Relativistic Invariance
(Lorentz invariance)
(Lorentz invariance)
The laws of physics are invariant under a transformation between two coordinate frames moving at constant
coordinate frames moving at constant velocity w.r.t. each other.
(The world is not invariant, but the laws (The world is not
invariant but the laws
of physics are!)
Review: Special Relativity
p
y
Einstein’s assumption: the speed of light is independent of the (constant ) velocity, l it v, v of the observer. It forms the basis for special relativity. f th b
It f
th b i f
i l 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 ).
It has the same value in all frames ( 0 ).
|dr| is the distance light moves in dt w.r.t the fixed frame.
A Lorentz transformation relates position and time in the two frames. A
L
t t
f
ti
l t
iti
d ti
i th t f
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
Only need two special cases.
Recall the picture
p
of the two frames
measuring the
speed of the same
light signal
light signal.
covariant and contravariant
components**
*For more details about contravariant
*For
more details about contravariant and covariant components see
and covariant components see
http://web.mst.edu/~hale/courses/M402/M402_notes/M402‐Chapter2/M402‐Chapter2.pdf
metric tensor relates components
metric tensor relates components
Using indices instead of x, y, z
Using indices instead of x, y, z
4‐dimensional
4
dimensional dot product
dot product
You can think of the 4‐vector dot product as follows: Why all these minus signs?
Why all these minus signs?
• Einstein’s
Einstein s assumption (all frames measure the assumption (all frames measure the
same speed of light) gives :
d(Ct)2 - dx
d 2 - dy
d 2 – dz
d 2 = 0 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 for t d f
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
partial derivatives
/xµ  µ
stands for (/(ct) , /x , /y , /z)
= ( /(ct)
( /(ct) , )
partial derivatives
partial derivatives
µ
stands for (/(ct) , ‐/x , ‐ /y , ‐ /z)
= ( /(ct)
( /(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
Any
four vector dot product has the same value
in all frames moving with constant
velocity w r t each other.
velocity w.r.t. each other
Examples:
xµxµ pµxµ
pµpµ
xµxµ
pµxµ xµAµ
Lorentz Invariance
Lorentz Invariance • Lorentz
Lorentz invariance of the laws of physics
invariance of the laws of physics
is satisfied if the laws are cast in terms of four‐vector dot products!
f f
d
d
!
• Four vector dot products are said to be
“Lorentz scalars”.
• In the relativistic field theories, we must
In the relativistic field theories we must
use “Lorentz scalars” to express the i
interactions. i