Contravatiant Covariant Components

In a previous example, https://rpubs.com/lizhi1800/Geometry_of_Relativity, we have seen that to get observations \(\{1.732051,3.464102\}\)vector in the traveling frame, we need to use a transformation matrix multiplying vector \(E=\{4,5\}\). Vector \(E=\{4,5\}\) is called a contravariant vector.

And we can also inverse the transformation matrix, and then multiply vector \(\{0,3\}\) to get the same result \(\{1.732051,3.464102\}\)vector. Vector \(\{0,3\}\) is called a covariant vector.

Lorentz transforms \[x′=\gamma(x−\beta ct)\\ ct′=\gamma(ct−\beta x)\] Matrix notation \[\frac{\partial \{x',ct'\}}{\partial x},\frac{\partial \{x',ct'\}}{\partial ct}=\left| {\begin{array} *{\gamma}&{-\beta\gamma}\\ {-\beta\gamma}&{\gamma} \end{array}} \right|\] Multiplication by contravariant components and covariant components \[\left| {\begin{array} *{x'}\\ {ct'} \end{array}} \right|=\left| {\begin{array} *{\gamma}&{-\beta\gamma}\\ {-\beta\gamma}&{\gamma} \end{array}} \right|\left| {\begin{array} *{x}\\ {ct} \end{array}} \right|\\**contravariants \{x,ct\}\] \[\left| {\begin{array} *{x'}\\ {ct'} \end{array}} \right|=\left| {\begin{array} *{\gamma}&{-\beta\gamma}\\ {-\beta\gamma}&{\gamma} \end{array}} \right|^{-1}\left| {\begin{array} *{\bar x}\\ {c\bar t} \end{array}} \right|\\**covariants \{\bar x,c\bar t\}\]

library(Deriv)
## Warning: package 'Deriv' was built under R version 4.0.5
Deriv("gamma*(x-beta*ct)","x")
## [1] "gamma"
Deriv("gamma*(ct-beta*x)","x")
## [1] "-(beta * gamma)"
Deriv("gamma*(x-beta*ct)","ct")
## [1] "-(beta * gamma)"
Deriv("gamma*(ct-beta*x)","ct")
## [1] "gamma"
beta=.5
gamma=1/sqrt(1-beta^2)

m=matrix(c(gamma,-beta*gamma,-beta*gamma,gamma),nrow=2)

Compute from contravariant components

contravariants=c(4,5)
m%*%contravariants
##          [,1]
## [1,] 1.732051
## [2,] 3.464102
P=c(1.732051,3.464102)

Compute from covariant components

covariants=m%*%P
covariants
##      [,1]
## [1,]    0
## [2,]    3
solve(m)%*%covariants
##          [,1]
## [1,] 1.732051
## [2,] 3.464102