here and now
"How is it that there are the two times, past and future,
when even the past is now no longer and the future is now not yet?"
Augustinus, Confessions, Book 11, chap. 14
Meanwhile, we know that in addition to events which happened in our past and events which will happen in our future, there are also events which are space-like (neither in the past nor in the future); And we know that the present is really a single point in space-time.
Of course, there is nothing mysterious about relativity, the math is easy enough that nowadays they teach it in high school, Minkowski diagrams and the Lorentz group as homework exercise.
But still - I wonder about the present and how the activities of my brain fit into a single space-time point when I experience the "here and now" ...
Perhaps I should think more about world lines.
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7 comments:
I do not understand what is your point when you ask how the activities of your brain fit into a single point. Of course they do not fit!
Yes, but this means that the immediate reality of my experience (the "here and now") is associated with a finite size 4-volume in space-time.
In other words, the sharp distinction between past and future (with the present a single point in between) does not apply, when I consider my own direct experience.
And I think this is an interesting (?) comment on the Augustinus paradoxon.
I always thought that physics encodes past and future experience through past and future Cauchy evolutions. On this picture, "here and now" is just a Cauchy initial data set -- which need not be an infinitely thin slice (or a "point," as you say).
Of course, an initial data set is often idealized that way -- but it can also be taken to be a "thick" slice, i.e., one with positive 4-volume. And Cauchy problems are perfectly sensible in this setting -- for example, generic hyperbolic Diff-EQs can be shown to have locally unique solutions. So I'm perfectly happy with answering Augustine's question with the following:
here and now = initial data set;
future now = future Cauchy evolution;
past now = past Cauchy evolution.
(Thanks for the interesting post, Wolfgang.)
Cauchy data are defined along space-like (hyper)surfaces and as such a highly idealized concept.
In order to actually realize this concept one would need an (infinite) number of observers who are synchronized (in advance) to set up or measure e.g. a field on this hypersurface.
My point is that the immediate experience of the "here and now" is associated with brain activities in a 4-volume (not a point and not a (hyper)surface) and thus I conclude that the reality of this experience is also associated with a 4-volume or to use a fancy word it is a non-local phenomenon.
Great post! One of my Professors was discussing about this concept in one of his classes. I am a college sophomore with a dual major in Physics and Mathematics @ University of Canterbury in Christchurch, New Zealand. By the way, i came across these excellent physics flashcards. Its also a great initiative by the FunnelBrain team. Amazing!!!
spambot Jerry,
these physics flashcards must come with a really high profit margin if you go to the trouble to place an ad for them on this mediocre blog...
So you guys are able to get around the captchas, lets see how you deal with my next move (turning comments mostly off) ...
I received an interesting comment by email from Bryan.
----
Cauchy initial data does not need to be defined on a hypersurface. It can be defined in a neighborhood of a slice -- a region with positive volume. The fact that we usually use a "slice" is just a mathematical convenience.
Some results for these "thick slice" Cauchy problems can be found in Courant and Hilbert (1962), vol. 2:
link
You can also find brief discussion in Weinstein's fqxi essay -- see section 2.2.3.
link
As Weinstein notes, these "thick slice" IVPs are actually much better behaved than their "thin slice" counterparts. They can have unique solutions, even when the corresponding "thin slice" IVP does not. This may actually provide good physical and mathematical reason to think that the fundamental unit of "now" is an epsilon neighborhood of spacelike slice -- and not the momentary slice itself.
Best wishes,
Bryan
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