The Vertical Projection Model of Hyperbolic Space and Geometric Illustration of Special Relativity ()
1. Introduction
Einstein’s special relativity theory is one of the foundation blocks of modern theoretical physics [1] [2] [3] and its rich mathematical structures have been intensively studied [4] [5] . The composition law for velocities in special relativity is intimately related to hyperbolic geometry, as was first pointed out by Sommerfeld [6] , Varičak [7] [8] , Robb [9] , and Borel [10] . More recently, the subject was elaborated on by Ungar [11] [12] . In light of an all-important observation described in Theorem 2.1 below, which involves geometric description of isometries in hyperbolic space, we obtain new intuitive understandings of Lorentz transformation and relativistic addition of velocities in special relativity. Though Minkowski had given a geometric meaning of Lorentz transformation [1] , ours seems to be more geometrically satisfactory.
The basic idea of this article stems from the geometric construction of inversions with respect to a circle on the complex plane, that is, given a point P inside a circle C, the inversion point of P with respect to the circle C can be obtained by geometric construction with ruler and compass as follows: Starting from the center O of the circle C and passing through P, create a ray OP; draw the perpendicular line of OP through P, which intersects with C at point Q; draw the tangent of C through Q, then the intersection point of this tangent with the ray OP is the inversion point of P. Since inversions, which are isometries of conformal ball model of hyperbolic space, have the above geometric constructions, it is interesting to establish the analogous geometric description of isometries of other hyperbolic models. We succeed in finding out the analogous geometric description of isometries of vertical projection model of hyperbolic space (Theorem 2.1) and, furthermore, use it to obtain geometric illustrations of Lorentz transformation and Einstein velocity addition because they can all be recognized as isometries of hyperbolic space.
2. New Geometric Interpretation of Isometries in Vertical Projection Model of Hyperbolic Space
Let n be a positive integer. Let
be the Lorentzian n+1-space. For
,
, denote
and
. Notations
and
signify the usual dot (scalar) and cross (vector) product between two vectors throughout. Let c be a positive number (or the speed of light). The hyperboloid
. The open c-ball in
is
. We use the following four models of hyperbolic n-space:
1) Hyperboloid model:
2) Conformal ball model:
3) Projective model:
4) Vertical projection model (abbreviated as VP model):
The following natural isometries among these models are needed below:
Remark 2.1. From the isometry
, we know that the vertical projection model is just the vertical projection of the hyperboloid model, that’s why we named it vertical projection model (VP model for short). We will see that the vertical projection model is particularly suitable to express the relations in special relativity.
We first introduce some isometries in conformal ball model
. Let:
(2.1)
which is the analogue of the mapping
on the unit disk of complex plane. This mapping
is of fundamental importance to us, because its physical contents are Lorentz boost and Einstein velocity addition as can be seemed in Sections 3 and 4.
Let
denotes a sphere with center
and radius r. The inversion with respect to
is the mapping:
(2.2)
Let
be the reflection with respect to the hyperplane
, that is:
(2.3)
For
,
is the unique sphere perpendicular to
, whose center lies on the ray
, and the inversion with respect to which sends
to the origin
. Denote the isometry
as
.
,
.
(2.4)
Since
and
are isometries of
,
are also isometries of
by (2.4).
Recall that k-dimensional totally geodesic submanifolds in hyperbolic space are called k-planes. It is well known that spheres of dimension k that meet the boundary orthogonally and k-dimensional vector subspaces represent totally geodesic hyperbolic k-planes in
. Especially,
-planes in
are
-dimensional vector subspaces and spheres of the form:
,
.
In the subsequent, we describe the above isometries in vertical projection model
.
Proposition 2.1. 1)
-planes in VP model
are
-dimensional vector subspaces and hyperboloid of the form
,
. Furthermore, the k-planes (
) in
are the nonempty intersection of
-planes with
-dimensional vector subspaces.
2) In
, the inversion with respect to
, denoted by
, is the mapping:
(2.5)
3) The corresponding isometry in
of
(see (2.1)), denoted as
, is:
(2.6)
4)
(2.7)
Proof: The isometry
maps
-plane
,
in
onto
-plane in
, which is a sheet of hyperboloid given by:
In
, the inversion with respect to
is:
n
These isometries
and
in VP model
have the following quite clear geometric descriptions, which are analogous to the geometric construction of inversion on Euclidean space
.
Theorem 2.1 (Geometric meaning of
and
) 1) The image point
of
can be drawn as follows (Figure 1): Make a sphere S with center
and radius
, whose intersection with hyperboloid
is a
-sphere, and the hyperplane on which the intersection
-sphere lies is denoted as
. Through
, we draw a line that is parallel to the vector
. This line intersects the hyperplane
at
. Extend the vector
by
times, we get a vector
, then
.
2) Furthermore, the symmetric point of
with respect to the vector subspace perpendicular to
is
(Figure 2).
Figure 1. Geometric meaning of inversion.
Figure 2. Geometric meaning of
.
Proof: A direct calculation shows that:
By (2.7),
, so the reflection of
with respect to the vector subspace perpendicular to
is
.n
Remark 2.2. It is worth pointing out that the geometric interpretation of inversion given in Theorem 2.1 survives only in vertical projection model of hyperbolic space. It can’t implement in other models of hyperbolic space. It has some interesting geometric features: the line
is parallel to the line
, in particular,
and
map every parallel line of
onto itself; the scale factor
is a constant.
In order to clarify the relation of Theorem 2.1 with special relativity, we introduce the notion of addition on hyperbolic space
(we suppress the subscript here) using triangular rule, parallel transport and exponential map. For curve
, let
be the parallel transport isomorphism. For
, let
be the exponential map at
, which is a diffeomorphism. For
, we denote the unique oriented geodesic segment joining
to
by
. Let
be the center point of
.
Definition 2.1. For
, the addition of
and
is
. For
and
, scalar multiplication is
.
Remark 2.3. These are the analogues of addition and scalar multiplication on Euclidean space, but the triple
is not a vector space but actually form a so-called Gyrovector space (for definition see [11] [12] ).
Introduce the hyperbolic translation in conformal ball model
,
for
,
(see (2.1)), (2.8)
which is an isometry of
by Formula (2.4) (see [13] ).
Lemma 2.1. In
, we have:
(2.9)
Proof: By direct calculation, we have:
On the other hand,
. The parallel transport vector field
along curve
satisfies:
(2.10)
where the Christoffel symbol:
The unique solution of Equation (2.10) for initial condition
is
. Hence, also:
n
Proposition 2.2. The explicit addition formulas of the above four models are as follows:
1) Conformal ball model: for
,
(2.11)
2) Hyperboloid model: for
,
(2.12)
3) Projective model: for
,
(2.13)
4) Vertical projection model: for
,
(2.14)
Proof: 1) Since
is an isometry of
(see [13] ), it maps geodesic to geodesic, so:
(2.15)
Hence, by Definition 2.1 and Lemma 2.1,
2)
3)
4)
n
Remark 2.4.
(2.16)
In later sections, we shall apply the established results of hyperbolic space in this section to special relativity.
3. Geometric Illustration of Lorentz Transformation
Suppose that a reference frame S' is moving relative to a similarly oriented frame S with velocity
; and suppose further that they both take the origin of their coordinate systems to be the event at which they pass each other. The standard 4-dimensional Lorentz boost relates space-time coordinates
of frame S' and
of frame S is [2] [3] :
(3.1)
(the exponent T indicates transposition) where:
(3.2)
and the gamma factor
is defined by:
(3.3)
When
, the corresponding matrix:
(3.4)
is often called a boost in the
-direction.
(3.5)
where A is the unimodular orthogonal matrix (rotation in real world space):
(3.6)
Lorentz boost, which acts on the 4-dimensional Minkowski space-time
, can be viewed as an action on the 3-dimensional real world space
by the following procedure.
Since
is Lorentz invariant, Lorentz boost (3.1) maps the hyperboloid
in Minkowski space-time
1-1 onto itself. Furthermore, since Lorentz transformations are linear and the hyperboloid
contains a basis of the Minkowski space-time
, the Lorentz boost (3.1) is completely determine by its action on the hyperboloid
. Denote
as
and use the notations in Section 2. We see that the Lorentz boost (3.1) acts as an isometry on hyperboloid model
.
is isometric to VP model
under the vertical projection
. Thus under
, Lorentz boost corresponds to an isometry on
. From the physical point of view, it is more natural to deal with physical objects such as Lorentz boost on VP model
, because the underlying space of
is the real world space
. Also, mathematically, Lorentz boost is simplified when converting to the VP model
, as can be seen from the following theorem.
Theorem 3.1. When transitioning to the VP model
, the Lorentz boost (3.1) becomes
(3.7)
where
.
Proof: For
on the hyperboloid
, we have:
And by (3.1), it follows that:
n
Remark 3.1. Lorentz transformations are essentially different from rotations in 4-dimensional Euclidean space, since by Theorem 3.1 Lorentz transformations can reduce to 3-dimension but rotations can’t.
Theorem 3.2. A Lorentz boost is the composition of an inversion with a reflection. More precisely,
.
Proof: Comparing Formulas (6) and (7), we have
.n
Remark 3.2. By Theorem 3.2, Remark 2.4 and Definition 2.1 of addition in hyperbolic space, Lorentz boost is a translation operation of hyperbolic space.
By Theorem 2.1 and Theorem 3.2, we get the geometric meaning of Lorentz boosts.
Theorem 3.3 (Geometric meaning of Lorentz boosts). The Lorentz boost
of
can be drawn as follows (Figure 3): Make a sphere S with center
and radius
, whose intersection with hyperboloid
is a circle, and the hyperplane on which the intersection circle lies is denoted as
. Through
, we draw a line that is parallel to the vector
. This line intersects the hyperplane
at
. Extend the vector
by
times, we get a vector
, then
. Furthermore, the symmetric point of
with respect to the vector subspace perpendicular to
is
.
Theorem 3.4. A proper orthochronous Lorentz transformation is the composition of a Euclidian rotation with an inversion and a reflection.
Proof: First, a proper orthochronous Lorentz transformation is the composition of a Lorentz boost and a Euclidian rotation [4] . Furthermore, by Theorem 3.2, a Lorentz boost is the composition of an inversion with a reflection.n
Remark 3.3. By Theorem 3.4, the geometric meaning of proper orthochronous Lorentz transformations is clear.
Figure 3. Geometric meaning of Lorentz boost.
Figure 4. Geometric meaning of velocity addition.
4. Geometric Meaning of Relativistic Addition of Velocities
Consider the worldline of a particle in uniform motion which has velocity
in inertial coordinate system S. Along the particle worldline, the inertial coordinates
are functions of the proper time
. The four-velocity of the particle is:
(4.1)
The proper velocity of the particle is:
(4.2)
The relativity admissible velocity of the particle is:
(4.3)
Since
(4.4)
it follows that:
(4.5)
and
(4.6)
Remark 4.1. The relations of these three kind of velocities are
,
. In view of (4.5), the space of four-velocities, endowed with metric
, is just hyperboloid model
. The space of relativity admissible velocities, endowed with metric
, is projective model
. The space of proper velocities, endowed with metric
, is VP model
.
Let
be the four-velocity of the particle with respect to the system S'. Then take the derivative with respect to proper time
in Equation (3.1), we have:
(4.7)
(4.8)
(4.9)
which, comparing with the non-relativistic case, may be heuristically understood as the relativistic “composition velocity” of four-velocities
(the physical reverse 4-velocity of 4-velocity
) and V. Notice Formula (2.12), it is recognized as the addition in hyperboloid model
:
(4.10)
Thus, the space of four-velocities forms hyperboloid model
and the relativistic addition of four-velocities is exactly the addition in
as defined in Section 2 Definition 2.1.
Since
, the relativistic addition of relativity admissible velocities
and
is given by:
(4.11)
which is the well-known Einstein velocity addition formula. By Proposition 2.2, the space of relativity admissible velocities forms projective model
and the relativistic addition of relativity admissible velocities is exactly the addition in
as defined in Definition 2.1.
Since
, the relativistic addition of proper velocities
and
is given by:
(4.12)
In view of Proposition 2.2, the space of proper velocities forms VP model
and the relativistic addition of proper velocities is exactly the addition in
as defined in Definition 2.1.
Remark 4.2. Comparing (3.7) and (4.12), it is interesting to notice that in VP model
Lorentz transformation and velocity addition are given by the same formula, so they are equivalent to each other.
Proposition 4.1.
(4.13)
(4.14)
By Formula (4.12) and Theorem 2.1, we get the geometric meaning of relativistic addition formula of velocities.
Theorem 4.1 (Geometric meaning of relativistic velocity addition). The relativistic velocity addition
of
can be drawn as follows (Figure 4): Make a sphere S with center
and radius
, whose intersection with hyperboloid
is a circle, and the hyperplane on which the intersection circle lies is denoted as
. Through
, we draw a line that is parallel to the vector
. This line intersects the hyperplane
at
. Extend the vector
by
times, we get a vector
, then
. Furthermore, the symmetric point of
with respect to the vector subspace perpendicular to
is
.
Proof: By Formulas (2.6) and (4.12), we have:
(4.15)
n
Remark 4.3. The dilation factor in Theorem 4.1 is the gamma factor plus one, i.e.
.
Remark 4.4. When
, Lorentz transformation turns into Galilean transformation and addition in hyperbolic space reduces to ordinary addition in Euclidean space.
5. Successive Lorentz Transformations and Thomas Precession
Consider composition of Lorentz boots given by (3.2), by polar decomposition, we have [4] :
(5.1)
where
(5.2)
where
is an unimodular orthogonal matrix, called Thomas rotation, and is given by:
(5.3)
and where:
(5.4)
(5.5)
(5.6)
(5.7)
The Thomas rotation takes place in the plane
. The sense of the Thomas rotation in the
-plane is negative (we orient this plane in the usual way, such that
defines the direction of the normal). And the cosine of the angle of rotation
is:
(5.8)
We have:
(5.9)
and for
,
(5.10)
The corresponding results in VP model are as follows:
(5.11)
(5.12)
(5.13)
(5.14)
Corollary 5.1 (Geometric meaning of
for fixed
). First, repeat the procedure of Theorem 4.1, we get
, then rotate an angle
, which is given by (5.14), in the
-plane in the negative direction yields
.
6. Conclusion
In conclusion, Lorentz transformation and relativistic addition of velocities in VP model
have fairly intuitive descriptions. Since the hyperboloid model
is the same as the VP model
up to a vertical projection, the geometric meaning of Lorentz transformation and relativistic addition of velocities in
are also clear now.