8Special relativity
IA Dynamics and Relativity
8.4 Geometry of spacetime
We’ll now look at the geometry of spacetime, and study the properties of vectors
in this spacetime. While spacetime has 4 dimensions, and each point can be
represented by 4 real numbers, this is not ordinary
R
4
. This can be seen when
changing coordinate systems, instead of rotating the axes like in
R
4
, we “squash”
the axes towards the diagonal, which is a hyperbolic rotation. In particular,
we will have a different notion of a dot product. We say that this space has
dimension d = 1 + 3.
The invariant interval
In regular Euclidean space, given a vector
x
, all coordinate systems agree on the
length |x|. In Minkowski space, they agree on something else.
Consider events
P
and
Q
with coordinates (
ct
1
, x
1
) and (
ct
2
, x
2
) separated
by ∆t = t
2
− t
1
and ∆x = x
2
− x
1
.
Definition
(Invariant interval)
.
The invariant interval or spacetime interval
between P and Q is defined as
∆s
2
= c
2
∆t
2
− ∆x
2
.
Note that this quantity ∆
s
2
can be both positive or negative — so ∆
s
might be
imaginary!
Proposition. All inertial observers agree on the value of ∆s
2
.
Proof.
c
2
∆t
02
− ∆x
02
= c
2
γ
2
∆t −
v
c
2
∆x
2
− γ
2
(∆x − v∆t)
2
= γ
2
1 −
v
2
c
2
(c
2
∆t
2
− ∆x
2
)
= c
2
∆t
2
− ∆x
2
.
In three spatial dimensions,
∆s
2
= c
2
∆t
2
− ∆x
2
− ∆y
2
− ∆z
2
.
We take this as the “distance” between the two points. For two infinitesimally
separated events, we have
Definition (Line element). The line element is
ds
2
= c
2
dt
2
− dx
2
− dy
2
− dz
2
.
Definition
(Timelike, spacelike and lightlike separation)
.
Events with ∆
s
2
>
0
are timelike separated. It is possible to find inertial frames in which the two events
occur in the same position, and are purely separated by time. Timelike-separated
events lie within each other’s light cones and can influence one another.
Events with ∆
s
2
<
0 are spacelike separated. It is possible to find inertial
frame in which the two events occur in the same time, and are purely separated
by space. Spacelike-separated events lie out of each other’s light cones and
cannot influence one another.
Events with ∆
s
2
= 0 are lightlike or null separated. In all inertial frames, the
events lie on the boundary of each other’s light cones. e.g. different points in
the trajectory of a photon are lightlike separated, hence the name.
Note that ∆s
2
= 0 does not imply that P and Q are the same event.
The Lorentz group
The coordinates of an event
P
in frame
S
can be written as a 4-vector (i.e.
4-component vector) X. We write
X =
ct
x
y
z
The invariant interval between the origin and
P
can be written as an inner
product
X ·X = X
T
ηX = c
2
t
2
− x
2
− y
2
− z
2
,
where
η =
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
.
4-vectors with
X ·X >
0 are called timelike, and those
X ·X <
0 are spacelike.
If X · X = 0, it is lightlike or null.
A Lorentz transformation is a linear transformation of the coordinates from
one frame S to another S
0
, represented by a 4 × 4 tensor (“matrix”):
X
0
= ΛX
Lorentz transformations can be defined as those that leave the inner product
invariant:
(∀X)(X
0
· X
0
= X ·X),
which implies the matrix equation
Λ
T
ηΛ = η. (∗)
These also preserve X · Y if X and Y are both 4-vectors.
Two classes of solution to this equation are:
Λ =
1 0 0 0
0
0 R
0
,
where
R
is a 3
×
3 orthogonal matrix, which rotates (or reflects) space and leaves
time intact; and
Λ =
γ −γβ 0 0
−γβ γ 0 0
0 0 1 0
0 0 0 1
,
where
β
=
v
c
, and
γ
= 1
/
p
1 − β
2
. Here we leave the
y
and
z
coordinates intact,
and apply a Lorentz boost along the x direction.
The set of all matrices satisfying equation (
∗
) form the Lorentz group
O
(1
,
3).
It is generated by rotations and boosts, as defined above, which includes the
absurd spatial reflections and time reversal.
The subgroup with det Λ = +1 is the proper Lorentz group SO(1, 3).
The subgroup that preserves spatial orientation and the direction of time is
the restricted Lorentz group
SO
+
(1
,
3). Note that this is different from
SO
(1
,
3),
since if you do both spatial reflection and time reversal, the determinant of the
matrix is still positive. We want to eliminate those as well!
Rapidity
Focus on the upper left 2
×
2 matrix of Lorentz boosts in the
x
direction. Write
Λ[β] =
γ −γβ
−γβ γ
, γ =
1
p
1 − β
2
.
Combining two boosts in the x direction, we have
Λ[β
1
]Λ[β
2
] =
γ
1
−γ
1
β
1
−γ
1
β
1
γ
1
γ
2
−γ
2
β
2
−γ
2
β
2
γ
2
= Λ
β
1
+ β
2
1 + β
1
β
2
after some messy algebra. This is just the velocity composition formula as before.
This result does not look nice. This suggests that we might be writing things
in the wrong way.
We can compare this with spatial rotation. Recall that
R(θ) =
cos θ sin θ
−sin θ cos θ
with
R(θ
1
)R(θ
2
) = R(θ
1
+ θ
2
).
For Lorentz boosts, we can define
Definition (Rapidity). The rapidity of a Lorentz boot is φ such that
β = tanh φ, γ = cosh φ, γβ = sinh φ.
Then
Λ[β] =
cosh φ −sinh φ
−sinh φ cosh φ
= Λ(φ).
The rapidities add like rotation angles:
Λ(φ
1
)Λ(φ
2
) = Λ(φ
1
+ φ
2
).
This shows the close relation betweens spatial rotations and Lorentz boosts.
Lorentz boots are simply hyperbolic rotations in spacetime!