Inverse Lorentz Transformation

The inverse Lorentz transformation, which would give the primed frame components in terms ofthe unprimed (fixed) frame components, can be obtained by replacing β with -β. This followsfrom the observation that (as viewed from the moving frame) the fixed frame is moving withspeed -v.

ctxyz

ctxyz

''''

=

−−

γ γβγβ γ

0 00 0

0 0 1 00 0 0 1

The inverse Lorentz matrix is then

.Λ− =

−−

1γ γβγβ γ

0 00 0

0 0 1 00 0 0 1

Now one can see that for the differentials

cdtdxdydz

cdtdxdydz

ctct

ctx

cty

ctz

xct

xx

xy

xz

yct

yx

yy

yz

zct

zx

zy

zz

''''

' ' ' '

' ' ' '

' ' ' '

' ' ' '

=

−−

=

γ γβ

γβ γ

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

0 00 0

0 0 1 00 0 0 1

cdtdxdydz

and the following partial derivatives

∂∂

ν

µν

µ µνx

xT

'

= =− −Λ Λ1 1

Thus we know how the partial derivatives transform:

∂∂

∂∂∂∂∂∂

γ βγβγ γ

∂∂

∂∂∂

∂∂

∂

( ) ( ' )

'

'

'

ct

x

y

z

ct

x

y

z

=

−−

0 00 0

0 0 1 00 0 0 1

To make things look simpler, we can define

and .

∂∂

∂∂∂∂∂∂

∂∂∂∂

( )ct

x

y

z

≡

0

1

2

3

ctxyz

xxxx

≡

0

1

2

3

Then we can write the transformation equations for the partial derivatives and the differentials ina neater way:

∂∂∂∂

γ βγβγ γ

∂∂∂∂

0

1

2

3

0

1

2

3

0 00 0

0 0 1 00 0 0 1

=

−−

''''

and dxdxdxdx

dxdxdxdx

0

1

2

3

0

1

2

3

0 00 0

0 0 1 00 0 0 1

=

γ βγβγ γ

''''

If we now let µ and ν be indices which take the values 0,1,2,3 then the matrix equations abovebecome even simpler:

where µ = 0,1,2,3 gives the four matrix rows∂ ∂ ∂µ µν

νν

µν

ν= ≡∑Λ Λ' '

where µ = 0,1,2,3 gives the four matrix rowsdx x dxµ µν

ν µν

ν

ν= ≡∑Λ Λ' '

These latter two equations are written in “Einstein summation notation”, in which arepeated index (one up and one down) implies summation over that variable.

The Metric Tensor

The metric tensor, gµα , is defined by

From this we can see that

g =−

−−

1 0 0 00 1 0 00 0 1 00 0 0 1

Using this metric tensor we can define covariant components of the differential 4-vector (notesubscript on dxµ)

Then

The expression above gives the “length squared” of the “dx” 4-vector in space and time. It is aninvariant under any Lorentz transformation. It has the same value in all Lorentz frames. Likewise

gµα dxα dxµ = c2 dt2- dx2- dy2- dz2 (sum over α)g = metric tensor

dxµ = gµα dxα (sum over α)

= covariant components of the differential 4-vector

dxµ = contravariants components of the differential 4-vector

dxµ dxµ = gµα dxα dxµ = c2 dt2- dx2- dy2- dz2 (sum over α and µ)

xµ xµ = x’µ x’µ

has the same value in all Lorentz frames

Proper Time and Spacelike or Timelike Intervals

.

This invariant, c2(dτ)2, is evaluated in the “rest” frame where dx’ = dy’ = dz’= 0. dτ is called the propertime interval. Note that

When v/c becomes very small (approaches 0 and the non-relativistic limit) dτ =dt the time interval weview in non-relativistic physics as a constant (not subject to change because of motion).

Spacelike and Timelike Intervals:

When (dτ)2 is positive (and dτ is real) the space-time separation dxµ is said to be timelike. When (dτ)2 isnegative (and dτ is imaginary) the space-time separation dxµ is said to be spacelike. When (dτ)2 =0, thespace-time separation is said to be singular. A singular space-time interval means that

dxµ dxµ = (cdt)2 - (dx)2- (dy)2- (dz)2 = 0

That is, the following is satisfied exactly:

(cdt)2 = (dx)2 - (dy)2 - (dz)2 = |dr|2 so that c = |dr|/dt .

The |dr| is just the distance light travels in time dt. Let dr=d(r2-r1) where the subscripts 1 and 2 refer tothe two points in this space. According to Einstein’s postulate, any frame moving with speed, v’, withrespect to the unprimed frame will also observe (for the same event, the motion of light from point 1 topoint 2):

c =|dr’|/dt’.

Proper time and particle motion:

When a particle is moving at a constant relativistic velocity, v, one can view the rest frame ofthe particle as the “moving frame”. Then τ, the time in the rest frame, is the “proper time”forthe particle motion and the relativistic three momentum, p, for the particle is given by

p = m dr/dτ = m γ dr/dt where dt = γ dτ.

dxµ dxµ = gµα dxα dxµ = c2 dt2- dx2- dy2- dz2 = c2(dτ)2

(sum over α and µ)has the same value in all Lorentz frames.

τ = proper time

dτ =dt[1-v2/c2]1/2 = dt/γ.