User:Count Iblis/Speed of light

The speed of light, usually denoted by c, is a conversion factor that crops up in certain equations for the purpose of allowing one to use inconsistent units (i.e., units that are different from natural units) for distances and time intervals. The factor of c is inserted in equations valid in natural units by replacing all time variables t by t = c t'. Under this change of variables, velocities transform according to:

${\frac {dx}{dt'}}=c{\frac {dx}{dt}}$

Since light always propagates at the speed of 1 in natural units, the value of c is identical to the speed of light in the new units.

The classical limit

The rescaling of the time variable t = c t' can be used to study the interaction of objects at extremely slow speeds. We can investigate this by studying the way we need to rescale energy and momentum when we rescale the time variable so that we don't get singular equations for energy and momentum conservation in the limit c to infinity. Before any rescaling, we have:

$E=\gamma (v)E_{0}$

${\vec {P}}=\gamma (v)E_{0}{\vec {v}}$

where

$\gamma (v)={\frac {1}{\sqrt {1-v^{2}}}}$

We put ${\vec {v}}={\vec {v'}}/c$ and assume that v' is kept finite while c is sent to infinity. This amounts to sending the velocity to zero while zooming in into the low velocity world so that it remains visible. To make the velocity dependence of the energy visible we need to expand it to second order in ${\vec {v'}}/c$, while the velocity dependence of momentum appears at zeroth order. Replacing the expressions for the energy and momentum by these expansions, gives:

$E=E_{0}\left(1+{\frac {v'^{2}}{2c^{2}}}\right)$

${\vec {P}}=E_{0}{\frac {\vec {v'}}{c}}$

Now, consider an elastic collision for which the above expressions for the energy and momentum are applicable, i.e. v' is finite. Then, demanding that we have conservation of momentum in any arbitrary frame, yields that both the sum of the rest rest energies and the sum of the momenta are separately conserved. Energy conservation then implies that that the the sum of the kinetic energies

$T=E_{0}{\frac {v'^{2}}{2c^{2}}}$

is conserved. The two terms in the energy equation scale in different ways, so we have to introduce a new variable to make sure that both terms remain visible in the c to infinity limit. We can e.g. make the kinetic energy finite by putting

$E_{0}=mc^{2}$

where m is assumed to stay finite in the c to infinity limit. We then have to rescale the momentum:

${\vec {P'}}={\frac {\vec {P}}{c}}=m{\vec {v'}}$

The total rescaled momentum, the kinetic energy and the "mass" m are then finite quantities that are conserved in the c to infinity limit. Note that we end up with this result irrespective of how we make the quantities finite. We could just as well have kept $E_{0}$ finite. If we then call this the mass m:

$E_{0}=m$

then we need to rescale the momentum according to:

${\vec {P'}}={\vec {P}}c=m{\vec {v'}}$

and the kinetic energy needs to be rescaled according to:

$T'=Tc^{2}={\frac {1}{2}}mv'^{2}$

What then happens is that the rest energy $E_{0}$ does not scale in the same way as the kinetic energy. If we then define a rescaled rest energy $E_{0}'$ that does scale in the same way as the kinetic energy, we get:

$E_{0}'=E_{0}c^{2}=mc^{2}$

So, we see that in the scaling limit we end up with three finite independent conserved quantities, mass, kinetic energy and momentum and that if we formally keep the rescaling parameter c, the relation between the rest energy expressed in the finite kinetic energy units and the mass is $E_{0}=mc^{2}$.

The classical perspective

From the point of view of a classical observer, there are three conserved quantities: mass, energy and momentum. Above we've considered the special case of elastic collisions in which total kinetic energy of conserved. The classical world is obtained from the relativistic world by applying an infinite rescaling of the time coordinate and then properly scaling the physical quantities so that they stay finite in the infinite scaling limit. Since different quantities can scale in different ways, this leads to certain relations between quantities being lost in this scaling limit. E.g. the properly rescaled rest energy is the mass and its relation with the rest energy is lost as it is E = m c^2. This has e.g. consequences for inelastic collisions where part of the energy ends up in the rest energy. In the relativistic case the total energy appears at the kinematic level, while at the classical level only the kinetic energy appears and thus conservation of energy is not a useful equation here.

Because the classical world is the infinite scaling limit of the real world, it follows that it stays invariant under any further rescaling. This means that there is no way a classical physicist can relate his units for e.g. distances and time, they are completely independent of each other. In reality such independent units are related to each other via powers of the infinite rescaling constant. The classical physicist will have to assign independent dimensions to such units. It then follows that when the not really 100% classical physicst discovers relativity, the constant c appears in equations, it will have dimensions of Length/Time, with Length and Time being supposedly incompatible quantities. Also this constant c will be the speed of light. Since the speed at which light travels in a vacuum seems to be a physical property of light, this then leads to opposition to adopting the correct point of view that c is really just an irrelevant scaling parameter that can be set equal to 1 and thus has no fundamental significance whatsoever.