The Limits of Special Relativity
THE LIMITS OF SPECIAL RELATIVITY
B.G. Sidharth
International Institute for Applicable Mathematics & Information Sciences
Hyderabad (India) & Udine (Italy)
B.M. Birla Science Centre, Adarsh Nagar, Hyderabad - 500 063 (India)
Abstract
The Special Theory of Relativity and the Theory of the Electron
have had an interesting history together. Originally the electron was
studied in a non relativistic context and this opened up the interest-
ing possibility that lead to the conclusion that the mass of the elec-
tron could be thought of entirely in electromagnetic terms without in-
troducing inertial considerations. However the application of Special
Relativity lead to several problems, both for an extended electron and
the point electron. These inconsistencies have, contrary to popular
belief not been resolved satisfactorily today, even within the context
of Quantum Theory. Nevertheless these and subsequent studies bring
out the interesting result that Special Relativity breaks down within
the Compton scale or when the Compton scale is not neglected. This
again runs contrary to an uncritical notion that Special Relativity is
valid for point particles.
1 Introduction
When Einstein proposed his Special Theory of Relativity, there were two
ruling paradigms, which continue to hold sway even today, though not so
universally. The first was that of point elementary particles and the second
was that of space time as a differentiable manifold.
Little wonder therefore that as the relativistic theory of the electron de-
veloped, there were immediate inconsistencies which were finally ostensibly
resolved only with the intervention of Quantum Theory. This was because,
http://arxiv.org/abs/0704.1517v1
historically the original concept of the electron was that of a spherical charge
distribution [1, 2, 3]. It is interesting to note that in the non-relativistic case,
it was originally shown that the entire inertial mass of the electron equalled
its electromagnetic mass. This motivated much work and thought in this
interesting direction. To put it briefly, in non relativistic theory, we get [1],
Kinetic energy = (β/2)
where R is the radius of the electron and β is a numerical factor of the order
of 1. So we could possibly speak of the entire mass of the electron in terms
of its electromagnetic properties.
It might be mentioned that it was still possible to think of an electron as
a charge distribution over a spherical shell within the relativistic context,
as long as the electron was at rest or was moving with a uniform velocity.
However it was necessary to introduce, in addition to the electromagnetic
force, the Poincare stresses - these were required to counter balance the
repulsive ”explosion” of the different parts of the electron.
When the electron in a field is accelerated, the above picture no longer holds.
We have to introduce the concept of the electron self force which is given by,
in the simple case of one dimensional motion,
ẍ+ γ
ẍ+ 0(R2) (1)
where dots denote derivatives with respect to time, and R is the radius of
the spherical electron. More generally (1) becomes a vector equation. As can
be seen from (1), as R the size of the electron → 0 the first term → ∞ and
this is a major inconsistency. In contrast the second term which contains the
non Newtonian third time derivative remains unaffected while the third and
following terms → 0. It may be mentioned that the first term (which → ∞)
gives the electromagnetic mass of the electron while the second term gives
the well known Schott term (Cf.ref.[1, 2, 4]).
Let us now see how it was possible to rescue the relativistic electron theory,
though at the expense of introducing some unphysical concepts.
2 The Advanced and Retarded Fields
To proceed, from a classical point of view a charge that is accelerating ra-
diates energy which dampens its motion. This is given by the well known
Maxwell-Lorentz equation, which in units c = 1, and τ being the proper
time, while ı = 1, 2, 3, 4, is (Cf.[8]), is
= eF ık
, (2)
The first term on the right side is the usual external field while the second
term is the damping field which is added ad hoc by the requirement of the
energy loss due to radiation. In 1938 Dirac introduced instead
= e{F ık +R
where
Rık ≡
{F ık(ret) − F
k(adv)} (4)
In (4), F(ret) denotes the retarded field and F(adv) the advanced field. While
the former is the causal field where the influence of a charge at A is felt by
a charge at B at a distance r after a time t = r
, the latter is the advanced
field which acts on A from a future time. In effect what Dirac showed was
that the radiation damping term in (2) or (3) is given by (4) in which an
antisymmetric difference of the advanced and retarded fields is taken. Let us
elaborate a little further.
The Maxwell wave equation has two independent solutions, one having sup-
port on the future light cone, this is the retarded solution and the other
having support on the past light cone which has been called the advanced
solution. The retarded solution is selected to describe the physical situa-
tion in conventional theory taking into account the usual special relativistic
concept of causality. This retarded solution is physically meaningful, as it de-
scribes electromagnetic radiation which travels outward from a given charge
with the speed of light and reaches another point at a later instant. It has
also been called for this reason the causal solution. On the grounds of this
causality, the advanced solution has been rejected, except in a few formula-
tions like those of Dirac above, or Feynman and Wheeler (F-W) to be seen
below.
In the F-W formulation, the rest of the charges in the universe react back
on the original electron through their advanced waves, which arrive at the
given charge at the same time as the given charge radiates its electromagnetic
waves. More specifically, when an electron is accelerated at the instant t, it
interacts with the other charges at a later time t′ = t + r/c where r is the
distance of the other charge–these are the retarded interactions. However
the other charges react back on the original electron through their advanced
waves, which will arrive at the time t′ − r/c = t. Effectively, there is in-
stantaneous action at a distance. It must be mentioned that in the F-W
formulation there is no self force or radiation damping. This is provided in-
stead by the action of all other charges in the universe on the original charge.
There is also no electromagnetic mass, like the first term on the right side of
It must also be mentioned that Dirac’s prescription lead to the so called run-
away solutions, with the electron acquiring larger and larger velocities in the
absence of an external force [7]. This he related to the infinite self energy of
the point electron.
To elaborate further we use the difference of the advanced and retarded fields
(that is (9) in (1), in the following manner: We use successively F(ret) and
F(adv) in (1) and take the difference in which case the self force becomes
(Cf.[4])
F = −
(ẍ) + 0(R)
In the above, the troublesome infinity generating term of (1) is absent, while
the third derivative term is retained. On the other hand this term is required
on grounds of conservation of energy, due to the fact that an accelerated
electron radiates energy (Cf.[5]). Except for the introduction of advanced
fields, we have infinity free results. However, in this formulation too, there is
no electromagnetic mass term, and further, as will be seen below, we have to
extend our considerations to a small neighborhood of the electron, and not
just the point electron itself. To see this in detail, we observe that the well
known Lorentz Dirac equation (Cf.[1]), can be written as
maµ(τ) =
Kµ(τ + ατ0)e
−αdα (5)
where aµ is the accelerator and
Kµ(τ) = F
in + F
ext −
∼ 10−23sec (6)
τ ′ − τ
where τ denotes the time and R is the total radiation rate. Incidentally this
is a demonstration of the non locality in Compton time τ0, referred to above.
It can be seen that equation (5) differs from the usual equation of Newtonian
Mechanics, in that it is non local in time. That is, the acceleration aµ(τ)
depends on the force not only at time τ , but at subsequent times also. Let
us now try to characterise this non locality. We observe that τ0 given by
equation (6) is the Compton time ∼ 10−23secs. So equation (5) can be
approximated by
maµ(τ) = Kµ(τ + ξτ0) ≈ K
µ(τ) (7)
Thus as can be seen from (7), the Lorentz-Dirac equation differs from the
usual local theory by a term of the order of
ȧµ (8)
the so called Schott term. It is well known that the time component of the
Schott term (8) is given by (Cf.ref.[1])
≈ R ≈
where E is the energy of the particle. Whence integrating over the period of
non locality ∼ τ0 the Compton time, we can immediately deduce that r the
scale of spatial non locality is given by
r ∼ cτ0,
which is of the order of the Compton wavelength.
So far as the breakdown of causality is concerned, this takes place within
a period ∼ τ , the Compton time as we briefly saw [1, 7]. It was at this
stage that Wheeler and Feynman reformulated the above action at a distance
formalism in terms of what has been called their Absorber Theory. In their
formulation, the field that a charge would experience because of its action
at a distance on the other charges of the universe, which in turn would act
back on the original charge is given by
(ẍ) (9)
The interesting point is that instead of considering the above force in (9)
at the charge e, if we consider the response at an arbitrary point in its
neighborhood as was shown by Feynman and Wheeler (Cf.ref.[19]) and, in
fact a neighborhood at the Compton scale, as was argued recently by the
author [13], the field would be precisely the Dirac field given in (3) and (4).
The net force emanating from the charge is thus given by
F ret =
F ret + F adv
F ret − F adv
which is the acceptable causal retarded field. The causal field now consists of
the time symmetric field of the charge together with the Dirac field, that is the
second term in (10), which represents the response of the rest of the charges.
Interestingly in this formulation we have used a time symmetric field, viz.,
the first term of (10) to recover the retarded field with the correct arrow of
time. Feynman and Wheeler stressed that the universe has to be a perfect
absorber or to put it simply, every charged particle in the universe should
respond back to the action on it by the given charge in our instantaneous
action at a distance scenario. In any case, it was realized that the limits
of classical physics are reached in the above considerations, at the Compton
scale.
There are two important inputs which we can see in the above more recent
formulation. The first is the action of the rest of the universe at a given
charge and the other is minimum spacetime intervals which are of the order
of the Compton scale. The minimum spacetime interval removes, firstly the
advanced field effects which take place within the Compton time and secondly
the infinite self energy of the point electron disappears due to the Compton
scale.
3 Quantum Mechanical Considerations
The Compton scale comes as a Quantum Mechanical effect, within which we
have zitterbewegung effects and a breakdown of Causal Physics [14]. Indeed
Dirac had noted this aspect in connection with two difficulties with his elec-
tron equation. Firstly the speed of the electron turns out to be the velocity
of light. Secondly the position coordinates become complex or non Hermi-
tian. His explanation was that in Quantum Theory we cannot go down to
arbitrarily small spacetime intervals, for the Heisenberg Uncertainty Princi-
ple would then imply arbitrarily large momenta and energies. So Quantum
Mechanical measurements are an average over intervals of the order of the
Compton scale. Once this is done, we recover meaningful physics. All this
has been studied afresh by the author more recently, in the context of a non
differentiable spacetime and noncommutative geometry [21].
Weinberg also notices the non physical aspect of the Compton scale [15].
Starting with the usual light cone of Special Relativity and the inversion of
the time order of events, he goes on to add, and we quote at a little length
and comment upon it, “Although the relativity of temporal order raises no
problems for classical physics, it plays a profound role in quantum theories.
The uncertainty principle tells us that when we specify that a particle is at
position x1 at time t1, we cannot also define its velocity precisely. In conse-
quence there is a certain chance of a particle getting from x1 to x2 even if
x1 − x2 is space-like, that is, |x1 − x2| > |x
1 − x
2|. To be more precise, the
probability of a particle reaching x2 if it starts at x1 is nonnegligible as long
0 ≤ (x1 − x2)
2 − (x01 − x
· · · (11)
where h̄ is Planck’s constant (divided by 2π) and m is the particle mass.
(Such space-time intervals are very small even for elementary particle masses;
for instance, if m is the mass of a proton then h̄/m = 2×10−14cm or in time
units 6× 10−25sec. Recall that in our units 1sec = 3× 1010cm.) We are thus
faced again with our paradox; if one observer sees a particle emitted at x1,
and absorbed at x2, and if (x1 − x2)
2 − (x01 − x
2 is positive (but less than
or = h̄2/m2), then a second observer may see the particle absorbed at x2 at
a time t2 before the time t1 it is emitted at x1.
“There is only one known way out of this paradox. The second observer must
see a particle emitted at x2 and absorbed at x1. But in general the particle
seen by the second observer will then necessarily be different from that seen
by the first.”
There is another way to view (11). The light cone of special relativity viz.,
(x1−x2)
2− (x01−x
2 = 0 now gets somewhat distorted because of Quantum
Mechanical effects.
Let us consider the above in the context of a non zero photon mass. Such
a mass ∼ 10−65gms was recently deduced by the author, and it is not only
consistent with experimental restrictions, but also predicts a new effect viz.,
a residual cosmic radiation ∼ 10−33eV , which in fact has been observed
[11, 12, 16, 17, 18]. Such a photon would have a Compton length ∼ 1028cms,
that is the radius of the universe itself.
This would then lead to the following scenario: An observer would see a
photon leaving a particle A and then reaching another particle B, while a
different observer would see exactly the opposite for the same event - that
is a photon leaves B and travels backward in time to A, as in the Weinberg
interpretation. This latter gives the advanced potential. The distinction
between the advanced and retarded potentials of the old electromagnetic
theory thus gets mixed up and we have to consider both the advanced and
retarded potentials. We consider this in a little more detail: The advanced
and retarded solutions of the wave equation are given by the well known
advanced and retarded potentials given by, in the usual notation, the well
known expression
ret(adv)(x) =
jµ(x′)
|r − r′|
δ (|r − r′| ∓ c(t− t′)) d4x′
(The retarded part of which leads to the Lienard Wiechart potential of earlier
theory).
It can be seen in the above that we have the situation described within
the Compton wavelength, wherein there are two equivalent descriptions of
the same event–a photon leaving the charge A and reaching the charge B
or the photon leaving the charge B and reaching the charge A. The above
expression for the advanced and retarded potentials immediately leads to the
advanced and retarded fields (4) and (10) of the F-W description except that
we now have a rationale for this formulation in terms of the photon mass
and the photon compton wavelength rather than the perfect absorber ad hoc
prescription. In fact there is now an immediate explanation for this of the
Instantaneous Action At a Distance Theory alluded to. In this case the usual
causal electromagnetic field would be given by half the sum of the advanced
and retarded fields. We note that as the photon mass is so small, the usual
theory is still a good approximation.
To sum up [19], the Feynman Wheeler Perfect Absorber Theory required
that every charge should interact instantaneously with every other charge
in the universe, that is that the universe must be a perfect absorber of all
electromagnetic fields emanating from within. If this condition were satisfied,
then the net response of all charged particles along the future light cone
of the given charge is expressed by an integral that converges. We have
argued that this ad hoc prescription of Feynman and Wheeler as embodied
by the inclusion of the advanced potential is automatically satisfied if we
consider the photon to have a small mass 10−65gms as deduced by the author
elsewhere, and which is consistent with the latest experimental limits-this
leading to the effect mentioned by Weinberg within the Compton wavelength,
which is really the inclusion of the advanced field as well.
4 The Limits of Special Relativity
What we have witnessed above is that it is still possible to rescue the clas-
sical relativistic theory of the electron, but at the expense of introducing
the advanced fields into the physics, fields which have been considered to be
unphysical.
Another perspective is, as seen above, that there is instantaneous action at a
distance, which apparently goes against relativistic causality. But let us now
note that in both the Dirac and the Feynman-Wheeler approaches, we are no
longer dealing with point particles alone, but rather with a small neighbor-
hood of such a point particle, a neighborhood of a Compton length dimension.
Furthermore within the Compton scale, relativistic causality breaks down.
We can then reformulate the above considerations in the following manner:
The limit of applicability or the limit of validity of the Special Theory of
Relativity is the Compton scale. The points within the Compton scale no
longer obey Special Relativity and see a non relativistic, instantaneous action
at a distance universe.
5 Discussion
1. We would like to sum up the foregoing considerations. In Classical Physics
the point electron leads to infinite self energy via the term e2/R, where R is
the radius which is made to tend to zero. If on the other hand R does not
vanish, in other words we have an extended electron, then we have to intro-
duce non electromagnetic forces like the Poincare stresses for the stability of
this extended object, though on the positive side this allows the radiation
damping or self force that is required by conservation laws.
Dirac could get rid of the infinity by introducing the difference between the
advanced and retarded potentials: This was the content of the Lorentz Dirac
equation. The new term represents the radiation damping effect, but we then
have to contend with the advanced potential or equivalently a non locality in
time. However this non locality takes place within the Compton time, within
which the electron attains a luminal velocity.
The Lorentz Dirac equation on the other hand had unsatisfactory features
like the derivative of the acceleration, the non locality in time and the run
away solutions, features confined to the Compton scale.
The Feynman-Wheeler approach bypasses the infinity and the extended elec-
tron. Moreover the net result is that there is a retarded potential. But an
instantaneous interaction with the rest of the charges of the universe is re-
quired. It is this interaction with the remaining charges which leads to the
point electron’s self energy. Surprisingly however the interaction with the rest
of the charges in the immediate vicinity of the given charge in the Feynman-
Wheeler formula gives us back the Dirac antisymmetric difference with its
non locality within the Compton scale. There is thus a reconciliation of the
Dirac and the Feynman Wheeler approaches, once we bring into the picture,
the Compton scale.
In the Feynman-Wheeler approach on the other hand the self force is dis-
pensed with but at the expense of invoking the instantaneous interaction of
the electron in question with the rest of the charges in the universe, though
even here the Compton scale of the electron comes into question. Outside
this scale, the theory is causal that is uses only the retarded potential be-
cause effectively the advanced potential gets canceled out as it appears as the
sum of the symmetric and antisymmetric differences. The important point
however is that all this can be explained consistently in the context of the
photon having a non zero mass, consistent with experiment ∼ 10−65gms.
The final conclusion was that in a Classical context a totally electromagnetic
electron is impossible as also the concept of a point electron. It was believed
therefore that the electron was strictly speaking the subject of Quantum
Theory. Nevertheless in Dirac’s relativistic Quantum Electron, we again en-
counter the electron with the luminal velocity within the Compton scale,
precisely what was encountered in Classical Theory as well, as noted above.
This again is the feature of a point space time approach. At this stage a
new input was given by Dirac - meaningful physics required averages over
the Compton scale, in which process, the unphysical zitterbewegung effects
were eliminated. Nor has Quantum Field Theory solved the problem - one
has to take recourse to renormalization, and as pointed out by Rohrlich, one
still has a non electromagnetic electron. In any case, it appears that further
progress would come either from giving up point space time or from an elec-
tron that is extended (or has a sub structure) in some sense [3, 2, 7, 1].
2. Nevertheless it is curious to notice that there is some convergence be-
tween the Dirac and the Feynman-Wheeler approaches if we consider the
fact that special relativity, as seen above, does not hold within the Comp-
ton wavelength. This explains the non locality in time. This justifies the
use of the advanced potential or non locality in time of the Lorentz-Dirac
approach or also the fact that a point inside the Compton wavelength sees a
non relativistic instantaneous action at a distance universe around it - this
is the instantaneous action at a distance of the Feynman-Wheeler approach.
Furthermore, the radiation of photons emitted by the accelerated electrons
(in the Dirac self force) are meaningful only if they impinge on other charges
as in the Field Theory.
We now briefly note the following.
3. As Rohrlich [20] observes, ”Classical Physics ceases to be valid at or below
the Compton wavelength and this cannot be valid for a point object.”
4. The self interaction we encountered above gives rise to radiation reaction
for an extended object which for a point charge appears as a self acceleration
or pre acceleration and an extra inertia, the electromagnetic mass whose be-
havior was thought to contradict special relativity [3].
5. This apart the contradiction of run away accelerations or a divergent elec-
tromagnetic mass do not apply for an extended electron.
6. If we take special relativity into account, we get the undesirable factor
4/3 indicating that part of the mass is not electromagnetic - the beauty of
getting a unified theory for mass is lost.
7. Even in the non relativistic theory, Poincare stresses are required for the
stability of the extended electron.
8. On the other hand Fermi and others showed that relativistically the elec-
tromagnetic momentum need not be associated with the Poynting vector, in
which case the undesirable 4/3 factor does not arise and there is no need for
Poincare stresses.
9. In classical relativistic theory, there appeared an impasse. We could get
a special relativistic electron with cohesive forces in an extended model but
at the expense of purely electromagnetic electron. On the other hand point
electrons were not meaningful as their self energy diverged. Consequently
the structure dependent terms had to be taken seriously.
10. We have arrived at the Compton scale from two different approaches.
Classically, there was the electron radius and Quantum Mechanically the
Compton length, both of the same order except for a factor of the order of
the fine structure constant:
h̄/mc ∼ β · e2/mc2
We could consider this to a derivation of the value of the Planck constant of
Quantum Mechanics, in an order of magnitude sense.
11. In any case the above considerations at the Compton scale lead in recent
studies to a noncommutative geometry and the limit to a point particle no
longer becomes legitimate. This has been discussed in detail in [21].
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[18] Sidharth, B.G., physics/0608222.
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http://arxiv.org/abs/physics/0608222
	Introduction
	The Advanced and Retarded Fields
	Quantum Mechanical Considerations
	The Limits of Special Relativity
	Discussion
