2.1  Time Travel.

To induce the motion of a mass within the spatial dimensions of D₀ it is necessary to apply a spatial force in the desired direction of travel. The relativistics of the motion so incurred were investigated in detail in [1]. It is consequently not unreasonable to suppose that to effect motion of a mass in the temporal direction, it is necessary to apply an accelerating force in that direction. To investigate this consider the Existence Momentum of an energy mass m in D₀, i.e. from [1], Eq.(3.1)

where

v_r is the spatial velocity. r is a unit vector in the direction of spatial motion

Expressing the temporal component of (2.1) as

vt = c æ è 1 - vr2 c2 ö ø 1/2

(2.2)

then v_r can clearly be expressed as

vr = c æ è 1 - vt2 c2 ö ø 1/2

(2.3)

which then allows (2.1) to be restated as

M = m ì í î c æ è 1 - vt2 c2 ö ø 1/2   r + j vt ü ý þ

(2.4)

Differentiating (2.4) with respect to time gives the rate of change of momentum thus

F = dM dt = ì ÷ í ÷ î c dm dt æ è 1 - vt2 c2 ö ø 1/2   - mvt c æ è 1 - vt2 c2 ö ø 1/2   dvt dt ü ÷ ý ÷ þ r + j  æ è dm dt vt + m dvt dt ö ø

(2.5)

If now F is specified as

F = 0 r + j Ft

(2.6)

then a temporal force is being applied to m in such a direction as to encourage motion into the past, i.e. in the same direction as the natural temporal flow. This can be simply illustrated in the following figure where nomenclature is identical to that in [2], Fig.2.3

Fig.2.1 The Application of a Positive Temporal Force.

Combining (2.5) and (2.6) gives Spatial

c dm dt æ è 1 - vt2 c2 ö ø 1/2   - m c æ è 1 - vt2 c2 ö ø 1/2   dvt dt = 0

(2.7)

Temporal

dm dt vt + m dvt dt = Ft

First consider the spatial part of (2.7). This can be integrated immediately to give

mc æ è 1 - vt2 c2 ö ø 1/2   = k

(2.8)

In (2.8) the constant of integration, k , is given by initial conditions which are, when t = 0, v_t=c and the mass m=m₀. This simply means that the mass m was spatially stationary at t = 0. These conditions give k = 0 so (2.8) becomes

mc æ è 1 - vt2 c2 ö ø 1/2   = 0

(2.9)

Now because m cannot be zero, (2.9) infers that

vt = c

(2.10)

This therefore shows that the temporal velocity of the mass m has not been changed by the application of the temporal force, i.e. there is no increased velocity along the temporal axis. From (2.3) this also shows that v_r is also zero, i.e. there is no spatial motion. Consequently, in accordance with the law of conservation of energy, the applied force has been absorbed in some other way. To determine this the temporal part of (2.7) is integrated with respect to the time to give

mvt = Ft t + k

(2.11)

Applying the above initial conditions yields

k = m0 c

(2.12)

so that

mvt = m0 c + Ft t

(2.13)

and therefore from (2.10)

m = m0 + Ft t c

(2.14)

Eq.(2.14) shows that the applied temporal force is absorbed as an increase in mass. This result is therefore identical in part to the application of a spatial force as analysed in depth in [1]. The only difference here is the absence of an accompanying spatial acceleration. Also it can be easily shown that if the above analysis were carried out using (2.1) instead of (2.4), the result would be the same. Thus it is clear that with time defined as in this series of papers, travel through time is not theoretically possible. Albeit this is so, a small modification of the above analysis could lead to a potentially more useful result. This is the subject of the next Section.

R3 Version 1.0.0
Ó P.G.Bass, December 2004

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