Although this website was designed for the nonscientist, we believe that with only an
elementary education in mathematics, an understanding of the underlying
mathematics of time travel should be fine. To keep things simple, we are only going to cover basic relativity, Lorentz transformations, Dirac's negative
mass energy and the time travel energy equation. If you feel this section should be expanded further please let me know and email me.
Pythagoras into Einstein in Two Easy Steps
So
where do we start? Well let us start with one of the greatest triumphs of the
human mind, the great theorem of Pythagoras, a true pillar of all mathematics
and physics. The theorem, which is applicable to right angled triangles in flat
Cartesian (Newtonian) space takes the form of:
c^{2} = a^{2} + b^{2}
where a, b and c are the lengths of the sides of the triangle.
Next we will jump straight to Einstein's theory of Relativity which states that neither time, length, or indeed mass remain constant
additive quantities when approaching the speed of light c. Our simple ideas of
time and space come from the fact the we are so used
to living in a three dimensional universe. Einstein showed that this was simply
not true and in fact all the "foundational" three laws of Newton have to be
fudged by the Lorentz factor
L_{f} = (1  v^{2}/c^{2})^{ 1/2}
Elementary Guide to Relativity
There
are, however, certain quantities that do remain constant. These constants are
related to fourdimensional quantities known as metric tensors. From this
Einstein proved that space and time are two aspects of the same thing and that
matter and energy are also two aspects of the same thing. From the second of
these concepts we get the most famous equation in physics
E = mc^{2}
Now since time and space are aspects of spacetime and we wish to travel through
time and not build atom bombs we will leave E=mc^2 for the moment. To illustrate
this, look at the extension of Pythagorean theorem for the distance, d, between
two points in space:
d^{2} = x^{2} + y^{2} + z^{2}
where x, y and z are the lengths, or more correctly the difference in the
coordinates, in each of the three spatial directions. This distance remains
constant for fixed displacements of the origin.
In Einstein's relativity the same equation is modified to remain constant with
respect to displacement (and rotation), but not with respect to motion.
For a moving object, at least one of the lengths from which the distance, d, is calculated is contracted relative to a stationary
observer. The equation now becomes:
d^{2} = x^{2} + y^{2} + z^{2} (1v^{2}/c^{2})^{1/2}
and this implies that the distances all shrink as one moves faster, so does this
mean there are no constant distances left in the universe? The answer is that
there are because of Einstein's revolutionary concept of spacetime where time
is distance and distance is time! So now
s^{2} = x^{2} + y^{2} + z^{2}  ct^{2}
and this new distance s (remember s stands for Spacetime) does indeed remain
constant for all who are in relative motion. This distance is said to be a
Lorentz transformation invariant and has the same value for all inertial
observers. Since the equation mixes time and space up we have to always think in
terms of this new concept: spacetime! This means that time isn't constant and
that by simply increasing the velocity (to close to the speed of light for it to
have an affect) significant time dilation effects can be seen. It will be a very
long time indeed before we have the capability to build a time machine and
travel the universe.
The Lorentz Transformations
According to relativity
theory the length of a body as measured by an observer in uniform relative
motion is less than that measured by an observer at rest with respect to the
body. This is not a physical change in the body, but a consequence of the
Lorentz transformation. For simplicity it has been assumed throughout that z is
the direction of motion, consequently
This Lorentz invariant
applies to the four vectors: distance, velocity, acceleration and momentum and
each will be discussed below:
Distance
Velocity
In Newtonian mechanics
velocity is derived by differentiating the position with respect to time. The
relative nature of time in Einstein's relativity appears at first glance to pose
something of a problem. The solution is to use the "proper time", i.e. the time
measured by an observer attached to the moving object. This has the advantage
that the differences in x, y and z are zero and also that the "proper time" is
orthogonal to the other three axes; which is an intrinsic property of "proper
time" Newtonian mechanics, considered as the fourth Euclidean ordinate.
Differentiating using the proper time gives the fourvelocity expression below,
where uu is the familiar threespace velocity of Newton
Acceleration
Again, this is obtained
by differentiating the four velocity with respect to the "proper time" giving
A=(a,0) where a is the three space acceleration of Newton.
Momentum
The Lorentz
transformation invariant four momentum expression where p is the magnitude of
the three momentums of Newtonian mechanics is shown below. The conservation both
of the three momentums and of mass energy is contained within the conservation
of four momentums.
Dirac Negative Energy Equation
In classical Newtonian
physics the energy E of an object is given by:
where m is mass and v
is velocity. As both mass and any quantity squared must be positive, the energy
also must be positive. Classical Newtonian momentum is simply the product of the
mass and the velocity.
Both classical energy and momentum are conserved. In relativistic Einstein
physics, four momentum, P, is of the form (px,py,pz,iE/c). It is now this four
momentum, momentum energy that is conserved. As a vector its magnetude is
Lorentz invariant. If in one frame of reference, the rest frame, an object is at
rest then
where m is now the rest
mass. In a frame in which it is moving then
where p is just the
magnitude of Newton's three momentum, and E is the corresponding energy. Each
component of P is conserved, which consequently implies the conservation of both
mass energy and momentum, similarly to Newtonian mechanics. Additionally, as P
is invariant, the last two equations must be equal, i.e. after rearranging
We are at liberty to
take either the positive or negative square root of the right hand side for the
energy; the latter of these gives rise to negative mass energies.
The Time Travel Energy Equation
Below is a photograph
of the original chalk and blackboard derivation of the now infamous Time Travel
Energy Equation. This equation determines the maximum possible energy that one
can squeeze out of a rotating Black Hole. It is an expansion of Einstein's E=mc^{2}.
Points to note:
r in the integral is the Schwartzchild radius of the black hole,
The S_{i} are
elements of the 11x11 dimensional superstring tensor. These elements
incorporate the factor of c^{2}.
The Hawking Hamiltonian is an extension of the Newtonian Hamiltonian H_{N} multiplied by the product of the rotating vector mass M_{R} and the angular velocity w, which of course defines the vector angular momentum
of the black hole.
