The 1905 issues of the German journal Annalen der Physik contained four articles by Albert Einstein. One entitled in translation "On a Heuristic Viewpoint Concerning the Production and Transformation of Light" explained the photoelectric effect as being the result of photons of sufficient energy knocking electrons out special materials. Photons were identified as quanta of radiation having energy equal to hν where h is Planck's constant and ν is the freqency of the radiation. For this article he was awarded the Nobel Prize in Physics for 1921.

Another article, entitled in translation, “On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by the Molecular Kinetic Theory of Heat," explained Brownian motion.

A third was entitled in translation "On the Electrodynamics of Moving Bodies.” This article reconciled Maxwell's equations for the dynamics of electromagnetic fields with the laws of physical dynamics.

The fourth which history proved to be the most important of the four presents what Einstein called his Principle of Relativity which subsequently became known as Einstein's Theory of Special Relativity. It was entitled in translation "Does the Inertia of a Body Depend Upon Its Energy Content?”

1905 was Albert Einstein's Year of Miraculous Accomplishments. He was 26 years old.

The article presenting his Principle of Relativity was 31 pages long. In 1907 he published a paper, also in the Annalen der Physik, which was 59 pages long which elaborated the Special Theory of Relativity. It was entitled "On the Relativity Principle and the Conclusions Drawn from It." The 1907 article is fully as important for presenting the Theory of Special Relativity as the 1905 article. The concept of relativistic momentum is not mentioned until the 1907 article.

In Newtonian (non-relativistic) mechanics the linear momentum of a body is defined as the product of its mass m and its velocity v. It was found that this quantity is apparently conserved over time. Albert Einstein found that the apparent mass of a body in motion relative to an observer is given by

m = m₀/(1−β²))^½

where m₀ is the rest mass of the body and β is the velocity of the body relative to the speed of light c; i.e., β=v/c.

What Einstein wrote concerning momentum is

Thus the quantity ξ = μ(dx/dt)/(1 −(v²/c²))^½ plays the role of the momentum of the material point

where in his notation μ stands for rest mass.

Following Einstein it is apparently universally assumed that under Special Relativity linear momentum is given by the formula p=mv and the only relativistic correction is that the velocity-dependence of mass must be taken into account by taking

m = m₀/(1 − (v/c)²)^½

It will be shown below that according to Lagrangian analysis this is not the case. Instead there is an additional correction which must be taken into account.

Lagrangian Dynamics

Lagrangian dynamics provides a way to derive the formula for relativistic linear momentum rather than just assuming it. If K is the kinetic energy of a system and V is the potential energy then the Lagrangian of the system is defined as

L = K − V

If the Lagrangian of a system is a function of a set of variables {q_i; i=1,2,…,n} and their time derivatives {dq_i/dt; i=1,2,…,n} and the system is not subject to external forces then the dynamics of the system is given by the set of equations

d(∂L/∂v_i)/dt − (∂L/∂q_i) = 0

where v_i=dq_i/dt.

The expression (∂L/∂v_i) is the momentum with respect to the variable q_i.

The Newtonian Case

In Newtonian mechanics the kinetic energy of a body is ½mv² and v=dx/dt for some x. If the potential energy function does not depend upon x or v then

L = ½mv² − V
and therefore
∂L/∂v = mv

And since

∂L/∂x = 0
d(mv)/dt = 0
and thus linear momentum
is constant over time

The Relativistic Case

Now consider the relativistic case.

The expression ½mv² is not the full kinetic energy; it is just the first order approximation of the kinetic energy. The relativistic kinetic energy is instead

(mc²−m₀c²)
where m=m₀/(1−β²)^½
and β=v/c.

The Lagrangian for a particle moving in a potential field V is

L = (mc²−m₀c²) − V

If the potential energy is independent of v then the partial derivative of L with respect to v is

−½(m₀c²/(1−β²)^3/2)(−2β(1/c))
which reduces to
(m₀c²/(1−β²)^3/2)(v/c)(1/c)
or, equivalently
(m₀/(1−β²)^3/2)v
which can be expressed as
mv/(1−β²)

In other words, relativistic linear momentum requires not only the relativistic adjustment of mass but also division by a factor of (1−β²).

Thus if the kinetic and potential energies are independent of the spatial variable that defines velocity then it is mv/(1−β²) which is constant over time.

That is to say,

d(mv/(1−β²))/dt = 0

Apparently Einstein and the other physicists folllowing him gave some importance to the fact that as β=v/c goes to zero mv asymptotically approaches m₀v. But the expression m₀v/(1−β²)^3/2 also asymptotically approaches m₀v as β→0. In fact any expression of the form

f(β)mv + g(β)

asymptotically approaches m₀v as β→0 if f(β)→1 and g(β)→0 as (β→0.

Precedents

Herbert Goldstein's works on mechanics have long been considered the definitive source on the topic of classical mechanics. In the two editions of his Classical Mechanics, he refers to the longitudinal and the transverse masses for a body; i.e.,

m_l = m₀/(1−β²)^3/2
m_t = m₀/(1−β²)^1/2

where longitudinal means "in the direction of the motion" and tranverse is "perpendicular to the direction of motion." Transverse mass is just the relativistic mass and longitudinal mass is just the relativistic mass divided by (1−β²). This is the same results that came out of the analysis above. Another interpretation of the result of Lagrangian analysis is that the correct formula for relativistic momentum is longitudinal mass times velocity; i.e.,

p = m_lv

Joseph John Thomson developed these directional formulas for mass in 1881 and Oliver Heavyside worked out the mathematics in 1897. They were developed to explain why the acceleration of charged particles did not repond to force to the extent it was supposed to do so. Note that they were based upon empirical evidence.

Einstein's Justification of the
Conventional Formula

Einstein goes on in his 1907 article to represent the Hamiltonian total energy function H of a particle, in his notation, as

H = −μc²(1 −(v²/c²))^½ + const.

For H to equal zero when v=0 Einstein's constant must be μc². However this means at v=c the total energy is the finite value μc². Any finite value for total energy at v=c is contrary to the rest of Special Relativity.

Einstein goes on to say:

Further, one sees immediately that our equation of motion of the material point can be given the form of Lagrange's equations of motion [...]

Thus one sees that Einstein believed that relativistic momentum should be derived through Lagrangian analysis. But one also sees that Einstein so firmly believed in the formula for relativistic momentum being mv that he was willing to accept its derivation from a Lagrangian incompatible with relativistic kinetic energy.

Intellectually this is invalid. There is no support for the mv formula for relativistic momentum provided by showing that there is a pseudo-Lagrangian from which it can be derived by Lagrangian analysis. Any function of v can be justified by this procedure. One need only integrate the formula by velocity to get a supposed Lagrangian whose partial derivative with respect to velocity is the desired formula.

To illustrate, let s(v) be any function of v. Let S(v) be the indefinite integral of s(v) with respect to v; i.e.

∫s(v)dv = S(v) + C

where C is a constant. If (S(v) + C) is called a Lagrangian, say *L , then

(∂*L/∂v) = (d*L/dv) = s(v)

So s(v) is supposedly the momentum conjugate to the variable such that that v=(dx/dt). But a function is truly a Lagangian only if it is kinetic energy minus the potential energy.

Thus the v-dependent term of the Lagrangian is supposedly kinetic energy and when v=0 the kinetic energy is necessarily zero

S(0) + C = 0
C = −S(0)

The kinetic energy at of a particle moving at c, the speedd of light is then (S(c) − S(0))

Thus *L(c) = (S(c) − S(0)) which may or may not be infinite. But at v=c relativistic kinetic energy must be infinite because at v=c relativistic mass is infinite.

Thus the notion that the conventional formula for relativistic momentum is justified because there is some expression from which it can be derived by Lagrangian analysis is completely absurd. But this is what Einstein did to justify the conventional formula of mv for relativistic momentum.

Here is a graph of the comparison of relativistic kinetic energy to the supposed kinetic energy from Einstein's pseudo-Lagrangian as a function of relative velocity β. These kinetic energies are expressed in terms of their ratios to m₀c².

The values are so close for β<0.5 that they seem equal. Any empirical attempt to verify Einstein's formula for kinetic energy using values of β<0.5 would appear to succede, but only because they are so close to the true values for values of β in that range.

Here is a table of the errors in the Einstein formula for the entire range of β values.

The Treatment of the
Issue in Current Texts

Goldstein asserts that the Lagrangian of a particle is

L* = −m₀c²(1−β²)^½ −V
(page 206 equation 6-49)

Here is a graph of the supposed kinetic energy from Goldstein's pseudo-Lagrangian as a function of relative velocity β. expressed in terms of its ratio to m₀c².

Note that at β=0 this supposed kinetic energy is −m₀c² and at at β=1 this supposed kinetic energy is zero. Negative kinetic energy is of course complete nonsense. Apparently conventional physicists are so set on the formula mv for relativistic momentum that they are willing to accept a derivation of it from nonsense. The negativity and the nonzero value for the supposed kinetic energy at β=0 can be remedied by adding the term m₀c² to it, as does S.W. McCuskey in his An Introduction to Advanded Dynamics. But that does not take care of the finiteness of the value at β=1 and that finiteness is also nonsense.

Leonard Susskind and Art Friedman in their otherwise magnificent work Special Relativity and Classical Field have the same erroneous derivation of the conventional formula for relativistic momentum as do Einstein and the other authors of work dealing with relativistic dynamics; i.e., derivation of mv from a pseudo-Lagrangian involving negative kinetic energy or other nonsensical aspects. .

But Susskind and Friedman give what may be the reason they are apparently willing to accept an erroneous derivation of the convention formula rather than give it up. They say that Albert Einstein had a verbal argument justifying the conventional formula. Susskind and Friedman surprisingly do not use that argument of Einstein for the conventional formula. They cite the location of Einstein's argument as being in a 1905 article of Einstein.

A reading of all of the 1905 aricles of Einstein reveals that he did not mention relativistic momentum in any of them including the one in which he introduced his Principle of Relativity, now known as the Theory of Special Relativity. Incidentally that article notes the concepts of longitudial mass and transverse mass.

Einstein first mentions relativistic momentum in his long, 59 page article in 1907 which is entitled, "On the Relativity Principle and the Conclusions Drawn from It." (The original 1905 article for Relativity was only 31 pages long.)

Here is what Einstein wrote

Thus the quantity ξ = μ(dx/dt)/(1 −(v²/c²)^½ plays the role of the momentum of the material point

where in his notation μ stands for rest mass.

One also sees that Goldstein, Susskind and the other authors more or less blindly followed Einstein's treatment of the issue. Einstein apparently hoped that relativistic formulas could be found by substituting relativitic mass for rest mass in the Newtonian formula. This is obviously not true in the case of kinetic energy. Relativisic kinetic energy is not ½mv².

The issue can only be resolved empirically; i.e., which momentum, mv or mv/(1−β²) is conserved? In the limit as v→0 there is no difference between the two, so the settlement of the issue requires results for the case of β near unity. However if velocity does not change then any function of velocity is constant over time. Therefore the empirical test must involve a radical change in velocity as in a collision of particles of disparate masses.

Surprisingly the concepts and terms longitudinal mass and transverve mass preceded Special Relativity. As noted previously in the late 19th century various theorists noticed that charged bodies resisted acceleration more than is accounted for by their rest masses; i.e.,

m = F/a ≠ m₀

where F is force and a is acceleration

Conclusion

The Einstein formula of mv for relativistic momentum is simple and intuitive but wrong. Einstein's mistake in asserting that is easily corrected.

There appears to be no rigorous derivation of relativistic momententum as relativistic mass times velocity, mv. Based upon Lagrangian analysis relativistic linear momentum apparently requires not only the relativistic adjustment of mass but also division by a factor of (1−β²). The correct formula for relativistic momentum is then

P = m₀v/(1−β²)^3/2 = mv/(1−β²)

This is equivalent to relativistic momentum being equal to the product of longitudinal relativistic mass and velocity.

If the kinetic and potential energies are independent of the spatial variable that defines velocity then it is mv/(1−β²)=m_lv which is constant over time. If velocity does not change then any function of velocity is constant over time.

Not taking into account the division by (1−β²) means that a momentum of a really fast particle is drastically underestimated. The extent of the underestimation depends upon velocity.

Supposedly the conventional formula has been empirically verified. However it would indeed be strange if experiment truly verified a formula that has no rigorous derivation.

Thus while asserting that relativistic momentum is mv is a serious error on the part of Albert Einstein it in no way detracts from the validity of his Theory of Special Relativity. Unfortunately his prestige in physics has been such that even this error has been copied blindly.

(To be continued.)