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AN INTRODUCTION TO THE LORENTZ GROUP
DOLORES MARTÍN BARQUERO AND CÁNDIDO MARTÍN GONZÁLEZ
1. I NTRODUCTION
In the General Relativity theory, the space is modeled as a 4-dimensional
manifold M in which the change of coordinates maps are differentiable with
differential in the Lorentz group. To be more precise, around each point p ∈
M there is a neighborhood U and a homeomorphism ϕ : U → V where V
is an open subset of R4 . The couples (U, ϕ) are called charts in differential
geometry and coordinate systems in Relativity. In fact, these maps can be
seen as coordinate systems (locally) and the manifold is covered by such
collection of coordinates systems. These different coordinate systems are
compatible in the following sense: if there is another chart (U 0 , ϕ0 ) such that
U ∩ U 0 6= ∅ then, it is required that the maps ϕ0 ◦ ϕ−1 : ϕ(U ∩ U 0 ) → ϕ0 (U ∩
U 0 ) are differentiable. Furthermore, the tangent spaces Tϕ(p) (ϕ(U ∩ U 0 ))
and Tϕ0 (p) (ϕ(U ∩U 0 )) are Minkoswki spaces (R4 , q) relative to the quadratic
form q(x, y, z, t) := x2 + y 2 + z 2 − c2 t2 (as usual c = speed of light) and
the differential d(ϕ0 ◦ ϕ−1 )(ϕ(p)) : R4 → R4 is required to be a Lorentz
transformation. The space around a massive star of a black hole is modeled
under such scheme and the big problem of nowadays Physics is to make
this scenario compatible with Quantum Mechanics (a problem that Einstein
himself tried to solve unsuccessfully).
2. O N THE M INKOWSKI S PACE
Minkowski space was initially developed by the German mathematician
Hermann Minkowski. The mathematical structure of Minkowski spacetime
was shown to be an immediate consequence of the postulates of special relativity and it is the most common mathematical structure on which special
relativity is formulated.
2.1. On the Minkowski Space. In a given system of referencee R, we
have a particle moving from P = (x, y, z) (at time T = t) to Q = (x0 , y 0 , z 0 )
(when T = t0 ). So the displacement is dr = (dx, dy, dz) in dt seconds,
where dx = x0 − x, dy = y 0 − y, dz = z 0 − z, dt = t0 − t, in a system of
reference R.
In a second system of reference R0 the coordinates of P are (a, b, c) (at
time T 0 = s) and the coordinates of Q are (a0 , b0 , c0 ) (at time T = s0 ), so
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2
DOLORES MARTÍN AND CÁNDIDO MARTÍN
Q
dr
P
F IGURE 1. dr = P Q = Q − P = (dx, dy, dz).
the displacement relative to R0 is dr0 = (dx0 , dy 0 , dz 0 ) where dx0 = a0 − a,
dy 0 = b0 − b, dz 0 = c0 − c and the time elapsed is dt0 = s0 − s.
We want to investigate the linear maps T : R4 → R4 such that
(dx, dy, dz, dt) 7→ (dx0 , dy 0 , dz 0 , dt0 )
That is, we want to determine linear maps T associated to the changes of
coordinates maps. Of course they must be invertible, so they are in the
group GL4 (R).
We are assuming the axioms of special relativity: ”The speed of light is
the same for any inertial system of reference”.
Thus, if a photon moves from P to Q, then in the system of reference R
we have dr = (dx, dy, dz),
c = speed of light, dt = time elapsed,
2
(dx) + (dy)2 + (dz)2
⇒ (dx)2 + (dy)2 + (dz)2 − c2 (dt)2 = 0.
dt2
Definition 1. Consider V a vector space over the field K. Let q : V → K
be a quadratic form. We say that a vector v ∈ V is isotropic relative to the
quadratic form q if q(v) = 0.
c2 =
So the vector v = (dx, dy, dz, dt) is isotropic for the quadratic form
q : R4 → R
given by
q(x, y, z, t) = x2 + y 2 + z 2 − c2 t2 .
In other words, particles moving at the speed of light are represented by
isotropic vectors of R4 relative to the quadratic form q.
If one particle moves at the speed of light relative to the system of reference R, then it must move also at the speed of light relative to the system of reference R0 . So, if v = (dx, dy, dz, dt) is isotropic for q, then
T (v) = v 0 = (dx0 , dy 0 , dz 0 , dt0 ) must be also isotropyic for q (and reciprocally).
AN INTRODUCTION TO THE LORENTZ GROUP
3
F IGURE 2. In https://en.wikipedia.org/wiki/World line.
The quadratic forms
x 2 + y 2 + z 2 − c2 t 2
x2 + y 2 + z 2 − t2
are equivalent in the sense that one can get one of then from the other applying a change of basis.
From a physicist viewpoint, we can change our scale so that the speed of
light c = 1. So we will replace the first one with the second in part of our
study.
Definition 2. Minkowski space is a 4-dimensional real vector space equipped
with a nondegenerate, symmetric bilinear form that provides of the quadratic form
q(x, y, z, t) = x2 + y 2 + z 2 − c2 t2 ,
simply called the Minkowski inner product, with signature either (+, +, +, −)
(in our case) or (−, −, −, +).
Definition 3. The isotropic cone of a quadratic form is the set of all its
isotropic vectors.
Proposition 1. If q : R2 → R is a quadratic form which vanishes on the
points of the curve x2 − y 2 = 0, then there is a real number k such that
q(x, y) = k (x2 − y 2 ) for any x, y ∈ R.
Proof. ∃ A, B, C ∈ R such that q(x, y) = Ax2 + By 2 + Cxy for any x and
y. If we take a point (x, y) such that x2 − y 2 = 0, then (x − y)(x + y) = 0
hence y = ±x. So
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DOLORES MARTÍN AND CÁNDIDO MARTÍN
0 = q(x, x) = Ax2 + Bx2 + Cx2 and
0 = q(x, −x) = Ax2 + Bx2 − Cx2 , therefore
(
A+B+C =0
A + B − C = 0.
This implies C = 0, B = −A, and q(x, y) = A (x2 − y 2 ).
Proposition 2. If q : R3 → R is a quadratic form vanishing on the points
of the surface x2 + y 2 − z 2 = 0, then there is a real number k such that
q(x, y, z) = k (x2 + y 2 − z 2 ), for any x, y, z ∈ R.
Proof. There are real numbers Aij such that
(1)
q(x, y, z) = A11 x2 + A22 y 2 + A33 z 2 + A12 xy + A13 xz + A23 yz.
The quadratic form q(x, 0, z) vanishes on the points of the curve x2 − z 2 =0
(which is in the plane y = 0 of R3 ). Thus, there is an k ∈ R such that
q(x, 0, z) = k (x2 − z 2 ). Taking into account equation (1), we can write
k (x2 − z 2 )=A11 x2 + A33 z 2 + A13 xz hence A11 = k = −A33 , A1,3 = 0.
Proof. Similarly q(0, y, z) = k 0 (y 2 − z 2 ) whence k 0 (y 2 − z 2 )=A22 y 2 +
A33 z 2 +A23 yz, thus A22 = k 0 = −A33 , A23 = 0. Summarizing, q(x, y, z) =
kx2 +√ky2 − kz 2 + A12 xy = k (x2 + y 2 − z 2 ) + A12 xy. But
√ the
point
2
2
2
1, 1, 2 is in the surface x + y − z = 0. So 0 = q 1, 1, 2 = A12
and we conclude that q(x, y, z) = k (x2 + y 2 − z 2 ).
Proposition 3. If q P
: Rn → R is a quadratic form vanishing on the points
xi 2 − xn 2 = 0, then there is real number k such
of the hypersurface n−1
i=1
Pn−1
that q (x1 , . . . , xn ) = k ( i=1 xi 2 − xn 2 ), for any (x1 , . . . , xn ) ∈ Rn .
Proof. As before, there are real numbers Aij such that
n
X
X
(2)
q (x1 , . . . , xn ) =
Aii x2i +
Aij xi xj .
i=1
i<j
Pn−1 2
The form q (0, x2 , .., xn ) vanishes on the points of i=2
xi − xn 2 = 0,
which can be considered as a hypersurface of Rn−1 , hencePapplying a suit2
2
able induction hypothesis, we have q (0, x2 , . . . , xn ) = k1 ( n−1
i=2 xi − xn ).
Taking into account equation (2) we get
n−1
n
X
X
X
2
2
k1 (
xi − xn ) =
Aii xi 2 +
Aij xi xj .
i=2
i=2
1<i<j
We deduce: Aii = k1 , (i = 2, . . . , n − 1), Ann = −k1 , Aij = 0, for i, j 6=
1, i 6= j. Repeating this argument for q (x1 , . . . , xi−1 , 0, xi+1 , . . . , xn ), we
AN INTRODUCTION TO THE LORENTZ GROUP
5
finally conclude Aij = 0 for any i 6= j, and all the Aii ’s are equal to k ∈ R
Pn−1 2
2
except for Ann = −k. Thus q (x1 , . . . , xn ) = k
i=1 xi − xn .
Corollary 1. If q 0 : R4 → R is a quadratic form vanishing on the isotropic
cone of q(x, y, z, t) = x2 + y 2 + z 2 − c2 t2 , then there exists k ∈ R such
that q 0 = kq. Moreover, if q 0 6= 0 has the same signature as q, the scalar k is
positive.
Proof. The quadratic form r(x, y, z, t) := q 0 (x, y, z, t/c) vanishes on the
points of the hypersurface x2 + y 2 + z 2 − t2 = 0. Applying Proposition 3
there is a real number k such that r(x, y, z, t) = k(x2 + y 2 + z 2 − t2 ). Therefore q 0 (x, y, z, t/c) = k(x2 + y 2 + z 2 − t2 ) and equivalently q 0 (x, y, z, t) =
k(x2 + y 2 + z 2 − c2 t2 ) for any x, y, z, t ∈ R. Now if the signature of q 0
agrees with that of q, the scalar k can not be k < 0 because in this case the
signature of q 0 would be different to that of q.
3. T HE L ORENTZ GROUP
Recall that the orthogonal group in dimension n, denoted O(n), is the
group of distance-preserving transformations of a Euclidean space of dimension n, where the group operation is given by composing transformations. Equivalently, it is the group of n × n orthogonal matrices, where the
group operation is given by matrix multiplication, and an orthogonal matrix
is a real matrix whose inverse equals its transpose.
The term orthogonal group may also refer to a generalization of the above
case: the group of invertible linear operators that preserve a non-degenerate
symmetric bilinear form or quadratic form on a vector space over a field.
In particular, when the bilinear form is the scalar product on the vector
space F n of dimension n over a field F , with quadratic form the sum of
squares, then the corresponding orthogonal group, denoted O(n, F ), is the
set of n × n orthogonal matrices with entries from F , with the group operation of matrix multiplication. This is a subgroup of the general linear group
GL(n, F ) given by
O(n, F ) = {M ∈ GL(n, F ) | M t M = M M t = I}
where M t is the transpose of M and I is the identity matrix.
An important subgroup of O(n) is the special orthogonal group, denoted
SO(n), of the orthogonal matrices of determinant 1. Both O(n, F ) and
SO(n, F ) are algebraic groups, because the condition that a matrix be orthogonal, i.e. have its own transpose as inverse, can be expressed as a set of
polynomial equations in the entries of the matrix.
When F is the field of real numbers, for any non-degenerate quadratic
form q, there is a basis, on which the matrix of the form is a diagonal matrix
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DOLORES MARTÍN AND CÁNDIDO MARTÍN
such that the diagonal entries are either 1 or −1. Thus the orthogonal group
depends only on the numbers of 1 and of −1, and is denoted by O(α, β) or
Oα,β (R), where α is the number of negative ones and β the number of ones.
Definition 4. The Lorentz group is the orthogonal group asociated to the
quadratic form q(x, y, z, t) = x2 + y 2 + z 2 − t2 and denoted by O(1, 3), that
is,
O(1, 3) := {X ∈ GL4 (R) : X I13 X t = I13 }


−1
where I1,3 = 
1
.
1
1
Theorem 1. Let q : R4 → R be q(x, y, z, t) := x2 + y 2 + z 2 − c2 t2 . If
T : R4 → R4 is an element of GL4 (R) such that q(v) = 0 if and only if
q(T (v)) = 0, then there is positive real number k such that kT ∈ O1,3 (R).
Proof. Define q 0 : R4 → R by q 0 (v) := q(T (v)) and apply Corollary 1 (take
into account that q 0 and q have the same signature!). Then there is positive
−1
real
√ number α such that αq(v) = q(T (v)). So q(v) = α q(T (v)) =
q( α−1 T (v)) = q(kT (v)) where k = α−1/2 . Consequently kT ∈ O1,3 (R).
4. O N THE L IE ALGEBRA ASSOCIATED TO THE L ORENTZ GROUP
Definition 5. A Lie algebra L over a commutative unitary ring K is a
K-module L with a binary operation L × L → L such that (x, y) 7→ xy
satisfying:
(
x2 = 0
(xy)z + (yz)x + (zx)y = 0.
for any x, y, z ∈ L.
For historical reasons the product in a Lie algebra is denoted [x, y] instead
of xy (and is called the Lie bracket of x and y).
Exercise. Prove that [x, y] + [y, x] = 0 for any x, y ∈ L (a Lie algebra).
Denote by Mn (R) the algebra of n × n matrices with real coefficients.
This is a Banach algebra, that is a real normed algebra which is a complete
normed space with the norm topology. The exponential map exp : Mn (R) →
Mn (R) is defined by the usual power-series
1
1 2
x + · · · + xn + · · · .
2!
n!
This series is convergent for any x.
exp(x) = 1 + x +
AN INTRODUCTION TO THE LORENTZ GROUP
7
4.1. Some results: exponentials. First: for any real number x ∈ R the
∞
X
xn
x
power series e =
is convergent.
n!
n=0
Second: let A ∈ Mn (R) (or in Mn (C)) and consider the series
∞
X
An
n=0
Since kAk ∈ R the series
exp(kAk) := e
kAk
=
P∞
n=0
is convergent and in fact
∞
X
kAkn
n=0
quence:
kAkn
n!
n!
k
X
kAkn
. Thus {
}k is a Cauchy sen!
n!
n=0
q
X
kAkn
∀ > 0, ∃n0 ∈ N such that ∀p, q ≥ n0 ,
< if q > p.
n!
n=p+1
Defining the partial sums Sk =
k
X
An
n=1
n!
, the sequence {Sk }k is a Cauchy
sequence since
q
q
X
X
An
kAkn
kSp − Sq k = k
k≤
<
n!
n!
n=p+1
n=p+1
Therefore the sequence in convergent and the series converges for any
matrix A. We use the notation
∞
X
An
.
exp(A) = eA :=
n!
n=1
Some properties of the exponential map are
(1) exp(0) = 1
(2) exp(x + y) = exp(x) exp(y) if x and y commute.
(3) exp(x) exp(−x) = 1 for any x.
(4) Thus exp(x) is invertible and
exp(x)−1 = exp(−x).
4.2. Some notations. GLn (R) denotes the group of invertible n × n matrices with coefficientes in R.
gln (R) denotes the Lie algebra of (all) n × n matrices with coefficientes
in R (with the Lie bracket [x, y] := xy − yx).
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DOLORES MARTÍN AND CÁNDIDO MARTÍN
Recall that a group is said to be a linear Lie group (over R) if it is a closed
subgroup of some general linear group GLn (R).
For a linear Lie group G ⊂ GLn (R) there is a Lie algebra
g := {x ∈ gln (R) : exp(λx) ∈ G, ∀λ ∈ R}.
The fact that this is a Lie algebra is not trivial. The Lie algebra g is also
denoted Lie(G).
For instance, fix a matrix m ∈ Mn (R) and consider the group
G := {x ∈ GLn (R) : xmxt = m}
where z 7→ z t denotes the matrix transposition operator.
Then x ∈ Lie(G) if and only if exp(λx) ∈ G for any λ ∈ R. Thus
x ∈ Lie(G) ⇔ exp(λx)m exp(λxt ) = m
So
d
dm
exp(λx)m exp(λxt ) =
=0
dλ
dλ
x exp(λx)m exp(λxt ) + exp(λx)mxt exp(λxt ) = 0
and for λ = 0 we get
xm + mxt = 0.
Summarizing x ∈ Lie(G) implies xm + mxt = 0.
Reciprocally, if xm + mxt = 0 then
Exercise. Prove that exp(λx)m = m exp(−λxt ), conclude that exp(λx)m exp(λxt ) =
m hence x ∈ Lie(G).
G := {x ∈ GLn (R) : xmxt = m}
Lie(G) = {x ∈ gln (R) : xm + mxt = 0}.


−1
In particular if I1,3 = 
1
, the Lorentz group
1
1
O(1, 3) := {X ∈ GL4 (R) : X I13 X t = I13 }
and its Lie algebra is
o(1, 3) = {X ∈ gl4 (R) : X I13 + I13 X = 0}.
Exercise. Prove that X ∈ gl4 (R) satisfies XI13 + I13 X = 0 if and only
if X is of the form:


0 x1
x 2 x3
 x1 0
x 4 x5 


 x2 −x4 0 x6 
x3 −x5 −x6 0
AN INTRODUCTION TO THE LORENTZ GROUP
9
Of course we can now do an abstraction process and define the Lorentz
type algebra over any field K to be the Lie algebra LK of all matrices


0 x1
x2 x3
 x1 0
x4 x5 


 x2 −x4 0 x6 
x3 −x5 −x6 0
with xi ∈ K (or using a commutative unitary ring of scalars R to define a
similar Lie algebra LR ).
5. S HORT D IGRESSION FOR S OPHISTICATED P EOPLE
For gourmets of Hopf algebras and Affine group schemes, there is an
alternative (and standard) form to define the Lie algebra associated to an
algebraic group (in the sense of affine group schemes). This is purely algebraic and therefore it does not involve analytical methods. Since the
(real) Lorentz group is an algebraic group, there is representing Hopf algebra (H, ∆, S, ) over the reals.
The ground field R is a H-module relative to the product
h · λ = (h)λ, h ∈ H, λ ∈ R
and so we can consider the derivation algebra der(H, R) which is a Lie
algebra relative to the product:
[f, g] := µ(f ⊗ g − g ⊗ f )∆
for any f, g ∈ der(H, R) (µ : R × R → R is the product).
The Hopf algebra H is the quotient of a polynomial algebra R[x1 , . . . , x16 ]
by an ideal generated by polynomials with integer coefficients. Thus it has
sense to consider H over any other field K. Let us denote it by HK . This enable us to define a Lorentz type algebraic group as the affine group scheme
represented by the Hopf algebra HK . Then we can define the Lorentz type
algebra LK as the Lie algebra of this algebraic groups, or equivalently, as
LK = der(HK , K)
with the Lie algebra structure as in the previous paragraph.
D EPARTAMENTO DE M ATEM ÁTICA A PLICADA , U NIVERSIDAD DE M ÁLAGA
E-mail address: [email protected]
D EPARTAMENTO DE Á LGEBRA , G EOMETR ÍA Y T OPOLOG ÍA , U NIVERSIDAD DE M ÁLAGA
E-mail address: [email protected]