4-vectors

From New IAC Wiki
Jump to navigation Jump to search
\underline{Navigation}

4-vectors

Using index notation, the time and space coordinates can be combined into a single "4-vector" xμ, μ=0, 1, 2, 3, that has units of length, i.e. ct is a distance.

[x0x1x2x3]=[ctxyz]


We can express the space time interval using the index notation

(ds)2c2dt2dx2dy2dz2=c2dt2dx2dy2dz2


(ds)2(dx0)2(dx1)2(dx2)2(dx3)2=(dx0)2(dx1)2(dx2)2(dx3)2


Since ds2 is nothing more than a dot product of a vector with itself, we should expect the components of the indices to follow a similar relationship.


(ds)2xμxν


(ds)2[dx0dx1dx2dx3][dx0dx1dx2dx3]


(ds)2(dx0)2(dx1)2(dx2)2(dx3)2=(dx0)2(dx1)2(dx2)2(dx3)2


The change in signs in the covariant term,

xμ=[dx0dx1dx2dx3]

To the contravarient term

xν=[dx0dx1dx2dx3]


Comes from the Minkowski metric


ημν=[1000010000100001]


[dx0dx1dx2dx3][1000010000100001][dx0dx1dx2dx3]



Using the Lorentz transformations and the index notation,

{t=γ(tvz/c2)x=xy=yz=γ(zvt)


[x0x1x2x3]=[γ(x0vx3/c)x1x2γ(x3vx0)]=[γ(x0βx3)x1x2γ(x3vx0)]


Where βvc

This can be expressed in matrix form as

[x0x1x2x3]=[γ00γβ01000010γβ00γ][x0x1x2x3]


Letting the indices run from 0 to 3, we can write

xμ=3ν=0(Λμν)xν


Where Λ is the Lorentz transformation matrix for motion in the z direction.






\underline{Navigation}