The Mass-Energy Relation

In 1873 and 1876, respectively, James Clerk Maxwell (using his electromagnetic theory) and Adolfo Bartoli (using the laws of thermodynamics) hypothesized, then proved and verified theoretically, the proposition that light exerts pressure upon any body that absorbs or reflects it. If a beam of light carrying energy at a rate of dE/dt’ joules per second (watts) falls upon a body, it exerts a force

(Eq’n 1)

if the body absorbs it completely and it exerts twice that much force if the body reflects it entirely back the way whence it came. The latter part of that statement reflects our understanding that we can conceive reflection of light as an absorption and a re-emission (inverse absorption) of the radiation.

Inspired by the imaginary experiment that Bartoli used to guide his theorizing, Ludwig Boltzmann and Wilhelm Wien used Equation 1 to calculate the amount of work done on light by a moving reflective surface. Applying that calculation to blackbody radiation enclosed within a cavity, they discovered new facts that they added to the growing theory of radiation thermodynamics. But they could have done more by asking and answering two simple questions: What happens if the cavity moves? and What happens if the cavity accelerates?

If the cavity moves.

Imagine a cylindrical tube closed with flat plates at both ends and imagine that the entire inner surface of the tube and its end plates have perfect reflectivity; that is, that they absorb none of the radiation inside the tube. Assume that we introduce radiation into the cavity in such a way that the radiation propagates only in the direction parallel to the cylinder’s axis of circular symmetry. Bouncing to and fro, the radiation exerts equal and oppositely directed forces on the end plates, thereby exerting no net force on the cylinder. In order to calculate the total amount of radiant energy contained in the tube we multiply the rate at which energy approaches one of the end plates by the time that a packet of energy takes to make the round trip from and to the plate. If the tube has length L, then

(Eq’n 2)

If the tube moves at a uniform speed v in the direction parallel to its axis, the rear plate will do work upon the radiation striking it and the radiation will do work on the forward plate. If the radiation strikes the rear plate with power dE’/dt, the force exerted upon it by the plate does work at the rate

(Eq’n 3)

In re-emitting the radiation the plate adds that work to the radiation, so the radiation comes off the plate with power

(Eq’n 4)

When that radiation bounces off the forward plate, it does so with power

(Eq’n 5)

That calculation went remarkably wrong. It tells us that the radiation loses energy as it cycles within the tube, presumably losing the energy to the tube if the law of conservation of energy remains a valid part of Reality. But the radiation can’t lose energy to the tube in the form of heat, because our assumption of perfect reflectivity of the inner surface of the tube won’t allow the tube to absorb radiation in the necessary manner. Likewise, the forces between the end plates and the radiation can’t transfer net energy to the tube, because the energy would have to take the form of kinetic energy of the tube, expressed as an increase in the tube’s velocity, and thereby violate the law pertaining to conservation of linear momentum. So Equations 4 and 5 give us an incorrect description of this little piece of Reality.

However, in devising those equations I neglected to account for an obvious geometric factor that affects the flow of energy in the system. After bouncing off the rear plate a packet of radiation must, because of the tube’s motion, travel more than the distance L before it strikes the forward plate and after bouncing off the forward plate the packet will travel less than the distance L due to the rear plate moving to meet it. We can still think of the light traveling the distance L each way, but only if we assert that it does so effectively at the speed c-v in the forward direction and at the speed c+v in the rearward direction. Those pseudo-velocities correlate with the compression of the radiation’s energy density by the motion of the rear plate and with the expansion of the radiation’s energy density by the motion of the forward plate. Energy flow correlates directly with energy density in a beam of radiation, so in order to calculate the correct energy flow we must multiply the raw energy flow coming off the plates by, respectively, the proportions

(Eq’ns 6)

Thus Equation 4 must properly take the form

(Eq’n 7)

In that equation the coefficient that multiplies the raw energy flow equals the square of the formula that describes the relativistic Doppler upshift. The Doppler shift denotes the change in the frequencies of waves emanating from a moving source relative to the frequencies that an observer moving with the source would measure. That the square of the Doppler factor appears in Equation 7 tells us that the rate of energy flow has the nature of a squared frequency. We know from the quantum theory that the energy in a beam of radiation stands in direct proportion to the frequency of the radiation. And the time derivative itself has the mathematical form of a frequency (an assertion only fortified by the fact that we can derive the relativistic Doppler shift directly from the time equation of the Lorentz Transformation).

Likewise, Equation 5 must properly take the form

(Eq’n 8)

Thus we describe the radiation coming back to the rear plate carrying the same power with which it first struck that plate. In that description the system obeys the conservation laws, so we accept our analysis as yielding a valid description of this piece of Reality.

To calculate the total amount of energy contained in the tube, conceive the radiation as manifested in N photons with frequency ν. In the tube at rest, at any given instant, we find the same number of photons, half of the total, in each of the beams. The beams differ from each other only in their directions of propagation, so we have the total energy as E=Nhν.

In the moving tube the temporal distortion of a moving
inertial frame has pushed the rear plate into the future relative to the forward
plate by an amount Δt=Lv/c^{2}.
That interval allows a certain number of photons to move from the rearward beam
into the forward beam, an amount equal to the product of the raw linear density
of the photons (N/2L), the speed of their propagation, and the time interval;
thus, in the moving tube the forward and rearward beams contain, respectively, a
number of photons equal to

(Eq’n 9)

Multiplying each of those equations by Planck’s constant and the appropriate Doppler shifted frequency and then adding together the resulting products gives us the total energy contained in the radiation;

(Eq’n 10)

So we see that the total radiant energy inside the moving tube equals the total radiant energy inside the stationary tube modified by a simple upshift. The second term in the coefficient on the right represents the work done by the rear plate on those photons that got shifted from the rearward beam into the forward beam by the temporal offset.

If the cavity accelerates.

If the tube accelerates at a uniform rate dv/dt=a in the direction parallel to its axis, the difference in the number of photons in the beams, as denumerated in Equation 9, must change. Again imagine looking at the tube from a position near the forward plate. Initially the tube floats motionless, so each beam contains N/2 photons of a certain energy hí. When the tube accelerates, the difference between the number of photons in the beams changes in accordance with

(Eq’n 11)

due to the changing temporal offset at the rear plate. If we multiply that result by the amount of energy carried by each photon, we get

(Eq’n 12)

the rate at which the growing temporal offset at the rear plate shifts energy from one beam to the other. Note that this is not the rate at which energy goes from one beam to the other due to ordinary reflection off the plate.

Equation 1 tells us that Equation 12 represents a force,

(Eq’n 13)

That force, as with the ordinary reflection force, pushes the rear plate in the direction opposed to that of the acceleration. Consequently, whatever accelerates the tube must add that much force to its thrust in order to maintain the acceleration. The system will respond as if the tube had become more massive by the amount

(Eq’n 14)

Thus we see that the radiation inside the tube manifests an inertial reaction to the tube being accelerated.

That inertial reaction is no mere illusion with a passing resemblance to Einstein’s mass-energy equivalence. Imagine what would happen if the tube were suddenly to absorb the radiation and incorporate its energy into itself. The inertia of the tube can’t change, because doing so would constitute a violation of the law pertaining to conservation of linear momentum. The tube behaves dynamically as if it had a mass M+m. For an observer watching the tube fly by at some speed v, the tube plus radiation carries linear momentum Mv+mv. The momentum mv associated with the radiation cannot simply vanish, not without violating the conservation law, so when the tube absorbs the radiation, the energy still gives the tube the additional mass m. Thus Equation 14 does actually express Einstein’s mass-energy equivalence.

Appendix:

Imagine a flat mirror with a beam of light striking it at an angle perpendicular to the plane of the mirror’s surface. The beam consists of monochromatic light of frequency ν carrying power dE/dT=Kν/T, in which K represents the product of Planck’s constant and the number of photons that strike the mirror in the interval T.

Next imagine an observer moving toward the mirror, moving alongside the beam at a speed v relative to the mirror. Thus moving away from the source of the beam, that observer measures the frequency of the incident beam Doppler downshifted to

(Eq’n A-1)

and measures the frequency of the reflected beam Doppler upshifted to

(Eq’n A-2)

That latter equation merely takes the frequency of the radiation reflected by the mirror in its rest frame and converts it into the frequency that would be measured in the moving frame. Time intervals t measured by the moving observer relate to time intervals T measured by someone standing by the mirror as

(Eq’n A-3)

in accordance with Special Relativity. So the moving observer measures the energy flow in the incident beam as

(Eq’n A-4)

Likewise, the energy flow in the reflected beam comes to the moving observer as

(Eq’n A-5)

Of course, the moving observer cannot measure dE/dT directly, but they can determine the energy flow in the reflected beam relative to the energy flow in the incident beam:

(Eq’n A-6)

which conforms to Equation 7 in the text above.

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