C:\Myfiles\Webfiles\max_vel3.doc Originated: Feb. 4, 04
Aug. 21, 04
Comments on the Maximum Velocity Achievable by a Mass
Edward M Billinghurst, 5132 Stone Canyon Ave, Yorba Linda, CA 92886
Doppler Effect; Special Relativity; Lorentz Transformation
And other concepts
The purpose of this paper is to show that gravitational masses can, in
principle, exceed the speed of light.
‘ In principle’ means that there is no known
physical law that limits the maximum velocity.
Narrative Discussion
Consider two masses, separated by some appropriate
distance. Newton’s law of gravity says
that there is a reciprocal, attractive, force between the masses which is
proportional to their product and inversely proportional to the square of the
separation distance, R.
Force = G (a constant) * (M1 * M2 )/R2. Note that the two masses are simultaneously
sources and targets of force impulses, because of the reciprocal relationship.
Since the force is attractive, the masses move towards each other, with
increasing velocity. Each mass speeds
up. If the initial separation ‘R’ is
great enough, then at least one mass, can in principle, exceed the speed of
light, which is a finite 300 million meters/second. There is no indication
in the gravitational law, as stated by Newton, that there is any limit on the
maximum velocity that can be achieved.
The velocities of the two masses do not have to be equal in
magnitude. Of course, if the two masses
collide, then they bounce apart with some velocity determined at the time of
impact, and they start moving away from each other. But they are still under the attractive force, so after some
amount of time, they begin to move towards each other again. If there are no other forces in their
vicinity, the two masses will continue the cyclic operation. The impact of collision will limit the
maximum velocity, but this is due to circumstances, not to any law which states
a limit.
Christian Doppler and Einstein showed that Newton’s Gravitational Force law must be modified to include the effect of the finite velocities of the masses, and to include the finite propagation velocity associated with any change in the position of either mass. For convenience, we will assume the propagation velocity is also equal to ‘c’. For the purpose of this discussion, the actual value is not important; only that it is finite, not instantaneous as Newton had implicitly assumed. We apply the modification to the ‘G’ term.
‘G’ as modified by the Doppler effect. ‘G’ becomes G multiplied by the relative velocity with respect to the speed of light ‘c’, of one mass divided by the relative velocity of the second mass. These two relative velocities may be different.
In symbolic terms, we get G * (c + v1)/(c – v2), for M1, and G * (c + v2)/(c – v1) for M2. Note that there are two separate transactions going on, because of the reciprocal nature of the force between the masses. Each velocity enters into the G coefficient, but with a different quantitative effect. The lighter mass will be accelerated faster than the heavier mass.
What this means is that one mass can exceed the speed of light while the second mass is still at a velocity less than the speed of light. If we consider the case where M1 is larger than M2, and focus on M2, we see that v2 can exceed c while M1 is still at a velocity less than c. This is crucial to the argument that a mass can exceed c. If the velocity of M1 becomes equal to or greater than c, then M2 will no longer receive any accelerating force. It is not apparent from the equations, but the physical significance of the velocity, v1, being greater than c means that no accelerating impulses can escape from M1; they will be swallowed up by M1. Mathematically, it looks like G would go to infinity if (c – v1) goes to zero, but physically it actually effectively goes to zero. This paradox illustrates the fact, often overlooked, that extreme care must be taken in interpreting mathematical results to make sure that they are consistent with physical reality. It is analogous to assuming that if G goes to infinity, then infinite acceleration and hence infinite energy is implied. But physically, we know that is not true. The target can never receive more energy than is emitted from the source. The justification appears in the discussion below in the derivation of the Doppler effect.
The Doppler effect modification given above applies to cases
of masses under attractive force. For
the case of repulsive forces, we get: G * (c - v2)/(c + v1). This is the case for which Einstein made his
famous statement that no material body can exceed the speed of light. That prediction followed from the
mathematical fact that the G function goes to zero when
the material body velocity equals c, so no further acceleration can
occur; the accelerating force goes to zero.
This discussion is not meant to prove that masses can, in practice, exceed the speed of light. Its main purpose is to show that the Lorentz Transformation cannot be used as a proof that the speed of light cannot be exceeded. Einstein’s statement does not follow from the LT as it applies to attractive, gravitational or like charge, systems.
Whether enough energy can be transferred from one gravitational mass to another to cause the target mass to exceed the speed of light requires a much more detailed analysis. This paper does not address that question.
Technical Discussion
From “The Meaning of Relativity”, Princeton University Press, A. Einstein, page 34
X’ = ((x – vt)/c)/(o(c – v)(c + v)), t’ = ((t – (v/c2)*x)/c)/(o(c – v)(c + v)), (Eqn 29), restated.
Note that (c – v)(c + v) = (c2 – v2).
The ‘primed’ terms, x’ and t’, relate to a moving body; unprimed terms relate to a stationary body.
These are the equations known as the Lorentz Transformation, developed by Einstein .
Footnote, page 38 “That material velocities exceeding that of the speed of light are not possible follows from the appearance of the radical, o((c – v)(c + v)), in the Special Lorentz Transformation, Eq (29)”. But, at that point, Einstein was discussing the repulsive case only. He did not comment on the attractive, gravitation or unlike electric charge, cases.
For the repulsive case, it does follow from the Lorentz equations that the speed of light cannot be exceeded.
After noting that (x = ct), algebraically restating the equations, and dropping the radical symbol, we get:
X’ = t*(c – vT)/(c + vS), ct’ = t*(c – vT)/(c + vS).
The radical is only relevant to a quantitative analysis, it is not relative to a qualitative discussion such as this.
The algebraic re-arrangement is made to get the x’ and t’ identities into the form that shows their dependence on the Doppler effect. Which should have been Einstein’s starting point. Einstein, in “The Principle of Relativity, Dover, page 57, ended up with the equivalent of the two identities immediately above, when he showed that the Lorentz Transformation is consistent with the Doppler effect equations.
The justification for dropping the radical is based on the fact that it is not needed. Einstein, in his development, only required that x’/t’ = c, for a moving body, to agree with the zero velocity equation x/t = c, and to agree with his first postulate that physical laws have to be the same everywhere and everywhen in the universe. The radical terms cancel out in the division operation; therefore the radical term can be anything without effect on x’/t’ = c.
The Doppler effect is usually stated as the frequency of radiation perceived at a moving target, compared to the frequency of emission from a moving source. However, the acceleration of a moving body depends on the frequency of the radiation, so the Doppler effect also applies to the acceleration. Einstein, in his initial paper, was concerned with the dynamics of two masses or entities, that is, the effect of velocity. That was his motivation for his dissertation. The subscripts on the velocity terms are added to emphasize that the two velocities involved are not the same. VT is the symbol for the velocity of the observer (target); vS is the symbol for the velocity of the source. Einstein, in his derivation of the Lorentz Transformation did not make this distinction, but as will be seen later, this distinction is crucial to an understanding of what is happening in any dynamic physical system.
Note that ‘c’ and ‘v’ are vectors, subject to the rules of vector algebra.
The symbolism of the term (c – v) is that the two vectors are in the same direction. The term (c + v) indicates that the two vectors are in opposite directions. The case when the two vectors are at some other angular relationship will not be discussed in this paper. It is not relevant to our purpose of showing that a mass can exceed the velocity of light.
Einstein quote “…radiation conveys inertia [momentum]
between the emitting [source] and absorbing [target] bodies.” Acceleration is caused by momentum
transfer. Dimensionally, momentum is
(Force * time). Light radiation
involves photons, which have momentum.
For gravity, the momentum particles are ‘gravitons’? Since gravitons are attractive, it would
appear that their momentum vector is opposite to the propagation vector. Efforts are on-going to attempt to detect
gravitons.
Conventional, well
established, Physics theory is used throughout the discussion. No ad hoc premises are invoked.
The Doppler Effect (shift) is the logical starting point for
any discussion involving moving bodies.
V1 à
O-----------à c ß---------------O ß V2
M1 M2
The defining equation relating frequency, wavelength, and propagation velocity is: f*8 = c.
The velocity terms, c and v, are
vectors. Terms of the form (c – v) indicate that
the vector, v,
is in the same direction as the radiation vector, c . Term (c + v) indicates
the velocity is opposed to c.
Time separation of Impulses: | < Delta TS >| < Delta TS >| etc
Momentum Impulses are separated in time at fixed
intervals of )TS, where )TS
= 1/fS . They propagate at
velocity c. The time
separation is invariant, set by the source, and never changes.
Spatial separation: |< vS*)TS >|< (c - vS)*)TS >|< (c - vS)*)TS >|< (c - <vS)*)TS >|
The wavelength, lambda, in space is: 8X = (c – vS)*)TS. Wavelength in space is set by the source velocity.
The time interval between impulses received at the target
is: )TT
= 8X
/(c + vT ); the wavelength divided by the
relative velocity of the target velocity referenced to c.
The Doppler frequency of Impulses received by the target is: fT = fS
*((c + vT )/ (c - vS)); the target acceleration is
proportional to fT .
Impulses can be received at the target at a greater burst rate than the
emission frequency, because the target is receiving impulses in the pipeline
between the source and target. The
target is moving into the emitted impulse train.
There are two independent causes of the Doppler effect; one caused by the
source vS, which sets the spatial wavelength, the other by the target vT, which determines the frequency of impulses
detected at the target.
Significance of the denominator term, (c - vS), in the fT equation. On its face, this term goes to zero when the relative velocity is zero, which would appear to make the frequency become infinite. But this is a case of the mathematical manipulations giving a result that is not physically realizable. Physically, the frequency cannot be infinite. The target cannot receive more energy or momentum impulses than are emitted by the source. The number of impulses received at the target in any given time interval is equal to the number emitted less the ones that are still enroute to the target. Bursts of target frequency may get fairly high, but the average number is limited as noted. One interpretation is that the wavelength between impulses approaches zero and then no impulses can escape the source. They would be swallowed up by the source, possibly. What actually happens is not clear to me; that determination perhaps is best left to quantum theory. However, for the purpose of this paper, we are not concerned with vS approaching c. I have defined the problem so that cannot happen. M1, being defined as greater than M2 for this case means that M1 cannot reach the velocity of light. When M2 exceeds c, it can no longer emit impulses in the direction of M1 , so M1 it no longer accelerates. It remains at the velocity it had when the velocity of M2 reached c. But M2 can continue to accelerate beyond c because it is still receiving impulses.
An alternate explanation of the significance of (c – vS). If vS becomes equal to c, the spatial separation between impulses becomes zero, all the impulses are together; propagating together. The impact at the target is one giant impulse; the two masses collide, in effect. If vS becomes greater than c, then successive impulses are emitted at a position closer to the target than preceding ones; the separation then is determined by the magnitude of (c – vS). This does not imply that successive impulses were released in time before the previously emitted impulses; the time separation of impulses is always )TS. The observer at the target only detects successive impulses, with no knowledge of the relative time of emission; only the time delta. I do not know what this means, but it is interesting to ponder. I mention it primarily to rebut the argument which I have seen made that a velocity of greater than c implies an effect which precedes its cause, in time. Not so.
From this, it is clear that one must consider the entire system of two masses. Two masses are required, since each one has to supply accelerating impulses to the other. The acceleration of each mass is a separate event; the two events cannot be multiplied together for one equation. The accelerations must be independently evaluated. The Lorentz Transformation is not applicable to acceleration analysis. The Doppler effect rules, separately, on each mass. One mass by itself does not receive any acceleration. By looking at this total picture, it is plausible that one mass can exceed the speed of light. It only takes one proposed case to refute Einstein’s statement that no mass can exceed the speed of light. One refutation invalidates a theory. No theory can be proved to be right unless all possible conclusions contained in the theory can be experimentally tested. Experiments that agree with a theory corroborate, but do not prove, a theory. Of course, it may be pragmatically impossible to experimentally test my theory for practical reasons. How could an earth bound astronomer ever find and measure the velocity of such an event? It may be reasonable to assume that objects being drawn into a very massive object could exceed c, but who knows?
And who cares?
Q. E. D.