The Mathematics Of Time
April 25, 2015 5:57 PM   Subscribe

So I have a few years to kill and I'd like to spend them fully understanding what physicists and mathematicians know about time. I'm not looking for any sort of summary, I want to understand the math from the bottom up. I once caught my father going through the Annus Mirabilis papers with a red pen; that's the sort of proficiency I have in mind.

Unfortunately my physics and calculus largely left off at the end of Freshman year, so I have a ways to go. I'm looking for some sort of hierarchy of the form:

In order to understand the math for Special Relativity you'll need to understand Z
In order to understand Z you'll need to understand X and Y
In order to understand X you'll need to understand U, V, W

Etc. I realize it's not as simple as a straight line, but that's the general idea.

I'd also like to find such a chart for the math that goes into Quantum Physics.
posted by Tell Me No Lies to Science & Nature (12 answers total) 48 users marked this as a favorite
Go through "The Road to Reality" by Roger Penrose. The book gives you a complete hierarchy of concepts (starts with algebra, works up through string theory) and explains the math for each concept. However, it has few example problems -- you'll want to get a textbook for each set of topics (few chapters) as you go if you're studying on your own.

Also, think about why you're doing this. If you want to feel smart and intellectual very quickly, go read the dialogue chapters from "Godel, Escher, Bach" by Douglas Hofstadter, ideally in a coffee shop or somewhere else public. Even physicists and mathematicians don't know "all" there is to know about anything, and the best ones study these things because they selfishly enjoy it. Chasing knowledge is great but chasing knowledge to validate yourself may never give you the validation you crave.
posted by sninctown at 6:50 PM on April 25, 2015 [2 favorites]

The Theoretical Minimum is a project of world renowned physicist Leonard Susskind.
These courses collectively teach everything required to gain a basic understanding of each area of modern physics including all of the fundamental mathematics.
Each area is covered with just enough theoretical and mathematical rigor to form a complete introduction to the subject. Although the courses stand alone, when taken in sequence they build upon each other to lay the foundation for an understanding of the most advanced theories in modern physics.
I think this is the best you're going to do short of enrolling in a PhD program.
posted by stuart_s at 6:54 PM on April 25, 2015 [17 favorites]

There are 15 courses: 6 "core" and 9 "supplementary." Each course has 10 lectures. Each lecture is a little short of 2 hours.

There are books that accompany two of the courses:The Theorectical Minimum and Quantum Mechanics: The Theoretical Minimum
The courses are specifically aimed at people who know, or once knew, a bit of algebra and calculus, but are more or less beginners.
posted by stuart_s at 7:09 PM on April 25, 2015 [1 favorite]

Response by poster: If you want to feel smart and intellectual

No worries there. In my own field I'm quite intellectually proficient.

I want to know some things about time that I'm pretty sure that humans don't know, but it's worth checking for what I'll learn on the way.
posted by Tell Me No Lies at 7:39 PM on April 25, 2015

Gerard t'Hooft has you covered.
posted by OnceUponATime at 7:50 PM on April 25, 2015 [1 favorite]

Best answer: Linear algebra is probably the best thing you could study. It's central to quantum mechanics. Some familiarity with differential equations is important, as they're the basic language of all physics. Linear algebra is also the basic building block for relativity. You can extend from linear algebra into differential geometry, which is the basic language for general relativity. Classical mechanics is useful for introducing a lot of the language and concepts for both quantum and relativity. (Studying classical mechanics more deeply is definitely worthwhile for deeper insight into quantum mechanics as well.) Knowing at least a bit of complex analysis is necessary for quantum at all, and more as you go deeper. Really, a bit of complex just makes everything simpler.

And if you want to study time, general relativity is one way you can do that. But classical mechanics into thermodynamics / statistical mechanics and into dynamical systems is another direction you should definitely consider. This is where I would think to talk about "time", myself, rather than relativity. For stat mech, what you want is just some basic probability. Dynamical systems is where one could talk about "chaos", and it follows from ordinary differential equations.

Quantum mechanics is essentially non-relativisitic. There was a bit of relativistic quantum theory, but it falls apart, and you study quantum field theory instead to combine QM and SR. This is getting fairly deep, and you need all the prerequisites you can bring to it. But if you get there, you can talk about time-reversal in particle physics, in the context of CPT-invariance. (Charge, parity [mirror-images], time.) Symmetry in general is the topic of "group theory" in mathematics, and is especially useful in quantum mechanics.

The trick is, you can study a lot of these topics at different levels, and with different mathematical tools. Physics students certainly don't tend to study all these as math courses, just learn on an as-needed basis in their physics courses. But I do think more than they get would be better.


I just want to say one word to you. Just one word.
Yes, sir.
Are you listening?
Yes, I am.
Linear algebra.
posted by solitary dancer at 10:06 PM on April 25, 2015 [3 favorites]

Isaac Asimov wrote an excellent nonfiction volume "The History of Physics" which explains a lot of important core physics concepts with minimal math. He does eventually get to einstein toward the end. I highly recommend it as a starting point.
posted by deathpanels at 3:43 AM on April 26, 2015 [1 favorite]

I want to know some things about time that I'm pretty sure that humans don't know

I don't believe there's ever been a person able to give a coherent account of time, or at least one that doesn't fundamentally amount to some kind of question-begging.

For most of my life I have been attempting to gain some kind of coherent reconciliation between the idea of the block universe, implied by the fact that only what has happened has happened, only what is happening is happening, and only what will happen will happen - and the interlinked ideas of self, time, change and motion. So far, any kind of intuitive grasp remains elusive.

Math and physics will teach you how to model reality in a way that allows time to be quantified, manipulated and related to other things - but I'm not sure that learning nuts-and-bolts methods to make specific, testable predictions is actually going to help much with the stuff that "humans don't know". Physicists seem about as likely as anybody else to devolve into nonsense after imbibing a few snifters of the old handwavium.

I'd be as pleased as hell to be wrong, so if you're ever in search of a skeptical ear with a fundamental sympathy for the project: email address in profile.

Be that as it may: Ilya Prigogine left behind plenty of good work to look into, and absolutely seconding Susskind above - he is very good at getting a point across in a way that makes it stick.
posted by flabdablet at 8:25 AM on April 26, 2015 [1 favorite]

Taylor and Wheeler's Spacetime Physics is fantastic, and assumes very little math background. Not strictly time-related, but it's an ideal introduction to special relativity. It's the self-study book I always recommend to first year college students who want to become professional physicists.
posted by eotvos at 9:55 AM on April 26, 2015 [2 favorites]

For a contemporary readable discussion try From Eternity to Here:The Quest for the Ultimate Theory of Time by the physicist Sean Carroll

The discussion may help point to the math and physics specialties you'll need to investigate.
posted by sammyo at 10:29 AM on April 26, 2015 [1 favorite]

Asking the math you need to understand special relativity and throwing in at the end that you'd also like to do this for Quantum Mechanics is a little like asking advice for going for a day-hike in the Catskills and then mentioning, "I'm also interested in climbing the Rockies."

The math you need for Special Relativity is very limited. A basic understanding of high-school level algebra and geometry is surely enough. In Special Relativity, the ideas are the things that will drive you crazy, not the math.

A very strong seconding to eotvos recommendation of Taylor and Wheeler. The book is a little . . . um. . . quirky and they name-drop Einstein in one of their chapters, but it's overall excellent. Just be aware that they way they do Special Relativity is fairly non-standard, but it's actually, I think, way better than the "standard" way.

Note that really understanding General Relativity is another matter altogether. It's not quite climbing Everest, but it's in that category. . .

If I were going to learn QM from scratch, I would make sure I really understood linear algebra and then would go and read the third volume of the Feynman Lectures.
posted by Betelgeuse at 6:01 PM on April 26, 2015

Feynman Lectures + Taylor And Wheeler's Spacetime Physics.
posted by wivy at 9:02 AM on May 6, 2015

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