Relativistic Invariance
(Lorentz invariance)
The laws of physics are invariant under
a transformation between two
coordinate frames moving at constant
velocity w.r.t. each other.
(The world is not invariant,
invariant but the laws
of physics are!)
Review: Special
p
Relativityy
Einstein’s assumption: the speed of light is independent of the (constant )
velocity,
l it v,
v off the
th observer.
b
It forms
f
the
th basis
b i for
f special
i l relativity.
l ti it
Speed
p
of light
g = C = ||r2 – r1| / (t
( 2 –t1) = ||r2’ – r1’ | / (t
( 2’ –t1‘)
= |dr/dt| = |dr’/dt’|
C2 = ||dr||2/dt2 = ||dr’||2 /dt’ 2
Both measure the same speed!
This can be rewritten:
d(Ct)2 - |dr|2 = d(Ct’)2 - |dr’|2 = 0
d(Ct)2 - dx2 - dy2 - dz2 = d(Ct’)2 – dx’2 – dy’2 – dz’2
d(Ct)2 - dx2 - dy2 - dz2
is an invariant!
It has the same value in all frames ( = 0 ).
|dr| is the distance light moves in dt w.r.t the fixed frame.
A LLorentz
t ttransformation
f
ti relates
l t position
iti and
d ti
time iin th
the ttwo fframes.
Sometimes it is called a “boost” .
•
http://hyperphysics.phy‐astr.gsu.edu/hbase/relativ/ltrans.html#c2
How does one “derive” the transformation?
Only need two special cases
cases.
Recall the picture
p
of the two frames
measuring the
speed of the same
light signal.
signal
covariant and contravariant
components**
*For more details about contravariant and covariant components see
http://web.mst.edu/~hale/courses/M402/M402_notes/M402‐Chapter2/M402‐Chapter2.pdf
metric tensor relates components
Using indices instead of x, y, z
4‐dimensional
4
dimensional dot product
You can think of the 4‐vector dot product as follows:
Why all these minus signs?
• Einstein’s
Einstein s assumption (all frames measure the
same speed of light) gives :
d(Ct)2 - dx
d 2 - dy
d 2 – dz
d 2=0
From this one obtains
the speed of light.
It must be positive!
C
= [dx2 + dy2 + dz2]1/2 /dt
Four dimensional gradient operator
4‐dimensional vector
component notation
• xµ stands
t d ffor
x0, x1 , x2, x3 for µ=0,1,2,3
ct, x, y, z = (ct, r)
• xµ stands for
x0 , x1 , x2 , x3 for µ=0,1,2,3
0123
ct, ‐x, ‐y, ‐z = (ct, ‐r)
partial derivatives
/xµ  µ
stands for
(/(ct) , /x , /y , /z)
= ( /(ct) , )
partial derivatives
µ
stands for
(/(ct) , ‐/x , ‐ /y , ‐ /z)
= ( /(ct) , ‐)
)
Invariant dot products using
4‐component notation
µ
xµxµ = µ=0,1,2,3
x
x
0123 µ
(repeated index summation implied)
= (ct)2 ‐x2 ‐y2 ‐z2
Invariant dot products using
4‐component notation
µ
µµ = µ=0,1,2,3


µ=0 1 2 3 µ
(repeated index summation implied)
= 2/(ct)2 ‐ 2
2
= 2/x2
+ 2/y2 + 2/z2
Any four vector dot product has the same value
in all frames moving with constant
velocity w.r.t.
w r t each other.
Examples:
xµxµ
pµpµ
pµxµ
xµxµ
pµxµ
xµAµ
Lorentz Invariance
• Lorentz invariance of the laws of physics
is satisfied if the laws are cast in
terms off four‐vector
f
d products!
dot
d
!
• Four vector dot products are said to be
“Lorentz scalars”.
• In the relativistic field theories,
theories we must
use “Lorentz scalars” to express the
i
interactions.
i