Relativistic Quadrics in Pseudo-Euclidean Spaces
Milena Radnović
Mathematical Institute SANU, Belgrade, Serbia
XVII Geometrical Seminar 2012
Main Reference
Vladimir Dragović and Milena Radnović
Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic
quadrics
Advances in Mathematics 231 (2012), 1173–1201
1
Pseudo-Euclidean space, ellipsoidal billiards and confocal quadrics
2
Relativistic quadrics
Pseudo-Euclidean space
Definition
Pseudo-Euclidean space Ek,l is a d-dimensional space Rd with
pseudo-Euclidean scalar product:
hx, y ik,l = x1 y1 + · · · + xk yk − xk+1 yk+1 − · · · − xd yd .
Here, k, l ∈ {1, . . . , d − 1}, k + l = d.
Pair (k, l ) is the signature of the space.
Denote Ek,l = diag(1, 1, . . . , 1, −1, . . . , −1), with k 1’s and l −1’s.
Then the pseudo-Euclidean scalar product is:
hx, y ik,l = Ek,l x ◦ y ,
where ◦ is the standard Euclidean product.
Lines and hyper-planes in the pseudo-Euclidean space
ℓ – a line in the pseudo-Euclidean space;
v – its vector
ℓ is called:
space-like if hv , v ik,l > 0
time-like if hv , v ik,l < 0
light-like if hv , v ik,l = 0
Orthogonal vectors
x ⊥ y ⇔ hx, y ik,l = 0
Light-like line is orthogonal to itself.
Hyper-planes and their orthogonal vectors
v 6= 0 – a vector; α : v ◦ x = 0 – the corresponding hyper-plane
Ek,l v ⊥ α
If v is light-like, then so is Ek,l v , and Ek,l v is parallel to the
hyper-plane.
Billiard reflection
v — a vector, α — a hyper-plane
Decompose vector v : v = a + nα , nα ⊥ α, a k α
Definition
v ′ = a − nα is the billiard reflection of v on α.
v = v ′ if v is contained in α.
v ′ = −v if it is orthogonal to α.
If nα is light-like, which means that it belongs to α, then the
reflection is not defined.
Reflection on a smooth surface
Line ℓ′ is a billiard reflection of ℓ on a smooth surface S if their
intersection point belongs to S and their vectors are reflections of
each other on the tangent plane of S at this point.
Confocal quadrics in pseudo-Euclidean space
Khesin, Tabachnikov, Advances in Mathematics (2009)
Ellipsoid
E :
x2
x12 x22
+
+ ··· + d = 1
a1
a2
ad
a1 , . . . , ad — positive constants
Family of quadrics confocal with E
Qλ :
2
xk+1
xk2
xd2
x12
+· · ·+
+
+· · ·+
= 1, λ ∈ R.
a1 − λ
ak − λ ak+1 + λ
ad + λ
Unless stated differently, we are going to consider the
non-degenerate case, when set {a1 , . . . , ak , −ak+1 , . . . , −ad }
consists of d different values.
Degenerate quadrics
λ ∈ {a1 , . . . , ak , −ak+1 , . . . , −ad }
Qεj aj is the intersection of the coordinate hyper-plane xj = 0 with
the cylinder:
X x2
i
= 0.
ai − εi λ
i 6=j
We denoted:
εi =
(
1,
1≤i ≤k
−1, k + 1 ≤ i ≤ d.
λ=∞
We added the following quadric to the family:
Q∞ is the intersection of the cone
2
− · · · − xd2 = 0
x12 + · · · + xk2 − xk+1
with the hyper-plane at the infinity.
Confocal quadrics and their types in the Euclidean space
Family of confocal quadrics in the d -dimensional Euclidean
space
xd2
x12
+ ··· +
= 1,
b1 − λ
bd − λ
λ ∈ R,
b1 > b2 > · · · > bd > 0.
Properties
E1 each point of the space Ed is the intersection of exactly d
quadrics from the family; moreover, all these quadrics are of
different geometrical types;
E2 the family contains exactly d geometrical types of
non-degenerate quadrics – each type corresponds to one of
the disjoint intervals of the parameter λ: (−∞, bd ),
(bd , bd−1 ), . . . , (b2 , b1 ).
Confocal quadrics and their types in the Euclidean space
Consider billiard within the ellipsoid λ = 0.
β1 , . . . , βd−1 – parameters of the caustics of a given trajectory.
{b̄1 , . . . , b̄2d } = {b1 , . . . , bd , 0, β1 , . . . , βd−1 },
b̄1 ≥ b̄2 ≥ · · · ≥ b̄2d .
Properties
E3 Along a fixed billiard trajectory, the Jacobi coordinate λi
(1 ≤ i ≤ d) takes values in segment [b̄2i −1 , b̄2i ];
E4 along a trajectory, each λi achieves local minima and maxima
exactly at touching points with corresponding caustics,
intersection points with corresponding coordinate
hyper-planes, and, for i = d, at reflection points;
E5 values of λi at those points are b̄2i −1 , b̄2i ; between the critical
points, λi is changed monotonously.
Properties E1–E5 represent the key in the algebro-geometrical
analysis of the billiard flow.
At the first glance, it seems that fine properties like those do not
take place in the pseudo-Euclidean case.
In a d-dimensional pseudo-Euclidean space, a general confocal
family contains d + 1 geometrical types of quadrics and, in
addition, quadrics of the same geometrical type have non-empty
intersection.
Confocal conics in the Minkowski plane
Relativistic conics in the Minkowski plane
Birkhoff, Morris, The American Mathematical Monthly (1962)
Confocal conics in the three-dimensional Minkowski space
Confocal family
Qλ :
x2
a−λ
+
y2
b−λ
+
z2
c+λ
= 1, λ ∈ R, with a > b > 0, c > 0.
Geometrical types of quadrics
1-sheeted hyperboloids oriented along z-axis, for
λ ∈ (−∞, −c);
ellipsoids, corresponding to λ ∈ (−c, b);
1-sheeted hyperboloids oriented along y -axis, for λ ∈ (b, a);
2-sheeted hyperboloids, for λ ∈ (a, +∞) – these hyperboloids
are oriented along z-axis.
Degenerate quadrics
Qa , Qb , Q−c , Q∞ , that is planes x = 0, y = 0, z = 0, and the
plane at the infinity respectively.
Tropic curves
Definition
On each quadric, the tropic curves represent the set of points
where the induced metrics on the tangent plane is degenerate.
Proposition [V. Dragović, M. R.]
The union of the tropic curves on all quadrics of a confocal family
is a union of two ruled surfaces Σ+ and Σ− . The two surfaces Σ+ ,
Σ− are developable as embedded into Euclidean space. Moreover,
their generatrices are all light-like.
The union of all tropic curves of a confocal family
Curved tetrahedra
Proposition [V. Dragović, M. R.]
Consider the subset T + of Σ+ determined by the condition
λ ∈ [b, a]. This set is a curved tetrahedron, with the following
properties:
its upper and lower edges represent self-intersection of Σ+ ;
other four edges are cuspidal edges of Σ+ ;
thus, at each vertex of the tetrahedron, a swallowtail
singularity of Σ+ occurs.
Curved tetrahedron T +
Light-like tangents to tropic curves
Proposition [V. Dragović, M. R.]
A tangent line to the tropic curve of a non-degenarate quadric of
the confocal family is always space-like, except on a 1-sheeted
hyperboloid oriented along y -axis.
Tangent lines of a tropic on 1-sheeted hyperboloids oriented along
y -axis are light-like exactly at four points, while at other points of
the tropic curve, the tangents are space-like.
Moreover, a tangent line to the tropic of a quadric from the
confocal family belongs to the quadric if and only if it is light-like.
Remark
In other words, the only quadrics of the confocal family that may
contain a tangent to its tropic curve are 1-sheeted hyperboloids
oriented along y -axis, and those tangents are always light-like.
Tropic curves and their light-like tangents on a hyperboloid
Generalized Jacobi coordinates
Definition
The generalized Jacobi coordinates of point (x, y , z) in the 3D
Minkowski space are the unordered triplet of solutions of
y2
z2
x2
+ b−λ
+ c+λ
= 1.
Qλ : a−λ
Note that any of the following cases may occur:
Generalized Jacobi coordinates are real and different.
Only one generalized Jacobi coordinate is real.
Generalized Jacobi coordinates are real, but two of them
coincide; or
All three generalized Jacobi coordinates are equal.
Relativistic types of quadrics – definition
A component of quadric Qλ0 is of
relativistic type E if, at each of its points, λ0 is less than the
other two generalized Jacobi coordinates.
relativistic type H1 if λ0 is between the other two generalized
Jacobi coordinates.
relativistic type H2 if λ0 is greater than the other two
generalized Jacobi coordinates.
relativistic type 0 if λ0 is the only real generalized Jacobi
coordinate.
Suppose (x, y , z) is a point of the 3D Minkowski space where the
generalized Jacobi coordinates are real and different. Decorated
Jacobi coordinates of that point are the ordered triplet of pairs
(E , λ1 ), (H1 , λ2 ), (H2 , λ3 ) of generalized Jacobi coordinates and
the corresponding types of relativistic quadrics.
Relativistic quadrics
z
y
x
Thank you!
Minkowski’s hand-coloured slide,
shown during his lecture Space and time (Cologne, 1908)