When I fall asleep, I wake-up later tomorrow morning for instance, not yesterday morning. It is interesting to see what kind of conditions this implies and whether it is possible to satisfy this easily and even more interesting is to see whether a time travel to the geometric past - maybe the Golden Youth - could be possible.
The following assumptions are made about what happens in BSFR.
- Causal diamond (CD) is a correlate for self. CD is obtained by gluing together two identical half-cones along their bottoms. Moment "Now" corresponds to the largest hyperplane Tnow=T (origin of time coordinate is at either (call it "lower") tip of CD) .
- During the sequence of SSFRs defining self, the 3-surfaces at the passive boundary of self are fixed although their 4-D tangent space changes and corresponds to the unchanging part of selfhood - soul one might say. The opposite active boundary of CD and 3-surfaces at it change and shift towards geometric future.This gives rise to wake-up consciousness involving sensory input and thoughts, emotions etc. induced by it. Each SSFR is preceded by the analog of unitary time evolution.
- BSFR means a death of self (subself) and its reincarnation with an opposite arrow of time. One can equally well speak about the analog of falling in sleep and waking up after that for some level of hierarchy of selves. The self born in the death of the self with an opposite arrow of time self has no direct memories about the state. Self can however have memories about dreams in which part of say brain is awake. These memories store information about what self experienced during the sleep.
In BSFR the active boundary of the CD becomes passive and is frozen. The size of CD is scaled down so that CD becomes small: this implies that the reincarnated self has a childhood and much of the memories - often not pleasant - stored near the active boundary as subselves living forth and back as conscious entities disappear. The surviving memories of self become "silent wisdom" of the reincarnated self.
- If CD belongs to a larger CD, call it CDsuper representing a larger unit of consciousness, the sub-CDs must shift to the same direction as the active boundary of CDsuper. Otherwise the sub-CDs would drop from the flow of consciousness. This is analogous to co-movement of matter in cosmology.
Note that the mental images of self correspond to sub-CDs around Tnow and shift towards geometric future as CD increases and new mental images emerges at Tnow plane: by M8-H correspondence these special moments in the life of self correspond to roots of the polynomial defining space-time surface and reside are the upper half-cone of the CD. As CD increases, new roots pop up inside the upper half-cone near the Tnow hyper-plane for some particular SSFRs. Completely counterintuitively, the mental images about past experiences are therefore in the geometric future of Tnow hyperplane!
- In each SSFR CD size increases - at least in statistical sense this implies that T grows. Each SSFR corresponds to a scaling for the CD shifting its active boundary towards the geometric future. During its life cycle CD experiences scaling Λ:
Tnow→ Tnow,sleep1= Λ(SSFR) Tnow , Λ(SSFR) >1 .
- When the system falls in sleep the size of CD is scaled down so that also the value of Tnow is scaled down by ΛBSFR<1:
Tnow,sleep2= (1- Λ(BSFR)) 2Tnow,sleep1 = (1- Λ(BSFR)) Λ(SSFR)2Tnow , Λ(BSFR)<1 .
After that the CD begins to increase in size by small scalings in SSFRs to opposite time direction and Tnow begins to decrease from its value Tnow,sleep begins to decrease.
- If CD belongs to a bigger CD - call it super-CD - representing a larger unit of consciousness with a longer life cycle, one can argue that the CD must shift to the same direction as the larger CD increases. Otherwise the CD would drop from the flow of consciousness defined by super-CD. This is analogous to co-movement of matter in cosmology.
Therefore a given life cycle corresponds also a shift Δ T of sub-CDs towards the growth direction of super-CD takes place and one has for the time coordinate Tsuper,now of the super-CD. Therefore one must perform shiftT→ T+ Δ T for Tnow,sleep1 and Tnow,sleep2 to take into account the drifting.
This gives for the moments "Now" before ad after the shrinking of CD in BSFR (falling asleep):
Tsuper,now,sleep1 = T0+ Tnow,sleep1 +Δ T ,
Tsuper,now,sleep2 = T0+ (1- Λ(BSFR)) 2Tnow,sleep1 +Δ T .
- Similar formula holds true for the moment of wake-up. In the previous formula Tnow is replaced with Tnow,sleep2 and one has
Tsuper,now,wakeup1 = T0+ Λ1)(SSFR) Tnow,sleep2 +Δ T1) ,Tsuper,now,wakeup2 = T0+ (1- Λ1)(BSFR)) Λ1)(SSFR)2Tnow,sleep2 +Δ T1) .
The parameter T0 depends on the choice of the origin of time for super-CD but is irrelevant.
One can deduce a consistency condition for the parameters of the model.
- During the sleep period the time coordinate Tsuper,now for moment "Now" in the coordinates of larger CD changes in the following manner:
Tsuper,now,sleep =T0+ Tnow,sleep1 → Tsuper,now,wakeup
=T0 + Λ1)(BSFR) Tsuper,now,sleep2 +Δ T1) .
T0 is an irrelevant parameter associated with super-CD. Note that there is breaking of time reversal symmetry since self associated with CDsuper has fixed arrow of time unlike CD. Hence Δ T has at least in a statistical sense the same sign irrespective of the arrow of time of self.
- This picture should be consistent with what we observe. When the tired average self fall a sleep at the evening, it wakes wake-up at the morning and is full of energy. Quite generally, wake-up occurs after time Δ T(sleep) meaning that the value of time Tsuper has increased by
Tsuper,now,wakeup= Tsuper,now(sleep1)+ Δ T(sleep) .
These two expressions for the value of Tsuper,now(wakeup) must be consistent and this gives a conditions on the parameters involved:
(1- Λ1)(BSFR)) Λ1)(SSFR)2Tnow,sleep1 +Δ T1)
= Tnow,sleep1 +Δ T + Δ T(sleep) .
Δ T(sleep) is given by
Δ T(sleep) =[(1- Λ1)(BSFR)) Λ1)(SSFR)2-1]Tnow,sleep1 +Δ T1) -Δ T .
Intuitively it seems clear that for a given arrow of time it is not possible to wake-up before one falls asleep, and the condition Δ T(sleep)>0 for the standard arrow of time gives a constraint on the parameters. One cannot however exclude the possibility of time travel without dying or falling asleep first of the duration of time travel is much longer than that of wave-up period: Δ T1) -Δ T.
A special solution corresponds to
Δ T(sleep)= Δ T1)- Δ T and (1- Λ1)(BSFR)) 2Λ1)(SSFR)=1