On Nonlinear Inertial Transformations
Nicholas Agia∗
arXiv:2512.25024v1 [physics.class-ph] 31 Dec 2025
University of New Hampshire, Department of Physics & Astronomy
Abstract
It is often assumed that the most general transformation between two inertial reference
frames is affine linear in their Cartesian coordinates, an assumption which is however not
true. We provide a complete derivation of the most general inertial frame transformation,
which is indeed nonlinear; along the way, we shall find that the conditions of preserving
the Law of Inertia take the form of Schwarzian differential equations, providing perhaps
the simplest possible physics setting in which the Schwarzian derivative appears. We then
demonstrate that the most general such inertial transformation which further preserves the
speed of light in all directions is, however, still affine linear. Physically, this paper may be
viewed as a reduction of the number of postulates needed to uniquely specify special relativity
by one, as well as a proof that inertial transformations automatically imbue spacetime
with a vector space structure, albeit in one higher dimension than might be expected.
Mathematically, this paper may be viewed as a derivation of the higher-dimensional analog
of the Schwarzian differential equation and its most general solution.
Contents
1 Introduction and Summary
1
2 Derivation of the Inertiality Constraints
3
3 Solving the Constraint Equations
6
3.1 Isolating f and the Schwarzian Differential Equation . . . . . . . . . . . . . . . . 7
3.2 Solving for g i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Properties and Significance of the Nonlinear Transformations
12
4.1 Mathematical Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Physical Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5 The Higher-Dimensional Schwarzian Derivative
1
17
Introduction and Summary
Frameworks of relativity based on inertial reference frames and the transformations therebetween
have a long history. Undoubtedly, the most famous and important such framework is that of
special relativity, whose inertial transformations were discovered by Lorentz and by Poincaré
and finally unified by Einstein in 1905 [1]. Despite the nonexistence of inertial reference frames
in our physical Universe, the concept continues to be fundamental to the current description
∗
Nicholas.Agia@unh.edu
1
of the Laws of Nature because general relativity is based on locally inertial frames through the
equivalence principle [2], which is also why special-relativistic quantum field theories such as the
Standard Model continue to be so useful in cases where the effects of gravitation are negligible.
One of the most basic questions one can ask is the following: what is the most general possible
inertial frame transformation between two inertial observers? To set up our conventions, let us
denote by (t, ⃗x) the spacetime coordinates of the first (“unprimed”) observer and by (t′ , ⃗x ′ ) the
spacetime coordinates of the second (“primed”) observer. We work in d − 1 spatial dimensions
which we take to be the Euclidean space Rd−1 equipped with the usual Euclidean metric. We
make no further assumptions about the structure of spacetime or about any dynamics of any
physical theory. The frame transformation (t, ⃗x) 7→ (t′ , ⃗x ′ ) allows us to transform the trajectory
⃗x(t) of any particle as measured in the unprimed frame, say, to the trajectory ⃗x ′ (t′ ) of the
same particle as measured in the primed frame. The only piece of information we are given is
that the two reference frames are inertial, which here means that a trajectory with vanishing
acceleration in one frame transforms to a trajectory with vanishing acceleration in the other
frame, i.e.
⃗a(t) = 0
⇐⇒
⃗a ′ (t′ ) = 0;
(1)
we use the Cartesian coordinates for ⃗x in which the Euclidean metric takes the standard form,
2
so that ⃗a(t) = d dt⃗x2(t) is the appropriate notion of acceleration. It is now a well-defined question
to ask: what is the most general such inertial transformation (t, ⃗x) 7→ (t′ , ⃗x ′ )? On account
of this being one of the most minimalistic questions one can ask in physics, it is somewhat
surprising that a rigorous derivation of the answer does not appear to be readily available in the
searchable literature. Often, it is simply assumed that an inertial transformation must be affine
linear in the spacetime coordinates, i.e. of the form t′ = c1 + ⃗c2 · ⃗x + c3 t and ⃗x ′ = ⃗c4 + A⃗x + ⃗c5 t
where the scalars c1 and c3 , the vectors ⃗c2 , ⃗c4 and ⃗c6 as well as the matrix A are all constant,
but this assumption is wrong. In fact, even Einstein in his derivation of the Lorentz boost from
postulates effectively assumed that the transformation had to be linear [1]1 . Of course, there
is no issue with the framework of special relativity itself or its assumptions, but this specific
assumption is actually unnecessary, as we shall see.
The most general inertial transformation is, in fact, nonlinear; as proven below, such a
transformation is not affine linear but rather projective linear in the spacetime coordinates,
i.e. it takes the form
c1 + ⃗c2 · ⃗x + c3 t
c6 + ⃗c7 · ⃗x + c8 t
⃗c4 + A⃗x + ⃗c5 t
⃗x ′ =
,
c6 + ⃗c7 · ⃗x + c8 t
t′ =
(2)
(3)
where all coefficients involved are constant. Certainly, that these nonlinear transformations are
indeed inertial is not new, and given that this question is hundreds of years old it seems likely
that the derivation showing that they are the complete set of inertial transformations would
have appeared somewhere at some point; nevertheless, the purpose of this paper is to provide
a rigorous proof of this fact that is accessible to the modern audience. As a consequence,
it demonstrates that the assumption of linearity (or, equivalently, homogeneity) in Einstein’s
derivation is unnecessary, as the nonlinear transformations are eliminated upon demanding the
preservation of the speed of light in all directions, thus decreasing the number of postulates
needed to derive special relativity by one. A more interesting consequence is that inertial
transformations can always be viewed as changes of basis in a (projectivized) vector space,
1
In the 1923 English edition, Einstein’s words were “In the first place it is clear that the equations must be
linear on account of the properties of homogeneity which we attribute to space and time.”, which only hides the
assumption in that of homogeneity; the point is that even this assumption is unnecessary.
2
automatically endowing spacetime with a vector space structure without first having to assume
that spacetime is already a vector space. On the mathematics side, this derivation constitutes
perhaps the simplest example in physics in which the famous Schwarzian derivative arises, a
single-variable nonlinear differential operator with numerous applications. We shall be able to
go even further and derive a suitable higher-dimensional version of the Schwarzian derivative
and its associated differential equation, whose solutions are the projective linear transformations
above. These constitute the main results of this paper.
Everything contained below is quite elementary, both conceptually and computationally.
The point is to provide a thorough proof of the foregoing statements, with all steps of the
derivations being shown; we also shall not assume that the reader is familiar with projective
geometry, so those and other mathematical details will be explained in greater detail than
they otherwise would be, with the intention that this paper be accessible to a wide range
of undergraduate students. The structure of this paper is as follows. In Section 2, we shall
derive the coupled set of nonlinear partial differential equations satisfied by all inertial frame
transformations, by all means a standard result. In Section 3, we shall obtain the most general
solution to these equations by decoupling the temporal and spatial transformation functions and
then solving them one at a time; it is here where the first decoupled equation for the temporal
transformation function will be seen to be the Schwarzian differential equation. In Section 4,
we shall detail some of the most important properties of the nonlinear inertial transformations,
both from the mathematics perspective and from the physics perspective. Lastly, in Section 5,
we shall prove that the inertiality constraints can be recast as a higher-dimensional Schwarzian
differential equation, introducing a rank-4 totally symmetric nonlinear differential operator
whose kernel consists of the projective linear transformations above.
2
Derivation of the Inertiality Constraints
Consider two inertial reference frames (t, ⃗x) and (t′ , ⃗x ′ ) whose coordinates are related to each
other by the transformation
t′ = f (t, ⃗x)
t = F (t′ , ⃗x ′ )
⃗ ′ , ⃗x ′ );
⃗x = G(t
′
⃗x = ⃗g (t, ⃗x)
(4)
(5)
by convention, let us say that each observer defines its coordinate system with itself forever at
the spatial origin, so that for instance the trajectory ⃗x(t) of the
primed observer as measured
by the unprimed observer is given by the solution to ⃗g t, ⃗x(t) = 0. In full generality, the only
condition placed on the above frame transformation so far is that it be invertible on some open
connected subset Ω ⊂ R × Rd−1 , as both (t, ⃗x) ∈ Ω and (t′ , ⃗x ′ ) ∈ Ω′ ≡ (f, ⃗g )(Ω) by definition
should provide unique coordinates to each spacetime point in the overlap of the domains of
validity of the respective observers. The fundamental task of interest to the relativity of point
particles is to transform any trajectory ⃗x(t) of a point particle measured in the (t, ⃗x) frame
to the analogous trajectory ⃗x ′ (t′ ) as measured in the (t′ , ⃗x ′ ) frame. This transformation is
effected by the fact that the new independent variable t′ is the time at which the particle
has coordinates (t′ , ⃗x ′ (t′ )) in the new frame corresponding to the original time coordinate t at
which this particle has coordinates (t, ⃗x(t)) in the original frame, which defines the function
t(t′ ) between the independent variables for this trajectory in the two frames. The physical
requirement that both t and t′ be valid time coordinates
is the condition
that this function
t(t′ ) also be invertible as a trajectory-dependent map t t, ⃗x(t) ∈ Ω → t′ t′ , ⃗x ′ (t′ ) ∈ Ω′ ,
namely
t′ (t) = f t, ⃗x(t)
⇐⇒
t(t′ ) = F t′ , ⃗x ′ (t′ ) .
(6)
3
Of course, this invertibility depends also on the trajectory ⃗x(t) under consideration, and so
the injectivity of t(t′ ) places restrictions on the allowed particle trajectories themselves if one
assumes that (t, ⃗x) 7→ (t′ , ⃗x ′ ) is a valid local frame transformation, providing one of the simplest
connections between kinematics and dynamics2 . Since we would like to construct the most
general possible inertial frame transformation, the only assumption we should make is that
sufficiently many linear trajectories with different velocities are allowed, to be clarified below.
As a result, that t′ (t) and t(t′ ) are true inverses here merely serves to eliminate the pathological
transformations (t, ⃗x) 7→ (t′ , ⃗x ′ ); for instance, the trivial trajectories ⃗x(t) = 0 and ⃗x ′ (t′ ) = 0
corresponding to the two observers always exist, which requires both f (t, 0) and F (t′ , 0) to be
invertible functions of their respective variables on their respective domains. We slightly expand
this condition by requiring that the constant trajectories ⃗x(t) = ⃗x0 and ⃗x ′ (t′ ) = ⃗x0′ are also
allowed for all fixed ⃗x0 and ⃗x0′ in the relevant domains, which imposes the invertibility of f (t, ⃗x)
and F (t′ , ⃗x ′ ) at constant ⃗x and ⃗x ′ , respectively.
Now, the transformation ⃗x ′ (t′ ) of any allowed trajectory ⃗x(t) can be obtained either as the
solution to
⃗x ′ (t′ ) = ⃗g F t′ , ⃗x ′ (t′ ) , ⃗x F t′ , ⃗x ′ (t′ )
,
(7)
or as the solution to
⃗x F t′ , ⃗x ′ (t′ )
⃗ t′ , ⃗x ′ (t′ ) ,
=G
(8)
which are equivalent due to the invertibility conditions above. While the latter equation is often
seen as more palatable from the point of view of computing the explicit form of the transformed
trajectory, it is the former equation which most easily lends itself to the computation of the
transformed velocity. From now on, we shall employ standard index notation and summation
conventions for spatial quantities, for instance writing the spatial coordinates ⃗x as xi where
i ∈ {1, 2, . . . , d − 1}. Furthermore, to derive the transformation of acceleration, we now assume
that the functions f (t, ⃗x) and each g i (t, ⃗x) are twice continuously differentiable, i.e. elements of
′
′
C 2 (Ω). In particular, the aforementioned
invertibility of t(t ) means that the map t 7→ t (t) is a
diffeomorphism on its domain t t, ⃗x(t) ∈ Ω . Then, directly differentiating (7) immediately
yields the familiar result3
∂g i
∂g i j
i
d h i ′
∂t + ∂xj v
′
= ∂f
v (t ) = dt′ (t)
g t(t ), ⃗x t(t )
,
dt(t′ )
+ ∂fk v k
′i
′
1
∂t
dt
2
(9)
∂x
For instance, this invertibility under arbitrary Lorentz boosts is the simplest way to derive that the speed
of any particle in special relativity must forever remain below the speed of light, forever remain at the speed
of light or forever remain above the speed of light, which does not rely on any dynamics whatsoever or energy
considerations.
3
It is a good exercise (in calculus and in index manipulation) for a student to derive this result the hard way
from (8) instead. From that equation, the transformed velocity is the solution to
∂Gi ′j
vi
∂Gi
v
=
−
.
∂f
∂x′j
∂t′
+ ∂fj v j
∂t
∂x
Then, one has to derive the spatial inverse
−1 i
1 ∂g i ∂f
∂g i
∂G
=
− ∂f
′
j
∂x
∂x
∂t ∂xj
j
∂t
and hence
∂G
∂x′
−1 i
∂Gj
1 ∂g i
=
−
,
∂f ∂t
′
j ∂t
∂t
after which (9) follows.
4
where we have suppressed most of the arguments for clarity but it is understood that each term
is a function of t′ alone, and one further derivative with respect to the sole independent variable
t′ yields the transformed acceleration. Specifically, the acceleration of the particle in the new
frame in terms of its acceleration and velocity in the old frame is
′i
a =
(
1
∂f
∂f m
∂t + ∂xm v
3
∂f ∂g i
∂f ∂g i
−
+
∂t ∂xj
∂xj ∂t
∂f ∂g i
∂f ∂g i
−
∂xj ∂xk
∂xk ∂xj
v k aj
∂f ∂ 2 g i ∂ 2 f ∂g i
∂f ∂ 2 g i
∂ 2 f ∂g i ∂ 2 f ∂g i
∂f ∂ 2 g i
+
− 2
+2
−2
vj
+
− 2
∂t ∂t2
∂t ∂t
∂xj ∂t2
∂t ∂t∂xj
∂t∂xj ∂t
∂t ∂xj
∂f ∂ 2 g i
∂ 2 f ∂g i
∂ 2 f ∂g i
∂f ∂ 2 g i
+
2
−
2
−
+
vj vk
∂t ∂xj ∂xk
∂xj ∂t∂xk
∂t∂xj ∂xk
∂xj ∂xk ∂t
)
∂f ∂ 2 g i
∂ 2 f ∂g i
+
−
v j v k v ℓ , (10)
∂xj ∂xk ∂xℓ ∂xj ∂xk ∂xℓ
where as usual all quantities are implicitly expressed as functions of t′ everywhere. Without
invoking any additional postulates, the only piece of information we know about two inertial
reference frames is that a vanishing acceleration in one frame implies a vanishing acceleration
in the other frame irrespective of velocity. That is, assuming both frames are inertial, ai = 0
must imply a′i = 0 for all allowed constant velocities v i . If the set of allowed constant velocities
has a continuous part, then this property is possible if and only if just the term linear in aj
survives; we shall assume either that such a continuous part exists or otherwise that this set is
discrete but contains sufficiently many non-collinear velocities so that the conditions below still
hold. Hence, we automatically know that the transformation of acceleration between inertial
frames must be given by
∂f ∂g i
∂f ∂g i
∂f ∂g i
∂f ∂g i
vk
−
+
−
∂t ∂xj
∂xj ∂t
∂xj ∂xk
∂xk ∂xj
′i
inertial transformation:
a =
aj .
(11)
3
∂f ℓ
∂f
∂t + ∂xℓ v
Inertiality requires that the sum of the other terms in the general acceleration transformation
∂f m
(10) vanish for all possible allowed constant velocities v i . Since the term ∂f
in the
∂t + ∂xm v
dt′ (t)
′
denominator is always strictly finite (because it equals dt and t (t) is necessarily differentiably
invertible), the constraints imposed by inertiality are the separate vanishing of the rank-1, -2,
-3 and -4 coefficients in the velocity expansion in (10). Therefore, the frame transformation
t′ = f (t, ⃗x) and x′i = g i (t, ⃗x) is inertial over its domain of validity if and only if the following
conditions hold:
∂f ∂ 2 g i ∂ 2 f ∂g i
− 2
∂t ∂t2
∂t ∂t
2
i
∂f ∂ 2 g i
∂ 2 f ∂g i ∂ 2 f ∂g i
! ∂f ∂ g
0=
+2
−2
− 2
j
2
j
∂x ∂t
∂t ∂t∂x
∂t∂xj ∂t
∂t ∂xj
2 i
∂f ∂ 2 g i
∂ 2 f ∂g i
∂ 2 f ∂g i
! ∂f ∂ g
0=
+
2
−
2
−
∂t ∂xj ∂xk
∂x(j ∂t∂xk)
∂t∂x(j ∂xk) ∂xj ∂xk ∂t
2
i
2
i
∂ g
∂ f ∂g
! ∂f
0=
− (j k ℓ) ,
(j
ℓ)
k
∂x ∂x ∂x
∂x ∂x ∂x
!
0=
(12)
(13)
(14)
(15)
where we use the notation (· · · ) to indicate total symmetrization in the enclosed indices with
5
unit weight, for example4 A(i B j) ≡ 12 (Ai B j + B j Ai ). The derivation of (12)-(15) is completely
standard and uninteresting; we include it in part for completeness but more importantly in
order to emphasize the precise assumptions made in this paper and how they enter. What is
not standard, and does not even seem to be readily accessible, is a rigorous derivation of the
most general solution to these constraint equations, which we provide below. It is admittedly
understandable that one may not wish to attempt to solve a system of coupled nonlinear partial
differential equations, especially when it is obvious that any f (t, ⃗x) and g i (t, ⃗x) consisting of
affine-linear functions in space and time yields a solution. Nevertheless, these equations are still
homogeneous and bilinear with constant coefficients, allowing us to decouple the equations and
solve them completely, which will yield the most general nonlinear solution.
3
Solving the Constraint Equations
We now proceed to solve the inertiality constraint equations (12)-(15) where f (t, ⃗x) and g i (t, ⃗x)
are now taken to be elements of C 3 (Ω) and not merely C 2 (Ω); the property of being thrice
continuously differentiable is what will enable us to decouple the equations.
On account of the physical condition that ∂f
∂t be nonzero everywhere on Ω, we first rewrite
the constraint equations (12)-(15) as
∂g i
∂ 2 g i ! ∂t ∂ 2 f
= ∂f
∂t2
∂t2
(16)
∂t
∂f ∂g i
∂f ∂g i 2
∂g i
2
! ∂t ∂xj − ∂xj ∂t ∂ f
∂t ∂ f
2
+
2
=
∂f ∂t∂xj
∂f 2
∂t∂xj
∂t2
∂t
∂t
∂ 2gi
∂f ∂g i
∂f ∂g i
∂g i
−
(j
(j ∂t
∂t
∂2f
∂2f
∂x
∂x
!
∂t
+
=
−
+
2
∂f ∂xj ∂xk
∂f 2
∂t2
∂xj ∂xk
∂t∂xk)
∂t
∂t
" ∂f ∂f
#
∂f
2
∂f ∂g i
∂f ∂g i
∂2f
1 ∂2f
!
∂xk ∂xℓ) ∂ f
∂xk
0=
− (j
− 2 ∂f 2
+ ∂f
.
k ∂xℓ)
∂f 3 ∂t2
∂t ∂x(j
∂x ∂t
∂t∂xℓ)
∂x
∂t
∂t
∂t
∂ 2gi
∂f
∂x(j
∂f ∂g i
∂f ∂g i
∂t ∂xk) − ∂xk) ∂t
∂f 3
∂t
(17)
∂2f
i
(18)
(19)
i
∂g
∂f ∂g
The physicality conditions on the frame transformation require that ∂f
∂t ∂xj − ∂xj ∂t be an
invertible matrix; denoting the spacetime Jacobian by
!
∂f
∂f
J≡
∂xj
∂g i
∂xj
∂t
∂g i
∂t
,
(20)
this invertibility follows from5
d−1
∂f ∂g i
∂f
∂f ∂g i
− j
=
det J.
det
j
∂t ∂x
∂x ∂t
∂t
4
(21)
The general rank-n index symmetrization is given by
i
1 X iσ(1) iσ(2)
(i
A1 1 Ai22 · · · Ainn ) ≡
A
A2
· · · Anσ(n) ,
n! σ∈S 1
n
where Sn is the group of permutations of the set {1, 2, . . . , n}.
5
It is worth noting that the nonvanishing of det J and of ∂f
does not by itself imply the invertibility of
∂t
i
i
∂g
∂g
the spatial Jacobian ∂x
j ; rather, it implies that the kernel of ∂xj is at most one-dimensional. However, the
i
∂g
and of ∂F
does indeed imply that ∂x
nonvanishing of det J, of ∂f
j is invertible, because
∂t
∂t′
∂g
det ∂x
∂F
= ∂f
j ,
∂g
∂t′
det
− ∂fi adj ∂g ij ∂g
∂t
∂x
∂x
6
∂x
∂t
Then, multiplying the final condition above by the inverse of this matrix yields
" ∂f ∂f
#
∂f
2
2f
2f
∂
1
∂
k
! i
∂xk ∂xℓ) ∂ f
∂x
− 2 ∂f
+ ∂f
,
0 = δ (j
2
∂f 3 ∂t2
∂t∂xℓ)
∂xk ∂xℓ)
∂t
∂t
(22)
∂t
and now contracting i with ℓ immediately says that the symmetric quantity in brackets identically vanishes, i.e.
∂f
∂f ∂f
2
2
∂2f
!
∂xj ∂xk ∂ f
∂x(j ∂ f
,
(23)
=
−
+
2
∂f ∂t∂xk)
∂f 2 ∂t2
∂xj ∂xk
∂t
∂t
on account of which the third condition above simplifies to
"
#
∂f
2f
2f
∂ 2 g i ! ∂g i
∂
2
∂
k)
− ∂x
.
+ ∂f
=
∂f 2 ∂t2
∂xj ∂xk
∂x(j
∂t∂xk)
(24)
∂t
∂t
In summary, so far the inertiality constraint equations are (16), (17), (24) and (23), which
require the differentiability class of all functions involved to be C 2 (Ω).
3.1
Isolating f and the Schwarzian Differential Equation
It is now simple to decouple the equations for f (t, ⃗x) via the Schwarz integrability conditions
of the commutativity of the third-order partial derivatives of f (t, ⃗x) and g i (t, ⃗x), which is why
we now need to restrict to functions of class C 3 (Ω).
Equating the spatial derivative of (16) with the temporal derivative of (17) results in
"
2 i
2 i 2 #
∂f ∂g i
∂f ∂g i
∂
∂ g
∂
∂ g
1 ∂t ∂xj − ∂xj ∂t ∂ 3 f
3 1 ∂2f
!
0=
−
=−
− ∂f
,
(25)
∂f 2
∂xj ∂t2
∂t ∂t∂xj
2
∂t3
2
∂t2
∂t
∂t
which on account of the fact that the prefactor here cannot vanish as a matrix at any point in
the domain requires that the term in brackets vanish. On account of the vanishing of this term,
equating the spatial derivative of (17) with the temporal derivative of (24) then yields
2 i 2 i ∂ g
∂
∂ g
∂
!
−
= Aijk + 2Aikj ,
(26)
0=
j
k
∂t∂x
∂t ∂xj ∂xk
∂x
where
i
i
∂f ∂g
∂f ∂g
1
j −
j ∂t
A jk ≡ − ∂t ∂x ∂f ∂x
2
2
i
∂t
"
∂3f
1
+
2
k
2
∂t ∂x
2 2
∂f
∂ f
2 ∂2f ∂2f
∂xk
−
∂f ∂t2 ∂t∂xk
∂f 2
∂t2
∂t
∂t
#
.
(27)
Of course, applying the condition above a second time but with j and k swapped results in
Aijk = −2Aikj = 4Aijk and hence Aijk = 0; as before, the prefactor in Aijk cannot vanish as a
matrix anywhere and thus the term in brackets is forced to vanish. Lastly, on account of the
3
3f
above expressions for ∂∂t3f and for ∂t∂2 ∂x
i , the spatial derivative of (24) reads
∂
∂xℓ
∂ 2gi
∂xj ∂xk
#"
"
#
∂f
∂f
2f
2f
2f
2f
3 ∂g i
∂
2
∂
∂
2
∂
ℓ)
k
∂x
=
− ∂f
− ∂x
+ ∂f
,
2 2 + ∂f
∂f 2 ∂t2
2 ∂x(j
∂t
∂t∂xk
∂t∂xℓ)
∂t
∂t
∂t
(28)
∂t
∂g i
∂g i
where adj ∂x
j is the adjugate of the matrix ∂xj . As such, we might as well say that the physicality conditions
∂g
consist of the nonvanishing of det J, of ∂f
and of det ∂x
.
∂t
7
which is already totally symmetric in j, k and ℓ, and similarly the spatial derivative of (23) is
found to be
2
∂f ∂f
∂f
∂f ∂f ∂f 2
2
3 ∂xi ∂xj ∂xk ∂ 2 f
∂3f
∂2f ∂2f
∂x(i ∂xj ∂ f ∂ f
∂x(i
=
−
6
+
6
,
(29)
∂f 4
∂f 3 ∂t2 ∂t∂xk)
∂f 2 ∂t∂xj ∂t∂xk)
2
∂t2
∂xi ∂xj ∂xk
∂t
∂t
∂t
which is again totally symmetric automatically.
Therefore, we have successfully isolated the function f (t, ⃗x) by showing that it obeys the
constraint equations
2
∂3f ! 3 1 ∂2f
= ∂f
(30)
∂t3
2
∂t2
∂t
2 2
∂f
∂ 3 f ! 1 ∂xi
∂ f
2 ∂2f ∂2f
=
−
+
(31)
∂f ∂t2 ∂t∂xi
∂t2 ∂xi
2 ∂f 2 ∂t2
∂t
∂t
∂f
∂f
∂f
2
∂2f
∂2f !
(i
i
j ∂ f
.
= − ∂x∂f∂x
+ 2 ∂x
2
i
j
2
∂f
j)
∂x ∂x
∂t
∂t∂x
∂t
∂t
(32)
The remaining constraint equations (16), (17) and (24) express the second-order derivatives of
g i (t, ⃗x) in terms of its first-order derivatives as well as f (t, ⃗x); hence, there are no constraints
on f (t, ⃗x) beyond (30), (31) and (32) because the latter set of equations imposes the Schwarz
integrability conditions of the former set of equations, which guarantees that a local solution
for g i (t, ⃗x) exists for any solution of f (t, ⃗x). As such, we now proceed to find the most general
solution f (t, ⃗x) to the three equations (30), (31) and (32).
The first of the isolated constraint equations we obtained above, (30), happens to be a
particularly famous one known as the Schwarzian differential equation. Consider for the moment
a function f (z) of a single real or complex variable z whose domain Ω is an open subset of either
R or C, and let f : Ω → C be of class C 3 (Ω) (which is equivalent to f being holomorphic on Ω
in the case that Ω ⊂ C); for this discussion only we let f ′ (z) denote the derivative of f (z) with
respect to z. Then, the Schwarzian derivative of f with respect to z is defined as [3]6
f ′′′ (z) 3 f ′′ (z) 2
Sz f (z) ≡ ′
−
.
(33)
f (z)
2 f ′ (z)
This nonlinear differential operator is familiar in mathematics for its role in complex analysis and
in projective geometry [4], and it is famous in physics for its appearances in two-dimensional
conformal field theory [5] and in the low-energy limit of Jackiw-Teitelboim gravity [6]. The
(homogeneous) Schwarzian differential equation is then
Sz f (z) = 0.
(34)
For instance, it is a standard mathematical result7 that the automorphism group of the Riemann
sphere, PSL(2, C), is the kernel of the Schwarzian derivative and conversely that the Schwarzian
derivative is the sole generator of the annihilator of PSL(2, C) [4], explaining its importance to
two-dimensional conformal field theory (and to projective geometry, since the Riemann sphere
C is equivalent to the complex projective line CP1 ). Moreover, the solutions to the Schwarzian
6
For historical reasons, the Schwarzian derivative Sz f is often denoted {f, z} instead; this is the notation
introduced by Cayley in [3], where he named the derivative after Schwarz.
7
The Schwarzian derivative is, more properly speaking, a (holomorphic) quadratic differential [4], and it is
for this reason that it plays a key role in the theory of the moduli space of Riemann surfaces (i.e. Teichmüller
theory) [7].
8
differential equation (34) are well-known; as long as the domain Ω is open and connected, the
most general solution f : Ω → C of class C 3 (Ω) to (34) is8
f (z) =
α + βz
,
γ + δz
(35)
α β
then provides
γ δ
a homomorphism with GL(2, C), which after quotienting by the C∗ which acts by multiplication
yields an isomorphism with9 PGL(2, C) ∼
= PSL(2, C). When describing (35) as the most general
automorphism of the Riemann sphere, it is understood that z is a stereographic coordinate and
that z = − γδ maps to the point at infinity; if one does not wish to include a point at infinity,
then one naturally excludes z = − γδ from the domain Ω. A function f (z) of the form (35)
is commonly called either a projective linear transformation or a Möbius transformation, with
PSL(2, C) being the Möbius group.
In our problem, the function f (t, ⃗x) has d independent (real) variables, so we define its
Schwarzian derivative with respect to each of them by replacing the ordinary derivatives in (33)
with the appropriate partial derivatives. With this notation, the third-derivative constraint (30)
at once reads
!
St f (t, ⃗x) = 0,
(36)
where α, β, γ, δ ∈ C are constants such that αδ − βγ ̸= 0. The map f 7→
which immediately determines the time dependence of f (t, ⃗x) to be
f (t, ⃗x) =
b1 (⃗x) + b2 (⃗x)t
,
b3 (⃗x) + b4 (⃗x)t
(37)
where b1 (⃗x), b2 (⃗x), b3 (⃗x) and b4 (⃗x) are purely-spatial functions satisfying b1 b4 − b2 b3 ̸= 0
at all points in the domain. In fact, (29) together with (32) implies as a special case that
f (t, ⃗x) also obeys the Schwarzian differential equation in each spatial coordinate xi separately,
!
i.e. Sxi f (t, ⃗x) = 0, but it does not behoove us much to use this fact at the moment; we shall
return to a more powerful statement along these lines later.
Next we must impose the remaining constraint equations, (31) and (32), on the function
f (t, ⃗x) given in (37); we shall primarily leave the spatial arguments of the functions bα implicit
x)
α (⃗
and use the standard index notation ∂i bα to represent the derivative ∂b∂x
i . From (37) we
straightforwardly compute
1
∂3f
+
∂t2 ∂xi 2
8
2 2
∂f
2 ∂2f ∂2f
2(b4 ∂i b2 − b2 ∂i b4 )
∂ f
∂xi
−
=
,
2
2
∂f
∂f
∂t
∂t2 ∂t∂xi
(b3 + b4 t)2
∂t
∂t
(38)
(z)
The general solution is almost trivial to obtain; as long as dfdz
̸= 0, one simply needs to note that


r
df (z) d2  1 
q
Sz f (z) = −2
.
dz dz 2
df (z)
dz
Indeed, it is this presentation of the Schwarzian derivative which appeared first historically, used not by Schwarz
(z)
but rather nearly a century earlier by Lagrange [8]. The assumption that dfdz
̸= 0 is then lifted by noting that
either f (z) is constant (which is precisely the function (35) in the degenerate case αδ − βγ = 0) or there exists
(z)
a neighborhood in which dfdz
does not vanish and hence the solution is (35), after which continuity implies the
df (z)
nonvanishing of dz everywhere in the domain.
9
The complex projective linear groups can be written either way, since PGL(n, C) ∼
= PSL(n, C) for all n;
however, for real projective linear groups, PGL(n, R) is isomorphic to PSL(n, R) if and only if n is odd, otherwise
the former is a cover of the latter.
9
so that the constraint (31) is equivalent to
!
b4 ∂i b2 − b2 ∂i b4 = 0.
(39)
If b4 ̸= 0, then this condition clearly just says that bb42 is a constant.
Continuing with the case b4 ̸= 0, and using the constancy of bb24 , a slightly more tedious
computation then yields
∂f
b1
b3
∂f ∂f
2f
2f
b
b
b
∂
∂
−
b
∂
∂
3
4
3
i
j
1
i
j
∂2f
∂
∂
(i
i
j
b
b
4
4
=
+ ∂x∂f∂x
− 2 ∂x
2
∂f ∂t∂xj)
∂xi ∂xj
∂t2
(b3 + b4 t)3
∂t
∂t
b1
b4 2b3 b4 ∂i ∂j b4 − (b1 b4 + b2 b3 )∂i ∂j bb34 t b24 b4 ∂i ∂j bb14 − b2 ∂i ∂j bb34 t2
+
+
, (40)
(b3 + b4 t)3
(b3 + b4 t)3
so that the constraint (32) imposes the conditions
b1
b3
!
− b 1 ∂i ∂j
=0
b3 b3 ∂i ∂j
b4
b4
b1
b3 !
2b3 b4 ∂i ∂j
− (b1 b4 + b2 b3 )∂i ∂j
=0
b4
b4
b1
b3 !
b4 ∂i ∂j
− b 2 ∂i ∂j
= 0.
b4
b4
(41)
(42)
(43)
Given the nonvanishing of b4 and of b1 b4 − b2 b3 , substituting the third condition into the second
and then back into the third again shows that these conditions are equivalent to
b1 !
∂i ∂j
=0
(44)
b4
b3 !
∂i ∂j
= 0.
(45)
b4
Thus, we have determined that the most general solution to the integrability conditions (30),
(31) and (32) in the case b4 ̸= 0 is given by (37) in which bb24 is constant and both bb41 and bb34 are
affine linear in ⃗x. Of course, since in this case we may divide both numerator and denominator
in (37) by b4 , this solution is equivalent to both b2 and b4 being constant and both b1 and b3
being affine linear in ⃗x.
Lastly we revisit the other original possibility that b4 = 0. In this case we simply rewrite
f (t, ⃗x) = c1 (⃗x) + c2 (⃗x)t for purely-spatial functions c1 and c2 ̸= 0. The constraint (31) is then
trivially satisfied, and we have
∂f
∂f ∂f
2
2
∂2f
2
2
∂xi ∂xj ∂ f
∂x(i ∂ f
+
−
2
=
∂
∂
c
−
∂
c
∂
c
+
∂
∂
c
−
∂
c
∂
c
i j 1
1
2
i j 2
i 2 j 2 t
∂f ∂t∂xj)
∂f 2 ∂t2
∂xi ∂xj
c2 (i j)
c2
∂t
∂t
c1
1
1
2
= c2 ∂i ∂j
− c 1 ∂i ∂j
− c 2 ∂i ∂j
t, (46)
c2
c2
c2
so that the constraint (32) now imposes
1
!
=0
c2
c1 !
∂i ∂j
= 0.
c2
∂i ∂j
10
(47)
(48)
Thus, c12 and cc12 are both affine linear in ⃗x, and hence even in the case b4 = 0 the temporal
2t
transformation function can still be written as f = c1 + c2 t = b1 +b
b3 , where b2 is constant and
both b1 and b3 are affine linear in ⃗x.
Therefore, we have shown that the most general thrice continuously-differentiable solution
to the integrability conditions (30), (31) and (32) is
f (t, ⃗x) =
b1 (⃗x) + b2 t
,
b3 (⃗x) + b4 t
where
∂i b2 = ∂i b4 = 0
and
∂i ∂j b1 = ∂i ∂j b3 = 0,
(49)
subject to the physicality condition b4 b1 (⃗x)−b2 b3 (⃗x) ̸= 0. That is, f (t, ⃗x) is necessarily projective
linear in spacetime.
3.2
Solving for g i
Having obtained the general solution for f (t, ⃗x) in (49), it is now simple to solve the remaining
constraints (16), (17) and (24) for the spatial functions g i (t, ⃗x). First, we note that the constraint
(16) is equivalently written as
!
∂g i
∂
!
∂t
= 0,
(50)
∂f
∂t
∂t
which says that there exist purely-spatial functions β i (⃗x) and di (⃗x) such that
g i (t, ⃗x) = β i (⃗x)f (t, ⃗x) + di (⃗x).
(51)
With this expression, the remaining two constraint equations, (17) and (24), become
b4 (b3 + b4 t)
f ∂j β i + ∂j di
b1 b4 − b2 b3
i
i + ∂ di ∂ b
2∂
β
(∂
b
−
f
∂
b
)
2
f
∂
β
1
3
(j
k)
k)
(j
(j
k) 3
!
f ∂j ∂k β i +
+ ∂j ∂k di = −
,
b3 + b4 t
b3 + b4 t
!
∂j β i =
(52)
(53)
which simplify to
!
0 = ∂j b2 β i + b4 di
(54)
!
0 = ∂j ∂k b1 β i + b3 di + ∂j ∂k b2 β i + b4 di t.
i
i
i
(55)
i
2 β +b4 d )t
We have previously parametrized g i = b1 β +b3 db3+(b
due to the first constraint equation
+b4 t
(16), and now the final two constraint equations (17) and (24) simply say that the numerator
coefficients bi5 (⃗x) ≡ b1 (⃗x)β i (⃗x) + b3 (⃗x)di (⃗x) and bi6 ≡ b2 β i (⃗x) + b4 di (⃗x) must satisfy
!
∂j bi6 = 0
!
∂j ∂k bi5 (⃗x) = 0.
(56)
(57)
Since these are the only conditions placed on the spatial functions β i (⃗x) and di (⃗x), it is clear
that we may choose them so as to make bi5 (⃗x) any affine-linear function and bi6 any constant.
In conclusion, we have exhaustively proven that locally the most general possible inertial
transformation of class C 3 (Ω) on an open connected domain Ω ⊂ R × Rd−1 is t′ = f (t, ⃗x) and
x′i = g i (t, ⃗x) given by
b1 (⃗x) + b2 t
b3 (⃗x) + b4 t
bi (⃗x) + bi6 t
g i (t, ⃗x) = 5
,
b3 (⃗x) + b4 t
f (t, ⃗x) =
11
(58)
(59)
where b2 , b4 and each bi6 are constants and b1 (⃗x), b3 (⃗x) and each bi5 (⃗x) are affine-linear spatial
functions, all independent apart from having to satisfy the physicality conditions that det J, ∂f
∂t
∂g and det ∂x
must all be nonvanishing everywhere in the domain of the transformation.
4
Properties and Significance of the Nonlinear Transformations
We have now proven that the most general thrice continuously-differentiable inertial transformation (t, xi ) 7→ (t′ , x′i ) in some open connected domain is given by
α + βi xi + γt
A + Bj xj + Ct
(60)
δ i + ϵ ij x j + ζ i t
x =
,
A + Bk xk + Ct
(61)
t′ =
′i
where all of A, Bi , C, α, βi , γ, δ i , ϵij and ζ i are now constants, which is subject to three minor
conditions in order to furnish a bona fide transformation between reference frames. The first
physicality condition is that
Aγ − Cα − (Cβi − γBi )xi
∂t′
=
̸= 0.
∂t
(A + Bj xj + Ct)2
(62)
The second physicality condition is then that
det
∂x′
∂x
=
(A + Ct) det ϵ − Bi (adj ϵ)ij (δ j + ζ j t)
̸= 0,
(A + Bi xi + Ct)d+1
(63)
where (adj ϵ)ij is the adjugate matrix of ϵij ; in full generality, we should express such quantities
in terms of the adjugate matrix instead of an inverse (ϵ−1 )ij because the matrix ϵij is not
required to be invertible (though it is obviously necessary, but not sufficient, for the adjugate
to be a nonzero matrix10 ). Nevertheless, the final invertibility condition is most easily written
in the case that ϵij actually is invertible11 , in which case it reads
[A − Bi (ϵ−1 )ij δ j ][γ − βk (ϵ−1 )kℓ ζ ℓ ] − [C − Bi (ϵ−1 )ij ζ j ][α − βk (ϵ−1 )kℓ δ ℓ ]
det J =
det ϵ ̸= 0.
(A + Bm xm + Ct)d+2
(64)
Of these physicality conditions, it is only the nonvanishing of this det J that truly constrains
the actual coefficients in the transformation; the roles of (62) and of (63), on the other hand,
are to restrict the domain of (t, xi ) so that their numerators never vanish. Likewise, if Bi and
C do not both vanish, then the hyperplane A + Bi xi + Ct = 0 is also supposed to be excluded
from the domain of the transformation.
4.1
Mathematical Structure
The general inertial map defined by (60) and (61) subject to (64) is a projective linear transformation in the d-dimensional space R × Rd−1 . In any framework of relativity, the set of
10
As a special case, the adjugate of any matrix in one dimension is the identity.
If det ϵ = 0 but ϵij + v i wj is invertible for fixed vector v i and one-form wj , then the inverse of ϵij + v i wj will
depend on the Jordan normal form of ϵij and so does not have an explicit general expression.
11
12
reference frame transformations always forms a group under composition. Here, the composition of the inertial transformation (t, xi ) 7→ (t′ , x′i ) defined by (60) and (61) with the inertial
transformation (t′ , x′i ) 7→ (t′′ , x′′i ) defined by
t′′ =
α′ + βi′ x′i + γ ′ t′
A′ + Bj′ x′j + C ′ t′
x′′i =
and
δ ′i + ϵ′i j x′j + ζ ′i t′
(65)
A′ + Bk′ x′k + C ′ t′
is the inertial transformation (t, xi ) 7→ (t′′ , x′′i ) defined by
t′′ =
α′′ + βi′′ xi + γ ′′ t
A′′ + Bj′′ xj + C ′′ t
and
x′′i =
δ ′′i + ϵ′′ij xj + ζ ′′i t
A′′ + Bk′′ xk + C ′′ t
,
(66)
where the parameters in this new transformation are
α′′ = α′ A + γ ′ α + βi′ δ i
(67)
βi′′ = α′ Bi + γ ′ βi + βj′ ϵj i
γ ′′ = α′ C + γ ′ γ + βi′ ζ i
A′′ = A′ A + C ′ α + Bk′ δ k
Bi′′ = A′ Bi + C ′ βi + Bj′ ϵj i
C ′′ = A′ C + C ′ γ + Bk′ ζ k
δ ′′i = δ ′i A + ζ ′i α + ϵ′i j δ j
ϵ′′ij = δ ′i Bj + ζ ′i βj + ϵ′i k ϵkj
ζ ′′i = δ ′i C + ζ ′i γ + ϵ′i j ζ j .
(68)
(69)
(70)
(71)
(72)
(73)
(74)
(75)
To the inertial transformation (60) and (61) we may associate a (d + 1)-dimensional matrix via


A C Bj
(t, xi ) 7→ (t′ , x′i ) 7−→  α γ βj  ,
(76)
δ i ζ i ϵij
which on account of the multiplication law above provides a homomorphism with GL(d + 1, R),
since the determinant of this (d + 1)-dimensional matrix is equal to the expression multiplying
1
in det J given in (64) and hence does not vanish. Of course, the only ambiguity
(A+Bm xm +Ct)d+2
in (60) and (61) is that all the parameters may be simultaneously scaled by the same nonzero
constant and still define the same inertial transformation, and quotienting by this multiplicative
R∗ provides an isomorphism of the general inertial transformation group with the projective
group PGL(d + 1, R) of dimension dim PGL(d + 1, R) = d(d + 2). Of course, the matrix in
(76) is in one-to-one correspondence with the automorphisms of a (d + 1)-dimensional vector
space R × R × Rd−1 , and quotienting this vector space (minus the origin and adding the points
at infinity) by the overall multiplicative R∗ means that the transformation (60) and (61) is
simply the most general automorphism of the projective space (R × R × Rd−1 )∗ /R∗ ∼
= RPd
[9]. Therefore, inertial frame transformations can always be identified with automorphisms of
spacetime by giving the latter the structure of the projective space RPd ; this is the central
result of this paper. It must be emphasized that viewing the general inertial transformation as
a projective transformation does not mean that we should actually identify spacetime with a
projective space itself; indeed, we certainly should not do so if we would like to have any notion
of causality. Instead, we do not add the points at infinity, and the truly nonlinear inertial
transformations are simply not global.
13
We have now proven that the unrestricted inertial transformation group on R × Rd−1 is
PGL(d + 1, R), but of course a specific framework of relativity based on inertial transformations
is supposed to select a subgroup of this PGL(d+1, R) corresponding to the actual symmetries of
spacetime. Specifically, the physical transformation between two inertial observers is supposed
to be determined by the trajectory of one observer relative to the other together with the relation
between their spatial frame at a fixed time. The most general inertial transformation (60) and
(61) also shows that the most general inertial observers necessarily travel at constant velocities
with respect to each other, a familiar foundational statement which nevertheless is usually just
assumed to be true; we have proven that its assumption is unnecessary. The independent types
of inertial transformations are thus always given by translations, boosts and constant spatial
frame transformations (almost always taken to be rotations, though this is not a requirement if
one does not wish to impose the cosmological principle). It is highly convenient subsequently to
view the selected subgroup of allowed inertial transformations as those automorphisms which
preserve an additional structure on the vector (or projective) space. For vector spaces, this extra
structure is most frequently taken to be an inner product; for instance, the famous example
of special relativity is built upon the equivalence of the Poincaré transformations with the
isometries of the Minkowski metric. Projective spaces, on the other hand, do not admit inner
products, but they do still admit metrics such as Cayley-Klein metrics [10] which may be used
for the same purpose.
If Bi and C vanish identically, then the inertial transformations (60) and (61) reduce to
the exceedingly familiar affine linear ones, corresponding to the subgroup of PGL(d + 1, R)
given by GL(d, R) ⋉ Rd of dimension dim GL(d, R) ⋉ Rd = d(d + 1). In the projective space
RPd , the generic GL(d, R) transformation fixes both an origin and the points at infinity, the
generic GL(d, R) ⋉ Rd transformation fixes precisely the points at infinity whereas the generic
PGL(d + 1, R) transformation does not fix any of the points at infinity. As such, the vector
space structure of spacetime endowed by a framework of relativity restricted to affine linear
inertial transformations can be embedded into a projective space simply by demanding that the
points at infinity are invariant under the inertial transformations. In fact, it was Felix Klein
who first pointed out that the Lorentz transformations of special relativity can be viewed in this
way as the isometries of a Cayley-Klein metric in projective space [11] (in which the invariant
quadric which defines the Cayley-Klein metric is essentially the lightcone). On the one hand,
the existence of the projective inertial transformations (60) and (61) is mathematically obvious
if one assumes a vector space structure of spacetime — specifically, if one assumes that the
spacetime R × Rd−1 can be viewed as an actual d-dimensional vector space (and not just the
Cartesian product of the temporal line R and the spatial vector space Rd−1 as sets), then zeroacceleration trajectories correspond to straight lines in this vector space, and the fundamental
theorem of projective geometry states that the most general map on a vector space which sends
lines to lines is a projective linear transformation12 [9]. Hence, the existence of projective linear
inertial transformations is neither surprising nor new. What does appear to be new, however,
is the proof that the most general nonlinear inertial transformation is in fact projective linear,
which automatically endows spacetime in any inertial theory of relativity with a vector space
structure, albeit in one higher dimension.
12
Strictly speaking, the most general map sending lines to lines is called a “collineation”, which in general
includes more than just the projective linear transformations. However, the quotient of the collineation group by
the projective linear group is related to field automorphisms, and so for real projective spaces all collineations
are projective linear transformations (whereas for complex projective spaces of dimension n ⩾ 2, the collineation
group is instead PGL(n + 1, C) ⋊ Z2 , where the Z2 refers to complex conjugation).
14
4.2
Physical Structure
The truly nonlinear inertial transformations, in which Bi and C are not both zero, have rather
bizarre properties compared to their affine-linear cousins, both on account of the nonlinearity
itself and of the non-global nature of the transformations. Here we record only the most basic
physical properties of (60) and (61).
We already know that the validity of the transformation begins or ends at a fixed time if
Bi = 0 or is restricted to one side of a spatial hyperplane whose normal direction is B i , and this
limiting spatial hyperplane further travels at constant velocity parallel to B i if C ̸= 0. The valid
spatial coordinates are further restricted by the first physicality condition (62), and the valid
time coordinates are further restricted by the second physicality condition (63). The presence of
these singular points of the transformations introduces (non-causal) horizons, and implies either
that the spacetime does not admit any global observers or that there exist preferred inertial
frames. These possibilities are familiar in other examples, though ones in which space is not
Cartesian: the non-existence of global observers is typical in black-hole spacetimes13 , and the
existence of preferred inertial frames often occurs in topologically-nontrivial flat spacetimes or in
æther theories. It should be noted that the limiting spatial hyperplane A+Bi xi +Ct = 0, which
would map to the points at infinity, can be excluded from the domain of the transformation
automatically by hiding it behind the spatial hyperplane and the time slice for which the
physicality conditions (62) and (63) are violated.
Let us now consider the general inertial transformation between two observers in relative
motion, for which we may set α = 0 and δ i = 0 in (60) and (61) without loss of generality.
Note that setting α = 0 and δ i = 0 does not in any way assume that translations are part
of the spacetime symmetry group of the relativity framework; rather, the trajectories of two
inertial observers in relative motion intersect at a unique spacetime point, and we simply define
the spacetime coordinates of the two observers such that this unique intersection occurs at
(t, xi ) = (t′ , x′i ) = (0, 0). Then, that the transformation be valid in a neighborhood of their
shared spacetime origins requires that A ̸= 0, which we may thus scale away and set A = 1.
Subsequently, the physicality conditions (62) and (63) applied at the spacetime origin here
require that γ ̸= 0 and that ϵij is in fact invertible. Lastly, let us call v i the velocity of the primed
observer as measured in the unprimed frame, so that ζ i = −ϵij v j . Thus, the transformation
under consideration here is of the form
t′ =
βi xi + γt
1 + Bj xj + Ct
and
x′i =
ϵij (xj − v j t)
,
1 + Bk xk + Ct
(77)
where now the invertibility condition (64) requires also that γ ̸= −βi v i . The inverse transformation is then given by
t=
βi′ x′i + γ ′ t′
1 + Bj′ x′j + C ′ t′
and
13
xi =
ϵ′ij x′j + ζ ′i t
,
1 + Bk′ x′k + C ′ t′
(78)
It should be noted that by “global observer” here we mean a single global observer, i.e. one observer whose
worldline intersects the causal past and the causal future of every point in spacetime.
15
where
βj (ϵ−1 )j i
βi′ = −
(79)
γ + βk v k
1
γ′ =
γ + βi v i
(80)
Bi′ = −Bj (ϵ−1 )j i +
C′ = −
C + Bi v i
γ + βj v j
′i
−1 i
ϵ j = (ϵ
ζ ′i =
)j−
C + Bk v k
βj (ϵ−1 )j i
ℓ
γ + βℓ v
(81)
(82)
v i βk (ϵ−1 )kj
(83)
γ + βℓ v ℓ
vi
.
γ + βk v k
(84)
The linear trajectory xi (t) = xi0 + v0i t in the unprimed frame becomes the linear trajectory
x′i (t′ ) = x′i0 + v0′i t′ in the primed frame, where the velocity transformation formula is
v0′i =
ϵij [(1 + Bk xk0 )(v0j − v j ) − (C + Bk v0k )xj0 ]
(85)
ℓ
m
(γ + βℓ v0ℓ )(1 + Bm xm
0 ) − (C + Bℓ v0 )βm x0
The nonlinearity of the transformation manifests itself in the dependence of the velocity transformation formula on the position of the particle at the time of intersection of the two observers.
Of course, this nonlinearity is completely invisible for all constant-velocity trajectories passing
ϵi (v j −v j )
j 0
through the shared spacetime origin, for which the velocity transformation v0′i = γ+β
coink
k v0
cides with that for linear inertial transformations. Moreover, since the velocity of the unprimed
ϵi v j
observer as measured in the primed frame is v ′i = − jγ , one often wishes to impose that
ϵij v j = γv i , which is always possible by suitably choosing the orientation and scaling of the
primed spatial coordinate axes; nevertheless, there is no reason to impose Einstein’s reciprocity
postulate in general, especially if one wishes to consider theories which are not symmetric under
spatial parity.
It is now simple to see that it is not possible to preserve the speed of light universally under
the above generalized boost unless Bi and C both vanish, i.e. constancy of the speed of light
automatically forces the inertial transformation to be linear. It suffices to show that the norm
of (ϵ−1 )ij v0′i cannot be made independent of xi0 except when Bi = 0 and C = 0. For the speed
of light to be preserved, the norm-squared of (1 + Bk xk0 )(v0j − v j ) − (C + Bk v0k )xj0 would have to
ℓ
m
be proportional to the square of (γ + βℓ v0ℓ )(1 + Bm xm
0 ) − (C + Bℓ v0 )βm x0 , where the constant
i
i
of proportionality cannot depend on x0 or on the direction of v0 . Identifying the coefficients of
the terms quadratic in xi0 results in the condition
|⃗v0 − ⃗v |2 Bi Bj − 2(C + Bk v0k )(v0,(i − v(i )Bj) + (C + Bk v0k )2 δij
!
2
= ϵ−1 v0′ [(γ + βk v0k )Bi − (C + Bk v0k )βi ][(γ + βℓ v0ℓ )Bj − (C + Bℓ v0ℓ )βj ]. (86)
If Bi and C are not both zero, then this equality would require the matrix δij + w(i Bj) , where
|⃗v0 −⃗v |2
2
Bi , to have rank 1 for different values of wi (corresponding
j (v0,i − vi ) +
C+Bj v0
(C+Bj v0j )2
to different directions of v0i ) but the same Bi , which is clearly impossible if there are at least
wi ≡ −
16
two spatial dimensions. That leaves just the case of one spatial dimension to consider, in which
all the coefficients are simply numbers, and the condition of proportionality reads
2 !
(v0 − v) =
!
−(vB + C)(v0 − v) =
2 !
(vB + C) =
1 ′
v
ϵ 0
2
1 ′
v
ϵ 0
2
1 ′
v
ϵ 0
2
(γ + βv0 )2
(87)
(γ + βv0 )(Bγ − Cβ)
(88)
(Bγ − Cβ)2 ;
(89)
however, equality for both signs of v0 = ±c would require γ = −vβ, which is the value forbidden
by invertibility. Therefore, it is possible to preserve the speed of light in all directions from all
starting points only if Bi = 0 and C = 0, irrespective of dimension; the rest of Einstein’s derivation then proceeds normally, with the assumption of linearity (or, equivalently, homogeneity)
now being eliminated.
Of course, one is free to consider Lorentz-violating theories of relativity [12], and one is wellmotivated to do so because special relativity is hardly guaranteed to be valid even locally at the
highest energies. The nonlinear transformations considered in this paper provide the complete
set of ways to break Lorentz symmetry while still preserving the Law of Inertia. In particular,
the natural vector space structure which appears then allows one to construct locally Lorentz
violating generalizations of general relativity by taking the equivalence principle to mean the
existence of the corresponding locally inertial frames in the same way as traditional general
relativity is constructed from local Lorentz frames. It is not the purpose of this paper to try
to classify the different nonlinear inertial transformation subgroups that would give rise to
viable Lorentz-violating but inertial theories of relativity, especially since it is not clear which
if any are physically well-motivated. Nevertheless, as pedagogical exercises, such explorations
are well-suited to beginning undergraduate researchers, as mostly what is required is algebraic
manipulation and a soupçon of ingenuity.
5
The Higher-Dimensional Schwarzian Derivative
Previously, we showed that the most general solution for f (t, ⃗x) satisfying the nonlinear differential equations (30), (31) and (32) is the projective linear transformation (49). Here, we show
that these nonlinear differential equations are equivalent to a suitable higher-dimensional generalization of the Schwarzian differential equation, which simultaneously shows that the general
solution of the latter is a projective linear transformation.
First, we note that (29) and (32) immediately imply
∂f
∂3f
3 ∂2f
∂2f
=
.
2 ∂x(i ∂xj ∂xk ∂xℓ)
∂x(i ∂xj ∂xk ∂xℓ)
(90)
The original equation (30) said that St f (t, ⃗x) = 0, and now this equation in the special case
i = j = k = ℓ says that Sxi f (t, ⃗x) = 0 for each separate spatial coordinate xi . The rank-4
spatial equation here is of course much stronger, and we now generalize it to all combinations
of spacetime coordinates as well.
17
Still from (29) and (32), we also compute the combinations
1
4
∂f
∂3f
∂f
∂3f
+
3
∂t ∂xi ∂xj ∂xk
∂x(i ∂t∂xj ∂xk)
!
∂f ∂f
∂f
2f ∂2f
2f
2f
3
∂
∂
∂
j
(i
(i
− ∂x∂f ∂x
=
+2 ∂x
2
∂f ∂t∂xj ∂t∂xk)
2
∂t2 ∂t∂xk)
∂t
∂t
∂2f
∂2f
3
(91)
2 ∂t∂x(i ∂xj ∂xk)
2
∂f
∂f ∂f 2
2
1
1 ∂xi ∂xj ∂ 2 f
∂2f ∂2f
∂f
∂3f
∂f ∂ 3 f
∂x(i ∂ f ∂ f
=
−
+
3
+
3
+
∂f ∂t2 ∂t∂xj)
6
∂t ∂t∂xi ∂xj
2 ∂f 2 ∂t2
∂t∂xi ∂t∂xj
∂x(i ∂t2 ∂xj)
∂t
∂t
∂2f ∂2f
3 1 ∂2f ∂2f
+2
(92)
=
2 3 ∂t2 ∂xi ∂xj
∂t∂xi ∂t∂xj
1
∂f ∂ 3 f
∂f ∂ 3 f
3 ∂2f ∂2f
3
+
=
.
(93)
4
∂t ∂t2 ∂xi ∂xi ∂t3
2 ∂t2 ∂t∂xi
=
Therefore, writing x0 ≡ ct, where the constant c is introduced purely for dimensional reasons,
and writing spacetime indices now as µ ∈ {0, i} as usual, we see that the constraint equations
(30), (31) and (32) imply
∂f
∂2f
∂3f
3 ∂2f
=
.
2 ∂x(µ ∂xν ∂xρ ∂xλ)
∂x(µ ∂xν ∂xρ ∂xλ)
(94)
Now let us show conversely that (94) is in fact equivalent to the constraint equations (30), (31)
and (32). To that end, we note that the five equations contained in (94) are
2
3 ∂2f
∂f ∂ 3 f
=
∂t ∂t3
2 ∂t2
3
3
∂f ∂ f
∂f ∂ f
∂2f ∂2f
3
+
=
6
∂t ∂t2 ∂xi ∂xi ∂t3
∂t2 ∂t∂xi
3
3
2
∂f ∂ f
∂2f ∂2f
∂f
∂ f
∂ f ∂2f
+
+
2
=
∂t ∂t∂xi ∂xj
∂t2 ∂xi ∂xj
∂t∂xi ∂t∂xj
∂x(i ∂t2 ∂xj)
∂f
∂3f
∂2f
∂2f
∂3f
∂f
=
6
+ 3 (i
i
j
k
∂t ∂x ∂x ∂x
∂x ∂t∂xj ∂xk)
∂t∂x(i ∂xj ∂xk)
3
∂f
∂ f
3 ∂2f
∂2f
=
.
2 ∂x(i ∂xj ∂xk ∂xℓ)
∂x(i ∂xj ∂xk ∂xℓ)
(95)
(96)
(97)
(98)
(99)
Isolating the third-order derivatives one at a time then yields
2 2
∂ f
(100)
∂t2
∂t
2 2
∂f
∂3f
1 ∂xi
∂ f
2 ∂2f ∂2f
=
−
+
(101)
2
∂f ∂t2 ∂t∂xi
∂t2 ∂xi
2 ∂f
∂t2
∂t
∂t
∂f
∂f ∂f 2 2
3
2
2
∂ f
1 ∂xi ∂xj ∂ f
2 ∂2f ∂2f
1 ∂2f ∂2f
∂x(i ∂ f ∂ f
=
−
2
+
+
,
∂f ∂t∂xi ∂t∂xj
∂f ∂t2 ∂xi ∂xj
∂f 2 ∂t2 ∂t∂xj)
∂t∂xi ∂xj
2 ∂f 3 ∂t2
∂t
∂t
∂3f
3 1
= ∂f
∂t3
2
∂t
∂t
(102)
18
followed by
∂f ∂f ∂f
∂3f
3 i j ∂xk
= − ∂x ∂x
i
∂f 4
2
∂x ∂xj ∂xk
∂t
−6
2 2
∂f ∂f
2
2
∂ f
∂x(i ∂xj ∂ f ∂ f
+
6
3
2
2
∂f
∂t
∂t ∂t∂xk)
∂t
∂f
∂f
2
2
2
∂ f ∂ f
6 ∂2f
∂2f
∂2f
∂x(i
∂x(i ∂ f
−
3
+
∂f ∂t∂x(i ∂xj ∂xk)
∂f 2 ∂t∂xj ∂t∂xk)
∂f 2 ∂t2 ∂xj ∂xk)
∂t
∂t
∂t
(103)
and
∂f ∂f ∂f ∂f
∂2f
∂2f
i
j
k ∂xℓ
= − ∂x ∂x∂f∂x
4
(i
ℓ)
j
k
∂x ∂x ∂x ∂x
2 2
∂f ∂f ∂f
2
2
∂ f
∂x(i ∂xj ∂xk ∂ f ∂ f
+
4
∂f 3
∂t2
∂t2 ∂t∂xℓ)
∂t
∂t
∂f ∂f
∂f ∂f
∂f
2
2
2
2
∂ f ∂ f
∂2f
∂2f
j
(i
∂x(i ∂xj ∂ f
∂x(i ∂ f
− 4 ∂x∂f ∂x
−
2
+
4
.
2
∂f ∂t∂xj ∂xk ∂xℓ)
∂f 2 ∂t2 ∂xk ∂xℓ)
∂t∂xk ∂t∂xℓ)
∂t
∂t
∂t
(104)
Next, note that
∂
∂t
∂f
∂f ∂f
2
2
∂xi ∂xj ∂ f
∂x(i ∂ f
− ∂f 2
+
2
∂f ∂t∂xj)
∂t2
∂t
∂t
!
∂f ∂f
1 i j
= − ∂x∂f∂x
3
2
∂t
2 2
∂ f
2 ∂2f ∂2f
+
,
∂f ∂t∂xi ∂t∂xj
∂t2
(105)
∂t
followed by
∂f
∂f ∂f
2
2
j
k ∂ f
∂x(j ∂ f
− ∂x∂f∂x
+
2
2
∂f ∂t∂xk)
∂t2
∂t
∂t
∂f ∂f
∂f ∂f
∂f ∂f ∂f 2 2
2
2
2
2
3 ∂xi ∂xj ∂xk ∂ f
∂xi ∂x(j ∂ f ∂ f
∂xj ∂xk ∂ f ∂ f
−
2
=
−
2
∂f 4
∂f 3 ∂t2 ∂t∂xi
∂f 3 ∂t2 ∂t∂xk)
2
∂t2
∂t
∂t
∂t
∂f
2
2
∂ f ∂ f
∂2f
2 ∂2f
(j
+ 2 ∂x
+
.
∂f ∂t∂x(j ∂xk) ∂xi
∂f 2 ∂t∂xk) ∂t∂xi
∂t
∂t
!
∂
∂xi
(106)
3
∂ f
The equation above for ∂t∂x
i ∂xj can hence be written as
∂
∂t
∂2f
∂f ∂xi ∂xj
1
∂t
!
2
∂f
∂f ∂f 2
2
1 ∂xi ∂xj ∂ 2 f
2
∂2f ∂2f
∂x(i ∂ f ∂ f
=
−
2
+
∂f 3 ∂t2 ∂t∂xj)
∂f 2 ∂t∂xi ∂t∂xj
2 ∂f 4 ∂t2
∂t
∂t
"
!# ∂t
∂f
∂f ∂f
2
2
∂
1
∂ f
(i
i
j ∂ f
− ∂x∂f∂x
+ 2 ∂x
=
.
2
∂f ∂t∂xj)
∂t ∂f
∂t2
∂t
(107)
∂t
∂t
We similarly compute
∂f
∂f ∂f
2
2
∂xj ∂xk ∂ f
∂x(j ∂ f
−
+
2
∂f
∂f ∂t∂xk)
∂f 2 ∂t2
∂t
∂t
∂t
∂f ∂f
∂f ∂f ∂f 2 2
∂f ∂f
2
2
2
2
3 i j ∂xk ∂ f
∂xi ∂x(j ∂ f ∂ f
∂xj ∂xk ∂ f ∂ f
= ∂x ∂x
−
−
2
+
∂f 5
∂f 4 ∂t2 ∂t∂xi
∂f 4 ∂t2 ∂t∂xk)
2
∂t2
∂t
∂t
∂t
∂
∂xi
"
1
!#
∂2f
∂2f
,
∂f 2 ∂t∂x(j ∂xk) ∂xi
2
∂t
(108)
19
so that
"
1
∂
i
∂f
∂x
∂t
∂f
∂f ∂f
2
2
1 ∂2f
∂xj ∂xk ∂ f
∂x(j ∂ f
− ∂f 2
+
2
−
∂f ∂t∂xk)
∂f ∂xj ∂xk
∂t2
∂t
∂t
∂t
∂f
∂f ∂f
∂f ∂f ∂f 2 2
2
2
∂ f
∂2f ∂2f
i
j ∂xk
∂x(i ∂xj ∂ f ∂ f
∂x(i
+
6
= 3 ∂x ∂x
−
9
∂f 5
∂f 4 ∂t2 ∂t∂xk)
∂f 3 ∂t∂xj ∂t∂xk)
∂t2
∂t
∂t
∂t
∂f
2
2
∂ f
3
∂2f
∂2f
(i ∂ f
+ 3 ∂x
−
.
∂f 3 ∂t2 ∂xj ∂xk)
∂f 2 ∂t∂x(i ∂xj ∂xk)
∂t
∂t
!
#
(109)
The solution to (107) is
∂f ∂f
∂f
∂t
∂t
2
2
∂2f
∂f
∂xj ∂xk ∂ f
∂x(j ∂ f
=
−
+
+
2
hjk (⃗x),
∂f ∂t∂xk)
∂f 2 ∂t2
∂t
∂xj ∂xk
where hjk = hkj with
(110)
∂hjk
∂t = 0. Substituting this expression into (109) then results in
∂f
2
∂hjk
1 ∂2f
∂x(i ∂ f
=
3
−
+
∂f ∂t∂x(i
∂f 2 ∂t2
∂xi
!
hjk) ,
(111)
∂t
∂t
which is automatically independent of time already. Finally, substituting (110) into the remaining equation (104) yields the simple condition
∂f
∂t
2
hi(j hkℓ) = 0.
(112)
In summary, the higher-dimensional Schwarzian differential equation (94) is equivalent to
2 2
∂ f
∂t2
∂t
2 2
∂f
∂3f
1 ∂xi
∂ f
2 ∂2f ∂2f
=
−
+
∂f ∂t2 ∂t∂xi
∂t2 ∂xi
2 ∂f 2 ∂t2
∂3f
3 1
= ∂f
∂t3
2
∂t
∂t
∂f
∂f ∂f
2
2
2
∂ f
∂f
∂xi ∂xj ∂ f
∂x(i ∂ f
=
−
+
2
+
hij
2
i
j
2
∂f
j)
∂f
∂x ∂x
∂t
∂t
∂t∂x
∂t
∂t
!
∂f
2
1 ∂2f
∂x(i ∂ f
∂i hjk = 3 − ∂f 2 2 + ∂f
hjk)
(i
∂t
∂t∂x
∂t
∂t
hi(j hkℓ) = 0.
(113)
(114)
(115)
(116)
(117)
However, the quadratic constraint hi(j hkℓ) = 0 is equivalent to the condition that hij be an
2
f
antisymmetric matrix14 , but hij is forced to be symmetric on account of the symmetry of ∂x∂i ∂x
j
14
The proof is elementary. Taking the special case j = k = ℓ = i of the quadratic condition hi(j hkℓ) = 0
immediately says that all diagonal elements of the matrix vanish, i.e. hii = 0 (no index summation). Then taking
the special case k = i and ℓ = j says that hij (hij + hji ) + hii hjj = 0 (no index summation), which on account of
the vanishing of all the diagonal elements determines hij (hij + hji ) = 0, whose most general solution is simply
hij = −hji (which subsumes the possibility that hij could vanish for a fixed i and j, since the condition with i
and j swapped, i.e. hji (hij + hji ) = 0, must hold as well), after which hi(j hkℓ) trivially vanishes identically.
20
in (110), and thus hij = 0. Therefore, we have proven the equivalence
2
∂3f
3 ∂2f
∂2f
∂3f
3 1 ∂2f
∂f
=
⇐⇒
= ∂f
2 ∂x(µ ∂xν ∂xρ ∂xλ)
∂t3
2
∂t2
∂x(µ ∂xν ∂xρ ∂xλ)
∂t
2 2
∂f
∂3f
1 ∂xi
∂ f
2 ∂2f ∂2f
=
−
+
∂f ∂t2 ∂t∂xi
∂t2 ∂xi
2 ∂f 2 ∂t2
∂t
∂t
∂f
∂f ∂f
2
2
∂xi ∂xj ∂ f
∂x(i ∂ f
=
−
+
2
∂f ∂t∂xj)
∂f 2 ∂t2
∂xi ∂xj
∂t
∂t
∂2f
(118)
between the higher-dimensional Schwarzian differential equation (94) and the original constraint
equations (30), (31) and (32).
We hence define an appropriate generalization of the Schwarzian derivative of a function
f (x) of any number of variables xµ to be the rank-4 totally symmetric nonlinear differential
operator
∂f
∂3f
3 ∂2f
∂2f
Sµνρλ (f ) ≡
−
.
(119)
∂x(µ ∂xν ∂xρ ∂xλ) 2 ∂x(µ ∂xν ∂xρ ∂xλ)
We have therefore shown in the course of this paper that the kernel of this higher-dimensional
Schwarzian differential operator precisely consists of the fractional linear transformations,
Sµνρλ (f ) = 0
⇐⇒
f (x) =
α + β µ xµ
,
γ + δ ν xν
(120)
for constants α, γ, βµ and δµ . In the case of a single dimension with variable z, the operator
df 2
Szzzz (f ) is of course dz
times the usual Schwarzian differential operator Sz f , but the only
difference in the kernels is that Sµνρλ (f ) = 0 allows the constant solutions as well. The reason
for this difference here is that (119) acts on a single function f (x) and not a vector of functions
f µ (x), but it is really just the first component of the full higher-dimensional Schwarzian operator
αµ +β µ xν
whose kernel is the full set of projective linear transformations f µ (x) = γ+δρ xνρ , for which we
can divide by the square of the determinant of the Jacobian so that we obtain only the invertible
projective linear transformations. Also, while in this paper we solved these
differential equations
∂g by using the additional physicality conditions ∂f
=
̸
0
and
det
=
̸
0, they are actually
∂t
∂x
unnecessary15 ; one still obtains the solutions (60) and (61) subject only to the invertibility
condition (64).
Finally, we should mention that the higher-dimensional Schwarzian derivative (119) is different than others given in the mathematical literature. Part of the variety of options which exist
is that usually there are many ways to generalize a differential operator in one dimension to an
analogous one in higher dimensions. The other reason is due to a coincidence — on CP1 , the
Möbius group is equal to PSL(2, C), the group of projective linear transformations, which is the
same as the group of (global) conformal transformations. In all higher dimensions, the Möbius
group is just the group of conformal transformations, which is not equivalent to a projective
linear group. In two real dimensions, the two concepts merge, but in higher dimensions they
differ, and many of the higher-dimensional Schwarzian derivative generalizations focus on its
connection to Möbius transformations (and generalizations thereof) [13]. The closest to the
higher-dimensional generalization presented here seems to be the one given by Ovsienko in [14],
which is applicable to Lagrangian subspaces of symplectic vector spaces.
i
∂g
∂f
At no point did we need to use the invertibility of the spatial Jacobian ∂x
j . As for ∂t , if there exists an open
neighborhood in the domain in which it does not vanish, then the preceding derivation goes through unaltered
and the solution in the full connected domain is the same by continuity. Otherwise, one merely looks for the
solutions to the original constraint equations with ∂f
= 0 everywhere, a shorter and very similar exercise which
∂t
again yields projective linear solutions.
15
21
Acknowledgments
I am grateful to Jacob Barandes for useful discussions and for encouraging me to write up this
derivation.
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