Thursday, 28 February 2019

Important Equations for General Relativity

Here are some important equations for General Relativity.
Babylonian equations
They are in Commentary Important Equations.pdf along with references and some notes.
Mathematics

  • Definition of ##{\partial }_{\mu }## (Carroll 1.54)
  • (Anti)symmetrisation operator (Carroll 1.79)
  • Vector as derivative (Carroll 2.16)
  • Commutator (Carroll 2.20/2.23)
  • Tensor transformation equation (Carroll 2.30)
  • Basis vectors (Physics Forums)
  • Covariant derivative / Christoffel symbol (Carroll section 3.2)
  • Torsion Tensor (C Eq 3.22)
  • The Christoffel connection Γ  (C Eq 3.27)
  • The geodesic equation (C Eq 3.44)
  • Directional covariant derivative (C Eq 3.38)
  • The parallel transport equation (C Eq 3.39, 3.40)
  • Riemann tensor (c Eq 3.112/3.113)
  • Bianchi identity (c Eq 3.140)
  • Ricci tensor and scalar, Weyl tensor (c Eq 3.144-3.147)
  • Einstein tensor and 'contravariant' derivative (c Eq 3.1452,2)
  • Killing's equation, Killing vectors (c Eq3.174)
  • Geodesic deviation equation (c Eq3.208)

Tensor tricks

  • Multi-dimensional Chain Rule
  • Partial derivative of components gives Kronecker delta (C Eq 1.152)
  • Partial derivatives commute
  • Metric is always symmetric (C section 2.5)
  • Contracting with metric lowers / raises index
  • Swap indices with metric or any similar tensor
  • A relationship for the derivative of the metric determinant
  • Fully contracted symmetric × antisymmetric tensor vanishes
  • Symmetrising a tensor equation
  • Two formulas involving four-velocity
  • The projection tensor
  • Contra / co-variant tensor transformation matrices
  • Tensor contractions using matrices

Physics

  • Electro Magnetic Field Tensor (C Eq 1.69)
  • Maxwell's equations (C Eq 1.96-1.98)
  • Energy Momentum tensor for a perfect fluid (C Eq 3.93 and 1.114)
  • Energy Momentum tensor for dust in SR (C Eq 1.110)
  • Energy-momentum tensor from action for matter (C Eq 4.75)
  • Energy-momentum conservation equation (C Eq 3.92 & 4.8)
  • Einstein's equation x 3 for general relativity (C Eq2.44-4.46)

More Maths


  • Differential and Integration on Web
  • Pullback / Pushforward operators (Carroll A.9, A.10)
  • Levi-Civita symbol and tensor (Carroll section 2.8)
  • p-forms (Carroll section 2.9)
  • Exterior derivative (Carroll 2.76)
  • Wedge product (Carroll 2.73)
  • Hodge star operator (Carroll 2.82)
  • Stokes's theorem (C Eq 3.35)
  • Euler-Lagrange Equation

All in Commentary Important Equations.pdf along with references and some notes.
Image from WikipediBabylonian equations.

Carroll's (1.68) where the Levi-Civita symbol is defined says "the Levi-Civita symbol is a ##(0,4)## tensor. It is NOT a tensor as he reminds us elsewhere.

Monday, 18 February 2019

Maxwell's equation with three-forms

James Clerk Maxwell was one of the greatest theoretical physicists of all time. He was up there with Newton and Einstein. He published his famous equations in about 1865. Quite a bit was known about electricity and magnetism and the relationships between them at the time but he spotted a missing component and was able to show that electromagnetic waves must exist and moreover travel at the speed of light, which was already known. Therefore light had to be electromagnetic waves. It must have made physicists feel queasy. I remember being astonished when we did Maxwell's equations and proved it at university. Soon after his equations were known, radio waves (1887, Hertz) and  X-rays (1895, Roentgen ) were discovered. Then came Einstein with Special (1905) and General (1915) Relativity.

Carroll writes "The other one of Maxwell's equations, (1.96) can be expressed as an equation between three-forms:
\begin{align}
\mathrm{d}\left(*F\right)=*J & \phantom {10000}(1) \\
\end{align}where the current one-form ##J## is the current four-vector with index lowered. Filling in the details is left for you, as good practice converting from differential form notation to ordinary index notation." I need the practice .....

##J## was given as\begin{align}
J^{\mu }=\left(\rho ,\ J^x,J^y,J^z\right) & \phantom {10000}(2) \\
\end{align}where ##\rho ## is the classical charge density and ##J^i## the current.

His equation (1.96) ("the other one of Maxwell's equations") was
\begin{align}
{\partial }_{\mu }F^{\nu \mu }=J^{\nu } & \phantom {10000}(3) \\
\end{align}This equation is really four equations one for each value of ##\nu = 0,1,2,3## and on the Left Hand Side we might have four terms for the summation over ##\mu## but one is zero because of the antisymmetry of ##F## (in the next equation). The four equations are written out in (8)-(10).

We have a new form of the electromagnetic field strength tensor:
\begin{align}
F^{\mu \nu }=\left( \begin{array}{cccc}
0 & E^1 & E^2 & E^3 \\
-E^1 & 0 & B^3 & -B^2 \\
-E^2 & -B^3 & 0 & B^1 \\
-E^3 & B^2 & -B^1 & 0 \end{array}
\right)=-F^{\nu \mu } & \phantom {10000}(7) \\
\end{align}
I am not certain if the indices on the ##B##'s should be up or down.

We expand (3) and we get the two relevant Maxwell's equations (one in three parts):
\begin{align}
{\partial }_1F^{01}+{\partial }_2F^{02}+{\partial }_3F^{03} & =J^0 & \phantom {10000}(8) \\
{\partial }_0F^{10}+{\partial }_2F^{12}+{\partial }_3F^{13} & =J^1 & \phantom {10000}(9) \\
{\partial }_0F^{20}+{\partial }_1F^{21}+{\partial }_3F^{23} & =J^2 & \phantom {10000}(10) \\
{\partial }_0F^{30}+{\partial }_1F^{31}+{\partial }_2F^{32} & =J^3 & \phantom {10000}(11) \\
\end{align}
We now convert (1) into index notation. It becomes
\begin{align}
{\widehat{\varepsilon }}^{\ \ \ \ \ }_{\beta \gamma \delta \iota }{\partial }_{\alpha }F^{\delta \iota }+{\widehat{\varepsilon }}^{\ \ \ }_{\gamma \alpha \delta \iota }{\partial }_{\beta }F^{\delta \iota }+{\widehat{\varepsilon }}^{\ \ \ }_{\alpha \beta \delta \iota }{\partial }_{\gamma }F^{\delta \iota }=2{\widehat{\varepsilon }}_{\kappa \alpha \beta \gamma }J^{\kappa } & \phantom {10000}(29) \\
\end{align}On the face of it there the free indices ##\alpha ,\beta ,\gamma ## will give 64 components of the equation to check and the double summation on the LHS gives ##16\times 64=1,024## terms and the summation on the RHS a further 256. In total 1,280 terms which is rather more than the 16 terms in Maxwell's equations. Conceivably we might need to write out every combination. Fortunately the number of terms will be reduced by the antisymmetry of the Levi-Civita symbol ##\widehat{\varepsilon }## and the electromagnetic field tensor ##F##.

We prove that all components of (29) vanish except ones where ##\alpha ,\beta ,\gamma ## all differ. We only have 24 components (or 96 terms) to check! After plodding through all of them, I found six of each of Maxwell's equations (8)-(11) as predicted. The other forty are all 0 = 0. (You can see my notes calculating ##\widehat{\varepsilon }## in the previous post also on the Levi-Civita symbol.)

Having done all that I did discover the easy way to get there without all the plodding tables and paper work. But it was worth it, because as Carroll said, I needed practice converting from differential form notation to ordinary index notation and moreover practice with manipulating indexed equations. The plodding inspired me.

The full six page story is here: Commentary 2.9 Maxwell equation with 3 forms.pdf

Sunday, 17 February 2019

The Levi-Civita Symbol

The Levi-Civita symbol, which is not a tensor, is defined as
\begin{align}
{\widetilde{\epsilon }}_{{\mu }_1{\mu }_2\dots {\mu }_n}=\left\{ \begin{array}{ll}
+1 & \mathrm{if\ }{\mu }_1{\mu }_2\dots {\mu }_n\mathrm{\ is\ an\ even\ permitation\ of}\ 01..(n-1)\  \\
-1 & \mathrm{if\ }{\mu }_1{\mu }_2\dots {\mu }_n\mathrm{\ is\ an\ odd\ permitation\ of}\ 01..\left(n-1\right) \\
0 & \mathrm{otherwise} \end{array}
\right. & \phantom {10000}(1) \\
\end{align}Instead of ##\widetilde{\epsilon }## it is also often written as ##\widehat{\varepsilon }##, which is what I usually do.

We usually only need it in three or four dimensions so it becomes much simpler:
\begin{align}
{\widehat{\varepsilon }}_{\alpha \beta \gamma \delta }=\left\{ \begin{array}{ll}
+1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ even\ permitation\ of}\ 0123\  \\
-1 & \mathrm{if\ }\alpha \beta \gamma \delta \mathrm{\ is\ an\ odd\ permitation\ of}\ 0123 \\
0 & \mathrm{otherwise} \end{array}
\right. & \phantom {10000}(2) \\
\end{align}or
\begin{align}
{\widehat{\varepsilon }}_{ijk}=\left\{ \begin{array}{ll}
+1 & \mathrm{if\ }ijk \mathrm{\ is\ an\ even\ permitation\ of}\ 123\  \\
-1 & \mathrm{if\ }ijk \mathrm{\ is\ an\ odd\ permitation\ of}\ 123 \\
0 & \mathrm{otherwise} \end{array}
\right. & \phantom {10000}(3) \\
\end{align}One simple thing is that if any two of the indices on ##\widehat{\varepsilon }## are the same, it vanishes.

Even in four dimensions it is fairly easy to calculate, for example
\begin{align}
{\widehat{\varepsilon }}_{1023}=-{\widehat{\varepsilon }}_{0123}=-1 & \phantom {10000}(4) \\
\end{align}It's easy to jot down the indices and determine the sign by walking indices in one direction or another. Here is an example from my real sad life

Thursday, 14 February 2019

Lorentz transformation for velocity

There's an equation on Wikipedia for the Lorentz transformation for velocity at
https://en.wikipedia.org/wiki/Lorentz_transformation#Transformation_of_velocities
which states
\begin{align} \large
{\boldsymbol{\mathrm{u}}}^{\boldsymbol{\mathrm{{'}}}}\boldsymbol{\mathrm{=}}\frac{\mathrm{1}}{\mathrm{1}\boldsymbol{\mathrm{-}}\frac{\boldsymbol{\mathrm{u}}\boldsymbol{\mathrm{\cdot }}\boldsymbol{\mathrm{v}}}{c^{\boldsymbol{\mathrm{2}}}}}\left[\frac{\boldsymbol{\mathrm{u}}}{{\gamma }_{\mathrm{v}}}\boldsymbol{\mathrm{-}}\boldsymbol{\mathrm{v}}\boldsymbol{\mathrm{+}}\frac{1}{c^2}\frac{{\gamma }_{\mathrm{v}}}{{\gamma }_{\mathrm{v}}+1}\left(\boldsymbol{\mathrm{u}}\boldsymbol{\mathrm{\cdot }}\boldsymbol{\mathrm{v}}\right)\boldsymbol{\mathrm{v}}\right]& \phantom {10000}(1)  \\
\end{align}
where the coordinate velocities and Lorentz factor are
\begin{align} \large
\boldsymbol{\mathrm{u}}=\frac{d\boldsymbol{\mathrm{r}}}{dt}\ ,\ \ \ {\boldsymbol{\mathrm{u}}}^{\boldsymbol{\mathrm{{'}}}}=\frac{d{\boldsymbol{\mathrm{r}}}^{\boldsymbol{\mathrm{{'}}}}}{dt^{'}}\ \ ,\ \ \ \ {\gamma }_{\mathrm{v}}=\frac{1}{\sqrt{1-\frac{\boldsymbol{\mathrm{v}}\boldsymbol{\mathrm{\cdot }}\boldsymbol{\mathrm{v}}}{c^2}}} & \phantom {10000}(2) \\
\end{align}
##\boldsymbol{\mathrm{v}}## is the relative velocity of the primed frame of reference. Instead of (1), it is sometimes useful to have the equation
\begin{align} \large
{\boldsymbol{\mathrm{u}}}^{\boldsymbol{\mathrm{{'}}}}\boldsymbol{\mathrm{=}}\frac{\mathrm{1}}{\mathrm{1}\boldsymbol{\mathrm{-}}\frac{\boldsymbol{\mathrm{u}}\boldsymbol{\mathrm{\cdot }}\boldsymbol{\mathrm{v}}}{c^{\boldsymbol{\mathrm{2}}}}}\left[\frac{\boldsymbol{\mathrm{u}}}{{\gamma }_{\mathrm{v}}}\boldsymbol{\mathrm{-}}\boldsymbol{\mathrm{v}}\boldsymbol{\mathrm{+}}\frac{1}{v^2}\left(1-\frac{1}{{\gamma }_{\mathrm{v}}}\right)\left(\boldsymbol{\mathrm{u}}\boldsymbol{\mathrm{\cdot }}\boldsymbol{\mathrm{v}}\right)\boldsymbol{\mathrm{v}}\right] & \phantom {10000}(3) \\
\end{align}
where ##v^2=\boldsymbol{\mathrm{v}}\boldsymbol{\mathrm{\cdot }}\boldsymbol{\mathrm{v}}##.
This form of the equation was useful in the first exercise in Sean M Carroll's Spacetime and Geometry : An Introduction to General Relativity. To get it we just have to show that
\begin{align} \large
\frac{1}{c^2}\frac{{\gamma }_{\mathrm{v}}}{{\gamma }_{\mathrm{v}}+1}=\frac{1}{v^2}\left(1-\frac{1}{{\gamma }_{\mathrm{v}}}\right) & \phantom {10000}(4) \\
\end{align}
We use ##\gamma ## instead of ##{\gamma }_{\nu }## because have terms like ##{\gamma }^2## where the 2 is an exponent.
\begin{align} \large
\frac{1}{c^2}\frac{\gamma }{\gamma +1} & =\frac{1}{c^2}\frac{1}{\gamma +1}\left(\frac{{\gamma }^2}{\gamma }\right)=\frac{1}{c^2}\frac{1}{\gamma +1}\left(\frac{\frac{1}{1-{v^2}/{c^2}}}{\gamma }\right) & \phantom {10000}(6) \\
 & =\frac{1}{c^2}\frac{1}{\gamma +1}\left(\frac{\left[\frac{1-1+{v^2}/{c^2}}{1-{v^2}/{c^2}}\right]c^2}{\gamma v^2}\right) & \phantom {10000}(7) \\
 & =\frac{1}{c^2}\frac{1}{\gamma +1}\left(\frac{\left[\frac{1}{1-{v^2}/{c^2}}-1\right]c^2}{\gamma v^2}\right) & \phantom {10000}(8) \\
 & =\frac{1}{c^2}\frac{1}{\gamma +1}\left(\frac{\left[{\gamma }^2-1\right]c^2}{\gamma v^2}\right) & \phantom {10000}(9) \\
 & =\frac{1}{c^2}\frac{1}{\gamma +1}\left(\frac{\left[\gamma +1\right]\left[\gamma -1\right]c^2}{\gamma v^2}\right) & \phantom {10000}(10) \\
 & =\frac{\gamma -1}{\gamma v^2} & \phantom {10000}(11) \\
\therefore \frac{1}{c^2}\frac{\gamma }{\gamma +1} & =\frac{1}{v^2}\left(1-\frac{1}{\gamma }\right) & \phantom {10000}(12) \\
\end{align}
                                         QED
I tried to post this on the Wikimedia page, but it was not allowed :-(
This proof is also available as a pdf here.

Wednesday, 6 February 2019

The second exterior derivative always vanishes

We investigate the claim that second exterior derivative of any form A always vanishes:
$$\mathrm{d}(\mathrm{d}A)\mathrm{=0}$$This is Carroll's equation (2.80). He then tells us that this is due to the definition of the exterior derivative and the fact that partial derivatives commute, ##{\partial }_{\alpha }{\partial }_{\beta }={\partial }_{\beta }{\partial }_{\alpha }##. From the definition we get
$$\large\mathrm{d(dA)}=\left(p+1\right)p{\partial }_{[{\mu }_1}{\partial }_{[{\mu }_2}A_{{\mu }_3\dots {\mu }_{p+1}]]}$$This contains nested antisymmetrisation operators which we met in Exercise 2.08. The expansion of the equation contains ##p!\left(p+1\right)!## terms in total containing permutations of the indices ##{\mu }_1{\mu }_2{\mu }_3\dots {\mu }_{p+1}##. If ##p## was ##10## that would be 39,916,800 terms.

First I exercised my permutation skills with ##2, 3, 4## and ##n## indices to get the drift of the proof. I was then able to expand groups of ##p+1## terms of ##\mathrm{d}(\mathrm{d}A)##. Each one vanished due to ##{\partial }_{\alpha }{\partial }_{\beta }={\partial }_{\beta }{\partial }_{\alpha }##. It did not depend on the antisymmetry of ##A##. This is rather like the fact that the wedge product of a 2-form and a 1-form does not depend on the antisymmetry of the 2-form as we discovered in Commentary 2.9 Differential forms.pdf.

We also note that
$$\large A_{[{\mu }_1}B_{[{\mu }_2}C_{{\mu }_3\dots {\mu }_{p+1}]]}=0$$for any rank 1 tensors  ##A,B## and any tensor ##C## and the up/down position of any index in the tensors is immaterial.

Read all 5 pages at Commentary 2.9 Second exterior derivative.pdf.