\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)2≡c2dt′2−dx′2−dy′2−dz′2=c2dt2−dx2−dy2−dz2
(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)2≡xμxν
(ds)2≡[dx0−dx1−dx2−dx3]⋅[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=[dx0−dx1−dx2−dx3]
To the contravarient term
xν=[dx0dx1dx2dx3]
Comes from the Minkowski metric
ημν=[10000−10000−10000−1]
[dx0dx1dx2dx3]⋅[10000−10000−10000−1]⋅[dx0dx1dx2dx3]
Using the Lorentz transformations and the index notation,
{t′=γ(t−vz/c2)x′=x′y′=y′z′=γ(z−vt)
[x′0x′1x′2x′3]=[γ(x0−vx3/c)x1x2γ(x3−vx0)]=[γ(x0−βx3)x1x2γ(x3−vx0)]
Where β≡vc
This can be expressed in matrix form as
[x′0x′1x′2x′3]=[γ00−γβ01000010−γβ00γ]⋅[x0x1x2x3]
Letting the indices run from 0 to 3, we can write
x′μ=∑3ν=0(Λμν)xν
Where Λ is the Lorentz transformation matrix for motion in the z direction.
\underline{Navigation}
⊲
△
⊳