Mechanics
Physics 151
Lecture 14
Special Relativity
(Chapter 7)
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Special Relativity: Lorentz Transformation and Minkowski Space, Study notes of Mechanics
The principles of special relativity, focusing on the lorentz transformation and its application in minkowski space. The invariance of spacetime distance, the characteristics of lorentz transformation, and the relationship between proper time and velocity. It also discusses the concept of 4-vectors and the minkowski metric tensor.
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Mechanics
Physics 151
Lecture 14
Special Relativity
(Chapter 7)
Special Relativity
You know it quite well already
Learned in Physics 15a/
We skip all the historical discussions
E.g. about non-existence of absolute time
Start from two principles and build a clean formalism
Laws of physics are the same in all inertial frames
Speed of light in vacuum is the same in all inertial frames
Maxwell’s equations are always correct
Spacetime Distance
Light is emitted at event 1 and detected at event 2
Speed of light, measured in
S
and
S’
, must be the same
Define distance in spacetime
2
2
2
2
2
2
x
y
z
x
y
z
c
t^
t
′^
′^
2
2
2
2
2
(^
)^
c t
x
y
z
2
2
2
2
2
(^
)^
c t
x
y
z
′^
′^
′^
2
2
2
2
2
2
(^
)^
(^
s
c t
x
y
z
We assume that the spacetime distance
s between any
two events is the same in all inertial frames
Light Cone
represents light
If an object slower than light travels from event 1 to 2, t^
must be larger
Æ
s
If (
s
< 0, there isn’t enough time for even light to reach
Interval between two events can fall into 3 regions
2
2
2
2
2
2
(^
)^
(^
)^
s
c t
x
y
z
2
(^
)^
s ∆
timelike
2
(^
)^
s ∆
spacelike
2
(^
)^
s ∆
lightlike
Reachable by ordinary objects
Reachable only by tachyons (
v >
c
)
Reachable by light
Time Dilation
Consider an object moving at a velocity
v
in
S
Define
S’
so that this object is constantly at its origin
Consider small movement from event 1 to event 2
Make it infinitesimal (^
,^
,^
)^
(^
,^
,^
x^
y^
z
dx dy dz
v
v
v
dt
2
2 2
2
2
2
(^
)^
(^
)^
in
(^
)^
(^
)^
in
s
c
t^
S
c
t^
x
y
z
S
′^
2
2
2
2
2
2
2
2
(^
)^
(^
)^
(^
)^
(^
cdt
cdt
dx
dy
dz
c
v
dt
′^
2 2
2
v c
dt
d
dt
dt
dt
Proper time
Moving clockappears to run
slower
4-Vectors
Express an event (
t, x, y, z
) as a
4-dimensional vector
c
is there to fix the unit
Length of this vector in “ordinary”4-d space would be
It’s more useful to define “length” as
ct
ct x y z ⎡
x
r
2
2
2
2
2
2
c t
x
y
z
x
xx
Totally useless for us
2
2
2
2
2
2
c t
x
y
z
⋅^
x
x x
x
x
Metric tensor
g
Lorentz Transformation
So far we have not specified how
S
and
S’
are related
The Question Of The Day is
We know the name: Lorentz transformation
What is the general form of the transformationLet’s figure out its characteristics
that keeps the spacetime distance
s
2
invariant?
(^
,^
,^
,^
)^
(^
,^
,^
,^
L
ct x y z
ct
x
y
z
′^
′^
′^
What is the general form of the transformation
that conserves the length in the Minkowski space?
Linearity
Physics does not depend on absolute position or time
Shifting the origin of
S
must simply shift the origin of
S’
Try
x
Æ
If we define
L
x
) must be a linear function of
x
+ fixed offset
x
x + a
′^
′^
x
x + a
(^
)^
L
L
x + a
x
a
i.e.
L
L
a
a
(^
)^
L
L
L
L
x + a
x
a
True for any
x
and
a
L
L
L
′^
x
x
(^
)^
L
L
L
′^
′^
x
a
x
a
L’
is linear
Lorentz Transformation
Now we have
What are the constraints on
L
L
must conserve the length of any 4-vector
In terms of components
g
is symmetric
Æ
Equation is symmetric
Æ
6 of 16 equations are duplicates
There are 10 constraints
′^
x
Lx
L
is a 4x4 real matrix
2
x
xgx
LgL
g
2 ′^
′^
x
x gx
xLgLx
^
=
ji^
jk
kl
il
L g
L
g
16 real equations
L
has 6 degrees of freedom
Rotation
Remember how rotation matrices were defined?
Conserve the length of any 3-vector in Cartesian space
In fact any rotation
A
in 3-space satisfies the condition
3-d rotation is a subset of HLT
Not the most exciting part of it
Rotation has 3 degrees of freedom (Euler angles)
There must be 3 more in HLT
2
2
2
2
2
2
(^
)^
const
c t
x
y
z
x
Unaffected
Conserved
Lorentz Boost
L
must satisfy
Origin of
S’
is
This must satisfy
Looks familiar except…
LgL
g
00
01
10
11
0
0
0
0
0
0
1
0
0
0
0
1
L
L
L
L
⎡^
⎤
⎢^
⎥
⎢^
⎥
=
⎢^
⎥
⎢^
⎥
⎣^
⎦
L
2
2
00
10
L
L
2
2
01
11
L
L
00
01
10
11
L
L
L L
x
′^
10
11
L ct
L x
x
vt
10 11 L
v c
L
β ≡
Solve!
00
01
10
11
L
L
L
L
2
1 1
γ
β
What are these
signs?
Sign Ambiguities
There are 4-fold sign ambiguities
Only 1 represents Lorentz transformation
Think about the low-velocity limit
(^01)
β γ
→→
This must be unit matrix
00
01
10
11
L
L
L
L
Sign of
ct’
is arbitrary
Sign of
x’
is arbitrary
2
1 1
γ
β
00
01
10
11
L
L
L
L
General Boost
We’ve got
L
for boost in
x
We can rotate
S
and
S’
to get
L
for boost in any direction
We can also use a bit of vector algebra
Split 3-vector
r
into two parts
Parallel component transforms like
x
above
0
0
0
0
0
0
1
0
0
0
0
1
γ
γβ
γβ
γ −
⎡^
⎤
⎢^
⎥
− ⎢^
⎥
=
⎢^
⎥
⎢^
⎥
⎣^
⎦
L
2
(^
r
β β
r &^
⊥
r
r
r &
Parallel
to
v
Perpendicular
to
v
(^
ct
ct
ct
r
β
r
&
2
(^
ct
ct
⊥
⋅^
r
β β
r
β
r
r
β
r
&
Proper Lorentz Transformation
Writing down explicitly for general
2 2
2
2
2
2
2
2
2
2
2
2
2
(^
(^
(^
(^
(^
(^
(^
(^
(^
(^
x^
y
x^
x^
z
y^
x^
y^
y^
z
z^
y
z^
x^
z
x^
y^
z
x y z
β β
β
β β
β
β
β
β β
β
β β
β
β
β
β β
β β
β
β
β
β
L
β
ββ
β
^
3 degrees of freedom (
, x
, y
) z
Æ
General form of Lorentz
transformation without rotation
Proper Lorentztransformation