-\gamma} \Theta##. I want to calculate its variation with respect to the induced boundary metric, ##h_{\mu \nu}##. The answer (given in eqns 6&7 of...
  42. D

    Question about Metric Tensor: Learn Differential Geometry

    Hey, I have not done any proper differential geometry before starting general relativity (from Sean Carroll's book: space time and geometry), so excuse me if this is a stupid question. The metric tensor can be written as $$ g = g_{\mu\nu} dx^{\mu} \otimes dx^{\nu}$$ and its also written as...
  43. I

    Cosmological constant times the metric tensor

    In the EFE, what does adding Λgμν mean and why is it not included in the Einstein tensor?
  44. S

    Riemannian Metric Tensor & Christoffel Symbols: Learn on R2

    Hi, Want to know (i) what does Riemannian metric tensor and Christoffel symbols on R2 mean? (ii) How does metric tensor and Christoffel symbols look like on R2? It would be great with an example as I am new to this Differential Geometry.
  45. S

    What is the transformation law for tensor components in differential geometry?

    I read in many books the metric tensor is rank (0,2), its inverse is (2,0) and has some property such as ##g^{\mu\nu}g_{\nu\sigma}=\delta^\mu_\sigma## etc. My question is: what does ##g^\mu_\nu## mean?! This tensor really confuses me! At first, I simply thought that...
  46. T

    Solving Exercise 13.7 MTW Using Light Signals

    I have managed to work out parts a and b of Exercise 13.7 from MTW (attached), but can't see how part c works. I can see how it could work in (say) the example of taking a radar measurement of the distance to Venus, where we have the Euclidian distance prediction and the result of the radar...
  47. 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.
  48. P

    Write Torsion Tensor: Definition, Metric Tensor & Equation

    Would it be possible to write the torsion tensor in terms of the metric? I know that a symmetric Christoffel Symbol can be written in terms of the partial derivatives of the metric. This definition of the christoffel symbols does not apply if they are not symmetric. Is it possible to write a...
  49. B

    Derivative of the mixed metric tensor

    So i am studying GR at the moment, and I've been trying to figure out what the derivative (not covarient) of the mixed metric tensor $$\delta^\mu_\nu$$ would be, since this tensor is just the identity matrix surely its derivative should be zero. Yet at the same time $$\delta^\mu_\nu =...
  50. U

    Energy-Momentum Tensor of Perfect Fluid

    Homework Statement I am given this metric: ##ds^2 = - c^2dt^2 + a(t)^2 \left( dx^2 + dy^2 + dz^2 \right)##. The non-vanishing christoffel symbols are ##\Gamma^t_{xx} = \Gamma^t_{yy} = \Gamma^t_{zz} = \frac{a a'}{c^2}## and ##\Gamma^x_{xt} = \Gamma^x_{tx} = \Gamma^y_{yt} = \Gamma^y_{ty} =...
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