Probabilistic Special RelativityJohn Wellings

email: wellingboy@hotmail.com

AssumptionsHeisenberg’s uncertainty principle is true.

Einstein’s Special Theory of Relativity is true and we are in Minkowski space and ignoring gravity.

Space time is discrete – consisting of a lattice of minimum distance (Planck length) and minimum time (Plancktime, which I will also call ‘Planck moments’).

There are two temporal variables: coordinate time in the sense of Einstein which is

the local time for any particle an evolution parameter, or parameterized time which

is a universal Planck time ‘refresh’, 1.8549 x 10^43times per second – similar to the Newtonian concept of time

A body cannot ‘jump’ in time or space – it must pass through contiguous moments of time and contiguous positions in space.

To keep the maths simple and diagrams easy to draw I willsuppress two dimensions and draw one dimension of space (horizontal axis) and one of time (vertical axis).

HypothesisA body cannot be absolutely at rest. There is no absoluteframe of reference to be ‘absolutely at rest’ relative to. If a body were absolutely at rest you would know bothits position and momentum and this would not agree with Heisenberg’s uncertainty principle. Thus a body has zero (or very low) probability of being in the same position from one Planck moment to the next. The body must have a probability of moving to the right or left in space, and a probability of moving forward or backward in time.

This would result in the body having a ‘Zitterbewegung’ constantly zig-zagging at light-speed in space-time.

a, b, c, d, R and T are all probabilities between 0 and 1.

Equationsa + b = 1

c + d = 1

R = a - b Resultant probability of moving through space = probability right – probability left

T = c – dResultant probability of moving through time = probability forwards – probability backwards

ab

d

cFigure 1

Proof R² + T² = 1

Below is the energy/momentum relationship from special relativity:

Divide all sides by E to give the below triangle:

Therefore (pc/E)² + (mc²/E)² = 1

As pc/E = v/c and 1/ γ = mc²/E

(v/c)² + (1/γ)² = 1

v = Rc (i.e. velocity in a particular direction is the resultantprobability of moving in that direction times the speed of light)

R = v/c

R² + (1/γ)² = 1

γ = 1/√1-v²/c²

v²/c² expressed as a resultant probability is R²

therefore γ = 1/√1- R²

T = 1/ γ

R² + T² = 1

γ = 1/T (which means that Einstein’s gamma factor is the reciprocal of a probability)

Thus we have a gamma factor which is proportional to (andentirely dependent on) the resultant time probability.

Space-time Probability

1

R

TFigure 6

R² + T² = 1

(a – b) ² + (c - d) ² = 1 a² + b² - 2ab + c² +d ² - 2cd = 1

As b = 1 - a and d = 1 - c

a² + (1 – a) ² - 2a*(1 - a) + c² + (1 - c) ² - 2c*(1 - c)= 1

a² + 1 + a² -2a - 2a + 2a² + c² + 1 + c² -2c - 2c + 2c² =1

1+ 4a² – 4a + 1 + 4c² - 4c = 1

2 + 4a² – 4a + 4c² - 4c = 1

a² – a + c² - c + ¼ = 0

(a - ½)² + (c - ½)² = ¼

1

½ 1

½

a (Spatial probability)

c (Temporal probability)

Figure 7Space-time Probability

Interpretation:The circumference of the circle represents 100% probability of movement through space-time. At each moment of Planck time a body must move through space-timewith 100% certainty which has spatial and temporal components such that:

R² + T² = 1

If a body moves through time with 100% certainty (T = 1) then it can only move through space with complete uncertainty (i.e. 50% left or right, so R = 0). And if a body moves through space with 100% certainty (i.e. light speed, R = 1) it can only move through time with completeuncertainty (i.e. 50% forward and backward, so T = 0). All other velocities comprise complementary probabilitiessuch that R² + T² = 1.

Example probability diagrams for different velocities

All observers will believe they are moving through ‘rest frame time’ and are entitled to assume a frame of reference in which they are at rest.

100%Forwards in time

50%To the right

0%Backwards in time

50%To the left

Figure 8Probabilities for a body ‘At rest’

50%Forwards in time

100%To the right

50%Backwards in time

0%To the left

Figure 9Probabilities for a body Moving at light speed (c)

93.301%Forwards in time

75%To the right

6.698%Backwards in time

25%To the left

Figure 10Probabilities for a body Moving at half light speed (½ c)

83.072%Forwards in time

87.5%To the right

16.928%Backwards in time

12.5%To the left

Figure 11Probabilities for a body Moving at three-quarters light speed (¾ c)

98.412%Forwards in time

62.5%To the right

1.588%Backwards in time

37.5%To the left

Figure 12Probabilities for a body Moving at one-quarter light speed (¼ c)

Addition of Relativistic velocities inSR

We have two observers A and B and a rocket C.

A’s frame of reference is at rest (lab frame).

In A’s frame of reference B is moving right at ¾c. In B’sframe of reference a rocket is moving right at ¾c.

What velocity is the rocket moving in A’s frame of reference?

w = (u + v) / (1+(u.v/c²))

w = (¾c+¾c)/(1+((¾c*¾c)/ c²))

w = 0.96c (velocity of rocket C in A’s frame of reference)

γ = 3.5714285714285714285714285714286

Recasting this scenario as astochastic model

We have two observers A and B and a rocket C.

A’s frame of reference is ‘at rest’, so moving with a probability 0.

In A’s frame of reference B is moving right with a probability of ¾ [R(B)]. In B’s frame of reference a rocket is moving right with a probability of ¾ [R(C)].

With what probability is the rocket moving right in A’s frame of reference? [R(A)]

You might think that the probability of the rocket C moving right in A’s frame of reference [R(A)] is simply…

R(A) = R (B) * R(C)

…because R(B) and R(C) are independent probabilities.

However this is not true because R(B) and R(C) are spatial components of probability which co-vary with temporal components of probability T(B) and T(C) by the relationship:

R(B)² + T(B)² = 1R(C)² + T(C)² = 1

And R(A) and T(A) will also have the same relationship:

R(A)² + T(A)² = 1Therefore you must construct a vector in which the temporal component of probability is an independent probability and the spatial component is co-variant.The independent temporal component probabilities are therefore multiplied and the spatial component probabilities are summed.

T(B) * T(C)R(B) + R(C)

The hypotenuse of this vector is then a non-normalised probability (i.e. it is greater than 1). You divide the hypotenuse by itself to give 1 (i.e. you normalize it to create a unit vector) and do the same for the spatial andtemporal component probabilities.

Therefore:

(R(B) + R(C))² + (T(B) * T(C))² = hypotenuse²

√((R(B) + R(C))² + (T(B) * T(C))²) = hypotenuse

T(B) * T(C)___________________________ = T(A)√((R(B) + R(C))² + (T(B) * T(C))²)

R(B) + R(C)_____________________________ = R(A)√((R(B) + R(C))² + (T(B) * T(C))²)

R(A) = normalized spatial probability of the rocket C moving right in A’s frame of reference.

T(A) = normalized temporal probability of the rocket C moving forwards in time in A’s frame of reference.

¾+¾R(A) = _______________________ √ ((¾+¾)² + (√7/16 * √7/16)²)

R(A) = 0.96 (probability of rocket C moving right in A’s frame of reference))

√7/16 * √7/16T(A) = ________________________ √ ((¾+¾)² + (√7/16 * √7/16)²) T(A) = 0.28

γ = 1/0.28 = 3.5714285714285714285714285714286 This gives exactly the same result as SR and this is demonstrated in the section ‘Equivalence of Einstein’s addition of velocities proof’ below.

Equivalence of Einstein’s addition ofvelocities proof

R3 = R1 + R2 _

(R1 + R2)2 + (T1 . T2)2

As R12 + T1

2 = 1 and R22 + T2

2 = 1

R3 = R1 + R2

_

((R1 + R2)2 + ( (1 - R12) . (1 - R2

2))2)R3 = R1 + R2 _

((R1 + R2)2 + (1 – R12) (1 – R2

2))R3 = R1 + R2

_

(R12 + R2

2 + 2R1R2 + 1 + R1

2 R22 - R1

2 - R22)

R3 = R1 + R2 _

(2R1R2 + 1 + R1

2 R22)

R3 = R1 + R2 _

(1 + R1R2) ( 1 + R1R2)

R3 = R1 + R2 _

1 + R1R2

This is the stochastic equivalent to Einstein’s equation (multiply probabilities by c to get in terms of velocities e.g. u = R1c, v = R2 c and w = R3c)

R3c = R1c + R2 c _

1 + R1 . R2 c2/c2

w = u + v _

1 + u . v / c2

Mass and Time Dilation

γ = 1/T

γ = 1/½

γ = 2

If a body moves with a 75% certainty forward in time and 25% backwards it is easy to see that on average you will ‘bump’ into two versions of the body at any moment in time. It will be squashed up in time and have a greater ‘time density’ – in this example double the ‘normal’ timedensity of a body moving in rest frame time (i.e. 100% certainty forwards in time).

(3 + 1) / 2 = 2

75%Forwards in time

93.301%To the right

25%Backwards in time

6.698%To the left

Figure 2Probabilities for a body with gamma factor 2 (0.86c or √¾c)

50%Forwards

25%Backwards

25%Forwards

Figure 3‘Time density’

Therefore the body will have double the mass/energy equivalence (on average). If you measure the mass over a very short period (Planck time scale) you will find it varies from 1 unit mass up to 3 unit masses or even more.This agrees with Heisenberg’s uncertainty principle with regard to measuring energy and time. It takes longer to get the average mass/energy equivalence measurement. If you measure for a very short time you will get wildly fluctuating mass/energy equivalence results.

Quote from Wikipedia:

An informal, heuristic meaning of the principle is the following: A state that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must be defined accurately, and this requires the state to hang around for many cycles, the reciprocal of the required accuracy.

If a body moves at light speed then you will have ‘unlimited’ versions of the body within a very small timeframe, so the mass / energy equivalence will approach infinity – or put another way, it will have a time density approaching infinity.

Length Contraction

When a body is ‘at rest’ it is actually moving randomly around at light speed with a 50/50 chance of going in twoopposing directions. This means the body will take up much more space in a way somewhat analogous to the idea of atoms in a hot material vibrating more and thus takingup more space (e.g. a metal expands when heated, contracts when cooled). Thus a body at rest takes up the maximum space – it is ‘smeared’ across space due to this opposing motion and we measure this as its length.

However if it is moving it now has more probability of moving in one direction than the other, so it moves in the opposing direction less and is therefore less smearedout over space. This only affects the length in the direction of movement, so its other dimensions at right angles to the direction of travel are unchanged – i.e. length contractions is only in the direction of movement.

For an object moving at light speed there no opposing motion at all so its length is at a minimum.

This relationship gives a gamma factor related to the resultant probability of movement in space such that:

γ = 1/√1- R² = 1/T

This is true because (assuming one Planck length is equalto one Planck time):tan T/R

Conclusion

If the uncertainty principle is correct then a body must have a Zitterbewegung and all motion must be at light-speed. All apparent sub-luminal velocities are just on-

T

R

average velocities dependent on the probability a body will move in a particular direction in space. As a body moves with more certainty through space it moves with less certainty in time, so the Zitterbewegung motion is increasingly in the temporal component of a space-time unit vector, such that R² + T² = 1

Special Relativity can thus be re-cast as a probabilistic model which I hope will enable it to be reconciled with Quantum Theories - giving finite predictions that agree with experiment and not meaningless infinities.