Time T(t) as seen by the patient
On 26/7/2006 I had some gall bladder surgery, where I was forced to undergo general anesthesia. I carefully monitored my consciousness during the procedure, so here I am attempting to mathematically model the situation.
Assume an "event" is a point in H, or a coordinate quaternion quadruple q=(t,x,y,z), with t,x,y,z>0, similarly to how an event is modelled in physics (so a movie or regular flow of consciousness for example, would be continuous streams of quadruples q(t)=(t, x(t),y(t),z(t)), with no relativistic effects present, so time t is, (for our purposes) linear. So we can assume that time and the corresponding streams of events are continuous.
The experience during the shut-down of consciousness was as follows: First, upon the injection of the anesthetic, there was a slight disorientation, which gradually increased. At some point, loss of consciousness occurred at t1. But the subject (me) never did "experience" actual loss of consciousness. Instead, I experienced an awakening into some other forward time t2, with t2 = t1 + 3 hrs, after all the operations were completed.
Using the above notation, and if we assume that the actual moment of unconsciousness occurs at t1=1 and the moment of awaking occurs at t2=t1+3, at loss of consciousness I was located at q(t1), with:
q(t1)= (t1,x(t1),y(t1),z(t1)) (in the anesthesiologist's room)
and suddenly I awoke to point q(t2), with:
q(t2)=(t2,x(t2),y(t2),z(t2)) (in the anesthesiologist's room, again).
My position was almost identical, so we can assume that x(t1)=x(t2), y(t1)=y(t2) and z(t1)=z(t2).
If we describe the time t of uninterrupted normal consciousness as a linear function T(t), the following graph models accurately what happens from the patient's perspective:
But the above function is:
T(t)={t, if t<1, t+3, if t>1}.
Converting in terms of Heaviside,
T(t) = t + 3*Heaviside(t-1)
In general, if the moment of unconsciousness occurs at t=t1 and if the total unconsciousness time is tTot=t2-t1 then the time of the event stream as experienced from the patient who "goes under" is described by the function:
T(t) = t + tTot*Heaviside(t-t1) (!!)
Some interesting facts: The patient always measures time along the y-axis. There is a jump discontinuity at t1=1, where the patient instantly jumps from time t1=1 hours to t2=4 hours. The derivative of T(t) is dT/dt=1+tTot*Dirac(t-t1):
In other words, from the patient's perspective, time passes at a constant rate dT/dt=1 hr/hr (for our function), except exactly at the point t1=1, when there is a singularity. q(t) for t1<=t<=t2 is undefined from the patient's perspective. This is non-existence for a duration of t2-t1 hours (so this time is actually lost to the patient), but the patient doesn't experience it, except as an instantaneous singularity, which gets lost in the subsequent stream of events!