A GROUP-THEORETIC CLASSIFICATION OF INERTIAL FRAMES
I. THE TWO-DIMENSIONAL CASE
George Svetlichny
∗July 24, 2002
Abstract
We classify inertial frames in two-dimensional space-time by deriving them from group theoretic assumptions. Besides the usual Einsteinian, Galilean, and Euclidian relativities, we also find solutions that allow anisotropies, and unconventional spacio-temporal translation subgroups.
1 Abstract Relativity
Frames of reference are used to describe physical processes. One believes that relative to any frame a complete description is possible. On the other hand, in order to concretely use a frame, physical processes are needed to set up its parameters and to relate the process under description to these. Thus under an operational viewpoint, frames are dependent on processes and their definition suffers from some circularity. We shall maintain such a bias and consider frames as special types of processes. Hopefully, the air of vagueness that this may lend to the exposition will in the end be somewhat dispelled by the precise math- ematical problems which we can distill from it. Physical processes are more often than not considered counterfactually. No engineering project can hope of any success without counterfactual considerations of physical processes and the corresponding mathematical analyses. Hence by a process we mean any actual state of affairs or one that can be admitted counterfactually. By aframe we mean any process relative to which any other process can receive complete specification. Apparently frames exist in abundance; this we take as a fact of nature. Given a set of frames we can hope for it to possess certain properties that enhance their usefulness. Again it is a fact of nature that sets of frames having such properties seem to exist. We focus on some of these properties:
∗Departamento de Matem´atica, Pontif´ıcia Universidade Cat´olica, Rua Marquˆes de S˜ao Vicente 225, 22453-900 G´avea, Rio de Janeiro, RJ, Brazil
e-mail: svetlich@mat.puc-rio.br
Universality: IfS is a specification of a process relative to some frame, then S is a specification of some process relative to any frame.
Now processes can be grouped into equivalence classes according to shared common description each one relative to some frame. What universality adds is that given such a class and a frame, some member of the class has the common description relative to this frame.
Transitivity : IfS is a specification of a frame relative to another, then all processes having specificationS are frames.
In other words, the equivalence class of a frame, contains only frames.
From transitivity one deduces that a specification S of a frame defines now a map, which we continue to call byS, from the set of frames into itself. Given a frameO, S(O) is the frame whose specification isS relative to frameO. We shall call maps so definedframe maps.
Composition : If O1, O2, and O3 are frames and S21, S32, and S31, are specification such thatSij is the specification of frameOi, relative to frameOj, then ifS21specifies frameP2relative to P1andS32 specifiesP3relative to P2, one again has thatS31 specifiesP3 relative toP1.
From composition and transitivity one deduces that composing two frame maps yields a new frame map: S31 =S32◦S21. This means that equivalence class of frames form a group where group product is defined via composition of frame maps. That the inverse of a frame map is also a frame map follows from composition takingO3to beO1 and noting that nowS31is a specification stating identity of frames.
By arelativistic theory we mean a theory of processes with a distinguished set of frames which satisfies the three properties of universality, transitivity, and composition. As part of such a theory one has the groupG of frame maps.
We call this group therelativity group. SinceG acts transitively on the set of frames, the latter is a symmetric space of the group.
Many mathematical problems related to relativistic theories thus reduce themselves to problems of group theory.
Abstract relativity would be the abstract study of relativistic theories. Such a study does not yet exist. What would give abstract relativity any mathemat- ical substance, and not allow just any group and any of its symmetric spaces as example, are constraints imposed on these by the requirements of interpretation.
Our next section exemplifies this.
2 Two Dimensional Inertial Frames
We shall now develop the theory of inertial frames in two-dimensional space- time. This is to be considered as sort of a minimal relativistic theory of kinemat-
ics, subject to a set of assumptions to be shortly spelled out. The assumptions are by no means uncontroversial or absolutely precise. They should be regarded as signposts leading to the precise group-theoretic problem that we resolve, and whose solutions we classify. There are by now many axiomatic derivations of the space-time structure of special relativity [1-7]. We feel our exposition proceeds with a minimum of physical assumptions, leading thus to a surprising variety of solutions. Physical assumptions can then be imposed a-posteriori to pick out the solution that seems to be relevant to the world as we know it. This procedure exemplifies thus our notion of abstract relativity.
Assumption 1 One is dealing with a relativistic theory. We shall denote byG the relativity group of the theory.
Assumption 2 Each frame provides a coordination of space-time.
Assumption 3 Coordinations provided by two different frames are related by C∞ maps.
Assumption 4 There are frames arbitrarily close to each other so that the coordinations are related by maps arbitrarily close to the identity.
Assumption 5 For two frames sufficiently close to each other, the affine part (first two terms of the Taylor series) of the transition map between the two coordinations, is sufficient, given the specification of some process in one frame, to provide a good approximation to the specification of the same process in the other frame.
From these we are lead to assume thatG is a Lie group with aC∞transitive action on a manifold of frames. The affine part of the group action on the set of frames constitutes an approximation of a neighborhood of the identity ofG and a neighborhood of some fixed frame. This affine action is the appropriate one in the tangent spaces of space-time when this is regarded mathematically as a differentiable manifold. By our adjective ”inertial” we mean to refer to such tangent space actions. Our last assumption means that an arbitrarily good specification of processes can be given using inertial frames, provided we restrict ourselves to a sufficiently local picture. For the rest of this exposition, we now assume global affine action and take a vector space model of space-time.
Let O be a frame and Z the space-time coordinates relative to O of some event. IfO0 =gO for someg∈ G, then there is an affine map W(g) such that Z0, the coordinates relative toO’ of the same event are given by:
Z0=W(g)Z =L(g)Z+I(g),
whereL(g) andI(g) are the linear and the constant part of the affine map.
FromW(gh) =W(g)W(h) we deduce:
L(gh) = L(g)L(h), I(gh) = L(g)I(h) +I(g).
Assumption 6 Space-time is two-dimensional.
Assumption 7 The specification of a frame relative to another consists of the space-time translation X =
µx t
¶
of the origin of one relative to the origin of the other, and the relative velocityuof one relative to the other.
Thus ifg∈ G we writeg= (X, u). If nowO0=gOthen the last assumption is taken specifically to mean: 1)X is the origin of O0 relative to O, and 2) the point moving with velocityurelative to Oand whose world line containsX, is at rest relative toO0.
From 1) one deduces: W(X, u)X = 0⇒I(X, u) =−L(X, u)X, and so W(X, u)Z =L(X, u)(Z−X).
From 2) one deduces that the line X+t µu
1
¶
transforms byW(X, u) to a line at rest and so:
L(X, u) =
µq −uq r s
¶ .
Assumption 8 If framesOandO0have coinciding origins, and likewise frames O0 andO00, then so do frames O andO00.
From this it follows that {(X, u) ∈ G|X = 0} is a subgroup, the velocity subgroup. We setL(u) =L(0, u) and specifically:
L(u) =n
µ1 −u c d
¶ .
One dimensional groups are commutative and soL(u)L(v) =L(v)L(u) which yields:
1−uc(v) = 1−vc(u), v+ud(v) = u+vd(u).
The general solution of the first equation isc(u) =au for some reala, and of the second equation isd(u) = 1 +bufor some realb. Thus:
L(u) =n
µ1 −u au 1 +bu
¶ .
Denote now byu⊕vthe group sum in the velocity subgroup. FromL(u)L(v) = L(u⊕v) one deduces the addition law of velocities:
u⊕v= u+v+buv 1−auv .
An analysis can now be made of the addition formula. Since it involves a rational function, its use can be extended to include∞as a velocity value. We shall call a valuecinvariant if as a function ofv,c⊕vis the constant functionc.
One now finds that there are three cases depending on the value ofD=b2−4a:
I: D >0, there are two invariant values:
−b±√ D 2a . II: D= 0, there is one invariant value:
−b 2a. III: D <0, there are no invariant values.
In terms of the compactified line ¯
R
=R
∪ {∞}, one can now classify the possible velocity subgroups:Case I: The complement of the two invariant values consists of two disjoint open intervals of ¯
R
. The interval containing 0 is the component of the identity and invariably belongs to the velocity subgroup. The other segment may or may not belong to the subgroup, and when it does, it constitutes the only other component.Case II: The velocity subgroup consists of ¯
R
minus the invariant value.Case III: The velocity subgroup is the whole of ¯
R
.Case I is to be identified with Einsteinian relativity. In general the two invariant velocities need not be negatives of each other. Case II is to be identified with Galilean relativity. Again in general the invariant value is not∞. Case III is to be identified with Euclidian relativity. In all three cases the group inverse of a velocity value u is given by−u/(1 +bu) and is thus in general not given by the negative. However, there is a certain amount of conventionality in the analysis up to now, and this non-standard behavior can be defined away. If we redefine the coordinates relative to each frame by the same linear transformation S:Z0 =SZ, thenL(u) suffers a similarity:
L0(u0) =SL(u)S−1. The particular choice
S= µ1 0
k 1
¶ .
redefines the temporal coordinate and results in:
u0 = u 1 +ku, a0 = k2−kb+a, b0 = b−2k.
Hence choosing k = b/2, one has b0 = 0 and a0 = a−b2/4. The further choice of
S = µ1 0
0 m
¶ ,
redefines the time scale, and gives u0 = u/m, a0 =m2a. Thus one can then finally bring L(u) to itsstandard form:
L(u) =n(u)
µ1 −u gu 1
¶
, g=−1,0,1.
To computen(u) one has fromL(u)L(v) =L(u⊕v):
n(u)n(v)(1−guv) =n(u⊕v).
Differentiating both sides with respect to v and settingv = 0, one obtains the differential equation.
n0(u)
n(u) =n0(0)−gu 1 +gu2 .
This differential equation can be easily solved; its solutions actually satisfy the original functional equation. Settingd=n0(0) and usingn(0) = 1 one finds the following solutions:
g=−1 : n(u) = (1−u2)−1/2[(1 +u)/(1−u)]d/2 g= 0 : n(u) =edu
g= 1 : n(u) = (1 +u2)−1/2edarctanu
Thus dis an anisotropy parameter, As already noted,g =−1 corresponds to Einsteinian relativity, g = 0 to Galilean relativity, and g = 1 to Euclidian relativity.
We are now ready to consider the translations. For this we impose our last assumption:
Assumption 9 If relative to frame O, frames O0 andO00 have coinciding ori- gins, thenO0 andO00 are not relatively displaced.
From this we conclude that (X, u) = (0, u0)(X,0) for some velocityu0. Hence W(X,0)W(0, u) =W(0, v)W(Y,0) for someY, v. Analyzing this relation, and settingL(X) =L(X,0) one finds:
Y = L(u)−1X, L(v)−1L(X)L(u) = L(L(u)−1X).
Set now
M(X) =L(X)−1=
µA 0 B C
¶ . One now has
L(u)−1M(X)L(v) =M(L(u)−1X).
This equation and the fact thatM(X) is zero in the (1,2) place, leads to:
v= Cu A+Bu.
Differentiating the next to last equation with respect tou, and settingu= 0 one has:
−L0(0)M(X) + (C/A)M(X)L0(0) =−DM(X)L0(0)X.
Now
L0(0) =
µd −1 g d
¶ .
The differential equations for M(X) can now be written as:
d(A−C)−B = Ax(dx−t) +At(gx+dt), (1) (A−C)[g(A+C) +dB] = A[Bx(dx−t) +Bt(gx+dt)], (2) d(A−C) +B = (A/C)[Cx(dx−t) +Ct(gx+dt)]. (3) From the fact that M(X) is zero in the (1,2) place, one deduces that the set{(X, u)∈ G|u= 0}forms a subgroup, called thetranslation subgroup. Thus there is a group law for translations, which we shall also designate by⊕, and we haveW(X)W(Y) =W(X⊕Y). This relation implies:
X⊕Y = Y +M(Y)X, M(X+M(X)Y) = M(X)M(Y).
Taking the Fr´echet derivative with respect toY of both sides of this equation and settingY = 0, one gets:
DM(X)M(X)h=M(X)DM(0)h.
If we now set
Mx(0) = µp 0
q r
¶
Mt(0) = µs 0
y w
¶ ,
then we have the following six differential equations for M(X):
AxA+AtB=pA, (i)
AtC=sA, (ii)
BxA+BtB=pB+qC, (iii)
BtC=sB+yC, (iv)
CxA+CtB=rC, (v)
CtC=wC. (vi)
Our problem now consists of solving equations (1)-(3) and (i)-(vi) subject to the conditions A(0) = 1, B(0) = 0, and C(0) = 1. One finds that these
solutions then satisfy the original functional equations from which the differen- tial equations arose. This of course is due to the group theoretic nature of the problem. The details of the solution to the differential equations we leave for the appendices, here we summarize and interpret the results.
Now equations (i)-(vi) are of interest in their own right, since they express two-dimensional translation relativity. One has ten distinct cases:
I: w6= 0, s6= 0, s6=w/2;
A=Cs/w, B = (r/w)(C−A), C=wt+rx+ 1.
II: w6= 0, s=w/2;
A=C1/2 B= (r/w)(C−A)−(r2−2qw)(x/2w)A, C=wt+ (r2−2qw)(x2/4) +rx+ 1.
III: w6= 0, s= 0;
A=px+ 1, B= (r/w)(C−A), C=wt+rx+ 1.
IV: w= 0, s6= 0;
A=epx+st, B= (r/w)(C−A), C= 1.
V: w= 0, s= 0, p6= 0, y=p, r6= 0;
A=px+ 1, B=p[t+ (q/r)(C−1)], C=Ar/p. VI: w= 0, s= 0, p6= 0, y=p, r= 0;
A=px+ 1, B=pt+ (q/p) lnA, C = 1.
VII: w= 0, s= 0, y= 0, r6=p, p6= 0;
A=px+ 1, B= [q/(r−p)](C−A), C=Ar/p. VIII: w= 0, s= 0, y= 0, r=p, p6= 0;
A=px+ 1, B= (q/p)AlnA, C =A.
IX: w= 0, s= 0, p= 0, r6= 0;
A= 1, B= (q/r)(C−A), C=erx. X: w= 0, s= 0, p= 0, r= 0, y= 0;
A= 1, B=qx, C= 1.
Imposing now upon each of the ten cases I−X equations (1)-(3), one finds besides the case M(X) = I, which is compatible with any value of d and all cases ofg, the following additional solutions:
Case 1: g=−1, d=±1 (anisotropic Einsteinian relativity)
M(X) = (wt+dwx+ 1)I, (E-1)
and
M(X) =
µ2dwx+ 1 0 w(dt−x) wt+dwx+ 1
¶
. (E-2)
Case 2 : g = −1, d = ±3 (anisotropic Einsteinian relativity); let H = wt+ (d/3)wx+ 1, then
M(X) =
µ H2 0
(d/3)(H−H2) H
¶
. (E-3)
Case 3: g= 0, d= 0 (isotropic Galilean relativity) M(X) =
µ(wt+ 1)s/w 0
0 wt+ 1
¶
, (G-1)
M(X) =
µest 0
0 1
¶
, (G-2)
and
M(X) =
µpx+ 1 0 pt (px+ 1)−1
¶
. (G-3)
Some of these forms imply the existence of spacio-temporal horizons. Thus in (E-2) withd= 1 one finds there is an invariant time displacement value of
−1/w. None of these new forms seem off hand to refer to the world as we know it. By assuming isotropy of space and the existence of a finite invariant velocity value, one is reduced to the unique solution of the usual Einsteinian relativity.
3 Extension To Higher Dimensions
A thorough higher dimensional study has not been carried out; the correspond- ing group theoretic problems are more difficult due to the presence of the spatial rotation group. The two dimensional results can be used though if we make some additional assumptions that insure us two-dimensional subspaces of space-time in which the two-dimensional relativities holds. This would be the case if the group sum of two colinear displacements were colinear with the summands, and if the group sum of two colinear velocities, were also colinear with the sum- mands. Under these conditions if the relativity type does not change under purely spatial rotations, the anisotropic forms would be ruled out since by a continuous rotation one can invert the space axis of a two-dimensional sub- space, and so change the sign of the anisotropy parameter. Since this can only take on discrete values once the invariant velocity has been standardized to have the same value in all directions, such a situation is impossible.
4 Acknowledgements
I thank Harvey Brown for helpful discussions on the axiomatic treatment of special relativity, and also Wolfson College and the Sub-faculty of Philosophy of Oxford University for their hospitality during my visit, when part of this work was done. This work received financial support from the Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) and the Finaciadora de Estudos e Projetos (FINEP), agencies of the Brazilian government.
References
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[2] A. G. Walker,Proc. Roy. Soc. Edinburgh,Sect. A, 62 319 (1948)
[3] A. G. Walker, “Axioms for Cosmology”, in L. Henkin, P. Suppes,and A.
Tarski, eds.,The Axiomatic Method,North Holland, Amsterdam, 1959 [4] W. Noll,Amer. Math. Monthly,71 129 (1964)
[5] G. Szekeres,J. Aust. Math. Soc. 8, 134 (1968)
[6] J. W. Shultz, “Foundations of Special Relativity: Kinematic Axioms for Minkowski Space-Time”,Lecture Notes in Mathematics,Vol 361, Springer, Berlin-Heidelberg-New York, 1973
[7] J. W. Shultz, “An Axiomatic System for Minkowski Space-Time”, Preprint No. MPI-PAE/Astro 181, Max-Plank-Instit¨ut f¨ur Physik und As trophysik, M¨unchen, 1979
A Solutions of (i)-(vi)
Case A:w6= 0
Now, (v) and (vi) are equivalent to B = (rC−CxA)/w, Ct =w, thus C = wt+f(x) andB= [rC−f0(x)A]/w. From (ii) we haveAt/A=s/[wt+f(x)] so A=h(x)(wt+f(x))s/w. UsingAtfrom (ii), equation (i) now givesAx+sB/C = p. After appropriate substitution this gives:
(sr/w)−p+h0(x)C= 0. (A-1)
Case AA:s6= 0
From (A-1) we conclude that sr/w =p and h0(x) = 0 so that h(x) = 1. By appropriate substitution into (iii) and after simplifying, using sr/w = p, one obtains:
(r2−pr−qw)C−f00(x)C2s/w= 0. (AA-1) Case AAA: s6=w/2
From (AA-1) one deducesr2−pr−qw= 0 andf00(x) = 0, thus f(x) =rx+ 1 andC=wt+rx+ 1. This concludes case I of our summary.
Case AAB:s=w/2
From (AA-1) one concludes r2 −pr −qw −f00(x) = 0, which when solved
givesf(x) = (1/4)(r2−2qw)x2+rx+ 1 and we obtain case II of our summary.
Case AB:s= 0
Equation (A-1) gives −p+h0(x) = 0, thus h(x) = px+ 1. Now (AA-1) is still valid in this case since its derivation did not use the hypothesis thats6= 0.
One concludes from it thatf00(x) = 0 and so f(x) =rx+ 1, and we have case III of the summary.
Case B: w= 0
From (vi) we get thatC=f(x); from (v) we have:
f0(x)/f(x) =r/A. (B-1)
We can integrate (ii) to giveA=h(x)est/f(x), and substituting this into (B-1) gives:
f0(x)/f(x) = (r/h(x))e−st/f(x). (B-2) Case BA:s6= 0
From (B-2) we get f0(x) = 0, r = 0, so f(x) = 1. Substituting into (i) one can solve for B to get B = [p−h0(x)est]/s. Substituting into (iii) gives an equation one of whose consequences ish00(x)h(x)−h0(x)2= 0. This integrates toh(x) =epx, and provides us with case IV of the summary.
Case BB:s= 0
From (B-2) we deduce f0(x)/f(x) = r/h(x); (i) can be integrated to give h(x) =px+ 1, and so:
f0(x)/f(x) =r/(px+ 1). (BB-1)
Case BBA:p6= 0
Integrating (BB-1) one has h(x) = (px+ 1)r/p. Integrating (iv) gives B = yt+k(x). Substituting into (iii) gives:
k0(x)(px+ 1) +y[yt+k(x)] =p[yt+k(x)] +q(px+ 1)r/p. (BBA-1) From thet2 terms one concludes that eithery=pory= 0.
Case BBAA:y=p
One hask0(x) =q(px+ 1)r/p−1. Case BBAAA :r6= 0
Integrating one getsk(x) = (pq/r)[(px+ 1)r/p−1] and establishes case V of the
summary.
Case BBAAB:r= 0
Integrating one gets k(x) = (q/p) ln(px+ 1) and establishes case VI of the summary.
Case BBAB:y= 0
From (BBA-1) we get k0(x)(px+ 1) = pk(x) +q(px+ 1)r/p. Setting k(x) = l(x)(px+ 1) one deducesl0(x) =q(px+ 1)r/p−2.
Case BBABA:r6=p
Integrating one gets l(x) = [q/(r−p)][(px+ 1)r/p−1−1] and establishes case VII of the summary.
Case BBABB:r=p
Integrating one gets l(x) = (q/p) ln(px+ 1) and establishes case VIII of the summary.
Case BBB :p= 0
One deduces h(x) = erx, A = 1, and B = yt+k(x). Equation (iii) leads tok0(x) +y[yt+k(x)] =qerxand so y= 0, andk0(x) =qerx.
Case BBBA:r6= 0
Integrating one getsk(x) = (q/r)(erx−1) and establishes case IX of the sum- mary.
Case BBBB:r= 0
Integrating one getsk(x) =qxand establishes case X of the summary.
B Imposing (1)-(3) upon cases I-X
Case I
Changing independent variables from (x, t) to (x, C) one finds from equations (1)-(3) that:
rd+wg−r(d−r/w) = 0 (a) and that taking this into account one further has:
(1) ⇔ (d+r/w)(Cs/w−C) = (s/w)(d−r/w)Cs/w−1(C−1), (2) ⇔ [g(Cs/w+C) + (dr/w)(C−Cs/w)](Cs/w−C) =
= (r/w)(d−r/w)Cs/w[1−(s/w)Cs/w−1](C−1), (3) ⇔ (d−r/w)(Cs/w−C) = (d−r/w)Cs/w−1(C−1) .
Assumings/w6= 1,2, one deduces from (1) and (3) thatd= 0, r= 0, and from (a) one gets g= 0. This provides us with case (G-1). If we now assume s/w= 2, then (1) implies d= 3r/w, (3) is already satisfied, and equating coef- ficients of powers ofC in (2) gives us the following possibilities: g= 0, d= 0, r = 0, and g = −1, d = ±3, r = dw/3. The first possibility reduces to the previous form withs/w= 2; the second possibility gives us case (E-3). Finally ifs/w= 1, thenB = 0 and (2) is satisfied. Equations (1) and (3) now lead to the following possibilities: g= 0,d= 0, r= 0, andg=−1, d=±1,r=dw.
The first possibility reduces to the first form withs/w= 1, and the second one leads to case (E-1).
Case II
We again transform to variables (x, C) instead of (x, t). By equating the coeffi- cients of thexCterm on both sides of equation (1) one deduces thatr2−2qw= 0 and this reduces this case to case I withs/w= 1/2.
Case III
Both sides of equations (1)-(3) are now rational functions ofxand t. By cross multiplying and equating coefficients, one easily finds now that there are two possibilities: g= 0, d=o,p= 0,r= 0, andg=−1,d=±1,p= 2dw, r=dw.
The first possibility is reduced to a form of case I, and the second gives case (E-2).
Case IV
This is easily analyzed to give the result g = 0, d = 0, p = 0. This gives case (G-2).
Case V
Here we transform to using (C, t) as independent variables instead of (x, t).
Equating the coefficients of t on both sides of equation (3) leads to p = −r.
With this, both sides of (1)-(3) are now rational functions ofCandt. One then readily finds thatd= 0, andg= 0. This gives case (G-3).
Cases VI-IX
These are all readily shown to be incompatible with equations (1)-(3).
Case X
The only solution here isM(X) =I, which is compatible with allL(u).
