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Units, Constants and Useful Formulas
Newton versus Einstein
------------------------
In flat space Newtonian mechanics, the motion of a uniformly accelerated object
is described by a parabola and follows the familiar SUVAT equation x = (1/2)at^{2}.
Things are more complicated in special relativity. Since v = at, as t
increases v would eventually be greater than the velocity of light, which
is not allowed.
Instead, we must think of the motion of an object in terms of hyperbolas
through flat space which are given by an equation of the form
x^{2}- c^{2}t^{2} = c^{4}/a^{2}. The hyperbola is asymptotic to the light cone* and
therefore exceeding the velocity of light cannot happen. An observer traveling
along the worldwide experiences the same acceleration at all times. We
will see that the hyperbola is Lorentz invariant.
* A light cone is the path that a flash of light, emanating from a
single event (localized to a single point in space and a single
moment in time) and traveling in all directions, would take
through spacetime.
Einstein (Special Relativity)
-----------------------------
The laws of physics are the same in all inertial frames of reference.
An inertial frame of reference (or Galilean reference frame) is a frame of
reference in which Newton's first law of motion applies. All inertial
frames are in a state of constant, rectilinear motion with respect to
one another; they are not accelerating (in the sense of proper acceleration
that would be detected by an accelerometer). Measurements in one inertial
frame can be converted to measurements in another by a simple transformation
(the Galilean transformation in Newtonian physics and the Lorentz transformation
in special relativity). In general relativity, an inertial reference frame
is only an approximation that applies in a region that is small enough
for the curvature of space to be negligible.
The speed of light in a vacuum is a universal constant, c, which is
independent of the motion of the light source/observer (Michelson and Morley). An
observer attempting to measure the speed of light's propagation will get exactly
the same answer no matter how the observer or the system's components are moving.
No physical object, message or field line can travel faster than the speed of light
in a vacuum. In the case of a mass it would take an infinite amount of energy
(see E = mc^{2}/(1 - √v^{2}/c^{2})) to accelerate the mass to that speed which is impossible.
Faster than Light?
------------------
FTL is the transmission of information or matter faster than c. This is not quite
the same as traveling faster than light, since:
In the following examples, certain influences may appear to travel faster than light,
but they do not convey energy or information faster than light, so they do not
violate special relativity.
- Light travels at speed c/n when not in a vacuum but travelling through a medium
with refractive index = n (causing refraction), and in some materials other
particles can travel faster than c/n (but still slower than c),
leading to Cherenkov radiation.
- If a laser pointed at the moon is rastered, the light spot on the moon can
almost certainly be made to move at a speed greater than c.
... Masses are not involved and there is no information carried.
- 2 galaxies moving away from each other can have combined velocities that
exceed the speed of light.
... It is space that is increasing because of the expansion of the universe.
The 2 galaxies can be considered as being stationary. It is the "empty"
space that is expanding.
- 2 fast-moving particles approaching each other from opposite sides. From
the point of view of an observer standing at rest relative to the
accelerator, this rate will be slightly less than twice the speed of light.
... Special relativity does not prohibit this. It tells us that it is
wrong to use Galilean relativity to compute the velocity of one of
the particles, as would be measured by an observer traveling alongside
the other particle. That is, special relativity gives the right
formula for computing such relative velocity.
- The phase velocity of an electromagnetic wave, when traveling through a
medium, can routinely exceed c. For example, this occurs in most glasses
at X-ray frequencies.
... The phase velocity of a wave corresponds to the propagation speed
of a theoretical single-frequency (purely monochromatic) component
of the wave at that frequency. Such a wave component must be infinite
in extent and of constant amplitude (otherwise it is not truly
monochromatic), and so cannot convey any information. Thus a phase
velocity above c does not imply the propagation of signals with a
velocity above c.
- EPR (quantum entanglements)
- Casimir