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Time Travel - Mathematics

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 covered. To keep things simple, we are only going to cover Lorentz transformations, Dirac's negative mass energy and the time travel energy equation. If you feel this section should be expanded further please e-mail.

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

Lorentz Transformation

This Lorentz invariant applies to the four vectors: distance, velocity, acceleration and momentum and each will be discussed below:

Distance

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

Velocity

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.

Momentum

Dirac Negative Energy Equation

In classical Newtonian physics the energy E of an object is given by:

maths

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

maths

where m is now the rest mass. In a frame in which it is moving then

maths

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

maths

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.

Time Travel Energy Equation