Consciousness Studies/The Philosophical Problem/Appendixt

Dynamics: Velocities in a Four Dimensional Universe

Many students of consciousness studies are unaware of the modern derivations of physical theories. Nowhere is this more evident than in the belief that Newtonian Physics is a complete description of physics. However, this belief is incorrect, the concept of Kinetic Energy that underlies Newtonian Dynamics is due to relativistic mass increases as a result of the existence of a four dimensional universe and the invariance of physical laws. The mass increase, although a relativistic effect, is noticeable as kinetic energy because it is equivalent to an energy of $E=mc^{2}$ where the speed of light squared is a huge number. What we call classical physics is in fact relativistic physics. (See special relativity for beginners for an introduction). Contrary to popular belief our observation must be described using four interrelated coordinate axes, not three.

Relativistic transformation of velocity

If time and space are affected by the four dimensional nature of the universe then velocity, which is the ratio of distance travelled to time taken, should also be affected. In the simplest case, where Bill and Jim are travelling towards each other they both obtain the same value for the velocity of approach. This is because Bill and Jim are symmetrical, either can consider either to be moving. The symmetry disappears however if both Bill and Jim are observing an object moving relative to both of them. Suppose there is an object moving at v relative to Jim and Jim is moving at J relative to Bill, what is the velocity of the object relative to Bill?

The illustration above shows that time dilation and phase are going to be involved in the comparison of Bill and Jim's estimate of the velocity of the moving object. If B is Bill's estimate of the velocity and J is Jim's estimate then, for an object moving along the x axis:

B = (J + v)/(1 + Jv/c²)

Derivation:

From the Lorentz transformation equations:

Distance travelled by object as observed by Jim = (x-vt)/√(1-v²/c²)

Time taken observed by Jim = (t - (v/c²)x)/ √ (1-v²/c²)

Where x and t are distance and time observed by Bill.

Velocity observed by Jim is distance/time: J = (x-vt) / (t - (v/c²)x)

Therefore x-vt = J (t - v/c²x)

So x/t, the velocity measured by Bill is: B = (J + v)/(1 + Jv/c²)

The equation given above only applies to observers moving in the directions shown. The involvement of time dilation and the fact that an object that is stationary for one observer will be moving for the other means that even velocities at right angles to the direction at which Jim and Bill approach each other will be affected (although a different equation must be applied).

Conservation of Momentum in a 4D Coordinate System

The conservation of momentum is a basic empirical law of Newtonian physics. It states that the products of the masses and velocities of a set of particles is constant. The way that Bill and Jim record different velocities for the same objects around them means that the Newtonian conservation of momentum law does not apply and is only an empirical approximation for low velocities. Most importantly, if Newtonian momentum is conserved for one observer it is not conserved for another observer moving relative to the first. Einstein originally formulated Relativity Theory using an axiom that the laws of physics are the same for all observers and amended the conservation of momentum law so that it applied in a 4D universe. Modern physics relies on Noether's Theorem:

To every symmetry corresponds a conservation law. Conversely, for every conservation law there is a corresponding type of symmetry.

The symmetrical nature of translations in space implies conservation of linear momentum.

Rotational symmetry implies conservation of angular momentum.

Time symmetry implies conservation of energy.

Invariance with respect to gauge transformations implies conservation of electric charge. Momentum is the product of mass and velocity. If Newtonian momentum, where mass is a constant, is not conserved because the velocity is variable between observers then perhaps the mass changes to compensate for the velocity.  If conservation of momentum is expressed as:

Mass(1) x Velocity(1) = Mass(2) x Velocity(2)

Then it can be shown that the conservation equation is true if:

Mass(2) = Mass(1)/√ (1-u²/c²)

Where u is the speed of the object, not the velocity.

In general the mass, m, of an object travelling at speed, u, is:

m = m₀/ √ (1-u²/c²)

Where m₀ is the mass of the object when it is at rest relative to the observer.

In other words, the momentum of an object is conserved but its mass is variable. The variation of mass with speed is a direct consequence of translational symmetry and the existence of a four dimensional universe. This variation in mass with speed is easy to demonstrate in relatively simple particle accelerators because even at half the speed of light the mass is 1.15 times the rest mass.

The new form of the conservation of momentum law suggests that the Newtonian formula for force must be replaced. Force is rate of change of momentum, in Newtonian physics this is the product of a constant mass times a change in velocity (acceleration). In 4D physics this must be calculated in terms of both the change in velocity and the change in mass.

Force = d(m₀u/√ (1-u²/c²))dt

Where u is velocity and u is speed.

The kinetic energy of a particle accelerated from rest to velocity u is:

K = ∫ F dx

But F = d(mu)/dt

K = ∫ d(mu)/dt dx

K = ∫ d(mu) dx/dt

dx/dt = u

K = ∫ (mdu + udm) u

K = ∫ (mudu + u²dm)

In the discussion of momentum above it was shown that the mass of a moving object is related to its mass at rest by:

m = m₀/ √ (1-u²/c²) This can be rewritten as : m²c² - m²u² = m₀²c²

Taking differentials:

2mc²dm - m²2udu - u²2mdm = 0

Dividing by 2m:

mu du + u² dm = c² dm

But:

K = ∫ (mudu + u²dm)

Therefore, integrating between mass at rest and mass at u:

K = ∫ c² dm = c² (m-m₀)

So :

Kinetic energy = mc² - m₀c²

The kinetic energy is present as the extra mass due to motion.

Total energy = m₀c² + Kinetic energy = mc²

The rest mass energy is given by:

E = m₀c²

A result that is amply demonstrated in atomic bombs where the mass of uranium is greater than the mass of the products generated by fission (cesium and rubidium). The excess mass being converted into a cataclysmic release of energy.

The Binomial Expansion of (1 - v²/c²)^-1/2

The binomial theorem can be used to expand any expression of the form (a + x )ⁿ so, (1 - v²/c²)^-1/2 can be expanded by substituting 1 for a and - v²/c² for x.

(a + x )ⁿ = aⁿ + na^n-1x + n(n-1)/2! na^n-2x² + ...

therefore:

(1 - v²/c²)^-1/2 = 1 + 1/2 v²/c² + (-1/2)(-3/2)/2! v⁴/c⁴ + ...

(1 - v²/c²)^-1/2 = 1 + 1/2 v²/c² + 3/8 v⁴/c⁴ + 5/16 v⁶/c⁶ ...

If the velocity is small compared with the speed of light the terms from 3/16 v⁴/c⁴ onwards become negligible so:

(1 - v²/c²)^-1/2 ≈ 1 + 1/2 v²/c²

The Newtonian Approximation for Kinetic Energy

The kinetic energy of the particle (the energy due to motion) is:

KE = m_vc² - m₀c² But it was seen earlier that m_v = m₀(1 - v²/c²)^-1/2

So:

KE = m₀(1 - v²/c²)^-1/2 c² - m₀c²

KE = m₀c² ((1 - v²/c²)^-1/2 - 1)

But (1 - v²/c²)^-1/2 ≈ 1 + 1/2 v²/c²

So

KE ≈ m₀v² /2

Which is the Newtonian empirical approximation for kinetic energy. Notice that the full result is derived from symmetries in a 4D universe whereas the Newtonian result is derived from empirical formulae.

Electricity and Magnetism in a 4D Universe

One of the greatest achievements of Relativity Theory was to unify electricity and magnetism. These two effects can be seen to be the same phenomena observed in different ways. This is shown in the illustration below.

It can be seen that once the idea of space-time is understood the unification of the two fields is straightforward. Jim is moving relative to the wire at the same speed as the negatively charged current carriers so Jim only experiences an electric field. Bill is stationary relative to the wire and observes the electrostatic attraction between Jim and the current carriers as a magnetic field. Bill observes that the charges in the wire are balanced whereas Jim observes an imbalance of charge.

Incidently, the drift velocity of electrons in a wire is about a millimetre per second but the electrons move at about a million metres a second between collisions (See link below).

Useful links:

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html">http://hyperphysics.phy-astr.gsu.edu/hbase/electric/ohmmic.html

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/releng.html">http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/releng.html