Monday, 14 December 2009

Momentum and Kinetic Energy

Fast car with motion blurKinetic energy is a slippery subject under relativity theory, so I try to give it a wide berth wherever possible. :) Here's how it behaved under C20th theory:
  • Under "early SR", kinetic energy seemed to appear in the equations in a quite an intuitive form … if we converted a body to light, and moved past the body at velocity v, different parts of the light-complex would be redshifted and blueshifted by Doppler effects, but the overall summed energy (according to the LET/SR relationships) would always end up being increased by the Lorentz factor, 1:SQRT[1 - vv/cc]. We could interpret the increase rather nicely by arguing that, since we could use SR to describe a “moving” body as being Lorentz time-dilated, this should translate into an apparent increase in the object's inertia (and its effective inertial mass), and by applying the E=mc^2 equation to that enhanced “relativistic mass” value, we arrived at the appropriate “enhanced” value for the total energy of the light-complex under SR. This additional energy due to motion wasn't the traditional “half em vee squared”, so our original schoolbook arguments and derivations for KE weren't quite correct in the new context except as a low-velocity first approximation.

  • With modern, “Minkowskian” SR, the subject of special relativity evolved. It cast off some earlier concepts from Lorentzian electrodynamics and Lorentz Ether Theory, and ended up as the theory of the geometrical properties of Minkowski spacetime. The idea of relativistic mass got downgraded, and some of the more mathy people started to say that the concept of relativistic mass was “bad” and shouldn't be taught. What was important (they said) were just two things: (1) the rest massenergy of a particle, and (2) its path through spacetime. Everything else was a derivative of these two things, and the conserved property wasn't either momentum or kinetic energy, but a new hybrid thing, called momenergy. But they kept the extra Lorentz-factor when calcuating things like SR momentum.
    So with the appearance of “modern SR”, our concept of what kinetic energy was (and what it ought to be) changed again. Taylor and Wheeler's “Spacetime Physics” (2nd ed., 1992) has a useful chapter on momenergy (§7) that works through this “Minkowskian” approach.

  • Under general relativity things got even more slippery, because we arguably moved from the concept of “simple” kinetic energy towards that of physical, recoverable kinetic energy that could be expressed as a change in shape of the metric. Defining the nominal energy contained in a region gets difficult unless we also define the properties of other neighbouring regions that it might interact with. Even if we know all the masses and velocities involved, the effective resulting energy can also depend on their distributions and arrangements.

If we switch to the “redder” equations suggested in the book, things change again. Because each ray is now redder than its SR counterpart by the Lorentz factor, that “nice” SR result that the totalled energy of the emitted light increases with velocity by the Lorentz factor vanishes. Now, the sum of all the energies of the rays gives exactly the same value regardless of the relative speed between the observer and the experiment. Adding the ray-energies together gives a fixed value that represents just the rest massenergy of the original body. So (as someone asked, after the Newton and E=mc^2 post) where did the kinetic energy component go?

Well, in this system, the thing that describes the original body's ability to do work due to its motion isn't just the total summed energy of the rays, but the total ray-energy multiplied by the asymmetricality of its distribution.
Suppose that we instead took an electrical charge and distorted its field – it'd now have have two energy components: the default energy associated with the electric charge, and an additional energy due to the way that the effect of that charge was distorted (like the energy bound up in a stretched or squashed spring). If an electrical charge is seen to be moving, its field seems distorted due to aberration effects, and it therefore carries an additional chunk of energy, even though the "quantity of field" is the same. Similarly, in our gravitomagnetic model, we have a moving gravitational charge whose "quantity of field" is the same for all velocities, but whose velocity-dependent distortion carries an additional whack. When we then convert the body to trapped light, the total energy of all the individual rays corresponds to just the "rest field" or the "rest massenergy", and the original body's kinetic energy shows up as the apparent additional energy-imbalance across the light-complex, due to the fact that it's moving.

Thought-experiments: If a “stationary” body is converted to light, the resulting distribution of light-energy is completely symmetrical. No asymmetry means no equivalent kinetic energy. Add energy symmetrically to the light-complex and then convert it back into matter, and because the resulting body still has zero overall momentum, the added energy has to translate into additional rest mass rather than KE. Add energy to the complex asymmetrically, and the imbalance means that the resulting mass now has to be “moving” wrt the original state in order to preserve our introduced asymmetricality, and appears as kinetic energy. Remove energy from the original balanced complex, asymmetrically, and you again create motion and KE in the resulting object (although the reconstituted body now has a smaller rest mass thanks to the energy that you stole).
In this model, the kinetic energy for a simple “moving” point-particle doesn't show up in a simple calculation of summed energy values for the equivalent light-complex. It appears as the energy-differential across the light-complex caused by the way that that rest energy is redistributed in the light-complex due to the original body's motion.

To find the asymmetry of energies in a light-complex, we can use a vector-summing approach, which allows ray-energies to cancel out if they're aimed in opposite directions, leaving us with a residual measure of the differential energy ... which relates to the net momentum of the light-complex.

So under the revised model, there are two energy values to consider, the "quantity of field" associated with a particle, and the angular distribution of how that first energy-field is arranged. The first one's the rest massenergy, the second's the kinetic energy.

This isn't the usual way of doing things, but it arguably gives us a more minimalist logical structure than the one used by special relativity. Under SR, the motional energy of a particle shows up in the geometry twice – first in the gross quantity of energy, and then a second time in the redistribution of that Lorentz-increased energy due to velocity – and we have to do some odd-looking four-dimensional things to cancel one from the other. With this system, the quantity only appears once, as the asymmetrical distribution of the fixed quantity of rest-energy.

To me this looks like it might be a more elegant way of doing physics.

1 comment:

Michael said...

The term "mass" has been clarified and is now distinguished from "relativistic mass", but kinetic energy has not been redefined, it is still the energy obtained in bringing a moving body to a halt. Whatever name is given to this energy, it is problematic for Newtonian optics and mass-energy equivalence. You point out that a moving body transformed into energy produces an unbalanced energy complex, but I still fail to see how this imbalance includes the kinetic energy as well as momentum. Momentum can be cancelled by opposing motion whereas kinetic energy cannot, as the following 2-part thought experiment shows:

Consider a system of 2 particles, each of (rest) mass m, on a collision course viewed in the centre of momentum frame of reference, so that the net momentum is zero. One way to use up the kinetic energy would be to have the particles operate a dynamo and we could use the generated electricity until the particles came to rest in our chosen frame of reference, after which the total mass of the stationary particles is 2m and the kinetic energy has been transformed into electricity.

Now we restart the experiment, but this time the moving particles self-annihilate into energy. Each energy complex is unbalanced to conserve the momentum of its original particle, but in combination there is a net momentum of zero as before. The problem with the Newtonian emission derivation that you outlined in the book is that the annihilation energy of each particle is the same (mc²) in any inertial frame of reference, including our chosen frame where there is no overall imbalance and no overall momentum. So if the energy condenses into mass once again, there is only enough to create a stationary particle of mass 2m. The kinetic energy that we used to generate electricity in the previous run has been lost here, so the question remains, where did the kinetic energy go?

There is a simple resolution to this dilemma; Newtonian emission theory and mass-energy equivalence are incompatible.

Happy New Year! - Michael