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Time Travel Maths

Although this website was designed for the non-scientist, 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 e-mail 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 = a2 + b2

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 - v2/c2) -1/2

Elementary Guide to Relativity

There are, however, certain quantities that do remain constant. These constants are related to four-dimensional 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 space-time 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 = x2 + y2 + z2

where x, y and z are the lengths,  or more correctly the difference in the co-ordinates, 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 = x2 + y2 + z2 (1-v2/c2)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 space-time where time is distance and distance is time! So now

s
2 = x2 + y2 + z2 - ct2

and this new distance s (remember s stands for Space-time) 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: space-time! 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 four-velocity expression below, where uu is the familiar three-space 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=mc2.

Points to note:
r in the integral is the Schwartzchild radius of the black hole,
The S
i are elements of the 11x11 dimensional super-string tensor. These elements incorporate the factor of c2.

The Hawking Hamiltonian is an extension of the Newtonian Hamiltonian H
N multiplied by the product of the rotating vector mass MR and the angular velocity w, which of course defines the vector angular momentum of the black hole.