From: Anders Sandberg (asa@nada.kth.se)
Date: Sun Jun 08 2003 - 01:48:03 MDT
On Sat, Jun 07, 2003 at 05:05:34PM -0700, Party of Citizens wrote:
> If C is defined as the fastest speed in the universe, does that change the
> result of E=MC2 calculations? Or could it be that they are invalid and
> have never been empirically validated?
No. The c of importance is the velocity that is left invariant by all
Lorenz transformations. It just happens historically that it was first
empirically observed that light moves at the same velocity regardless of
the speed of the observer, and then the attempts to explain this
observation led to the relativity postulates (roughly, that there exists
this kind of invariant velocity and that physics looks the same in all
inertial frames). From this one can then derive testable (and indeed
tested) predictions like length contraction, time dilation and
mass-energy equivalence. E=mc^2 is a consequence of the postulates.
Now, if one tries to use a c in the calculations that is not the
invariant speed then there will be a systematic discrepancy from the
momentum and time dilation effects observed in every particle
accelerator. It could of course be that physical light moves slightly
slower than the invariant velocity (light moves slower than c in many
media, and it is a common textbook exercise to show how the relativity
equations would look if everything was filled with a refractive ether
that made it slower), but in this case the difference has to be
extremely small and cosmological constraints (see below) limit it.
If something moves faster than the invariant speed there will be
reference frames where it moves at arbitrary speeds, including backwards
in time. So FTL gravity waves would likely end up having the same bad
effects on cosmology as finite speed photons, even when ignoring the
causality issue.
To sum up, special relativity has a very simple and elegant mathematical
form. The invariant velocity c is a bit like the invariant number 1 in
multiplication; it has to exist in the group structure of Lorenz
transformations. It does not say anything about whether there are things
that move at that speed or not in physics, or even whether physics obeys
the Lorenz group. That is just experimental observations, which seem to
fit extremely well.
[Cosmological constraints: Because if photons can move at other
velocities than the invariant velocity (if they move slowly we can
always set up a fast moving lab moving alongside to examine photons at
rest - this is actually a good Gedankenexperiment to see why
electromagnetic waves have to move at the invariant speed; you can't
have a stationary wave packet in vacuum) they would have a rest mass.
Given the relativity equations (still valid even if light is slow) and
the observable energy E of a photon at speed v we get:
m_0 = E^2/[v^2 c^2/(c^2-v^2) + c^4]
If v is close to c and E close to the observed pc value, then m_0 will
be extremely small. Putting E=h nu (Planck's relation) into it we get a
rest mass for photons of frequency nu:
m_0 = h^2 nu^2/[v^2 c^2/(c^2-v^2) + c^4]
Note that for small nu m_0 becomes even smaller.
Particles with a finite rest mass m_0 will be generated in processes
above energy 2 m_0 c^2 (another experimentally observed fact). This
means that during the big bang a certain fraction of the energy ought to
have ended up as photons just as it did with neutrinos and
electron/positron pairs. But adding a rest mass completely messes up the
amount of photons at different frequencies - the spectrum would not end
up as the blackbody spectrum we all know and love, but rather look like
the other particle spectra that are predicted in cosmology. In addition,
a lot of slow photons would have been able to manifest at lower
energies, cooling down the universe and objects in it much faster than
otherwise. In fact, it would change all blackbody spectra, so I guess
the constraint isn't really cosmological but experimental - if photons
could move slower than the invariant speed we would see a very different
world. ]
-- ----------------------------------------------------------------------- Anders Sandberg Towards Ascension! asa@nada.kth.se http://www.nada.kth.se/~asa/ GCS/M/S/O d++ -p+ c++++ !l u+ e++ m++ s+/+ n--- h+/* f+ g+ w++ t+ r+ !y
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