I’m not sure if this will have a GQ answer, but I figured I’d try the collective wisdom of the Dope. (And I tried to search for earlier threads, but the hamsters died. )
I’ve been reading HG Wells’ “The Time Machine” from audible, and the preamble made me wonder about the history of the concept of time as a fourth dimension. Doing quick google searches hasn’t found anything conclusive in terms of someone who wrote about that idea before Wells.
The ancient Incans actually had a word for spacetime, but I don’t know how specific that understanding went.
Newton believed that time was part of the fundamental structure of the universe, not just the way people perceived it, and this is consistent with interpreting time as a fourth dimension, but I don’t know if he ever actually called it that.
Can anybody help answer this question? Thank you very much.
It doesn’t look to me that Newton is describing time as a 4th orthogonal dimension to the three spatial dimensions. Rather he is describing it as an independent dimension.
The (modifed) quote from Jean le Rond d’Alembert
is clearly regarding time as a peer to the three spatial ones, which is much closer to the current view. The critical point being that we travel in all four dimensions, and that our speed through those four dimensions is constant. The four dimensions are interchangeable, something that Newton did not contemplate. Jean le Rond d’Alembert’s acquaintance, it seems, got part of the way there.
The temporal dimension is not entirely interchangeable with spatial dimensions… the whole reason we bother distinguishing them is because there is, in fact, a distinction. Just not as much as we might have traditionally thought.
(Specifically, in Minkowski spacetime, spacelike directions and timelike directions have oppositely signed squared-norm; beyond that, there is a complete symmetry, so far as I know, except that any orthogonal basis contains three spacelike directions and one timelike direction. Still, these are differences.)
I don’t know the whole history of the notion of time as a dimension, but Hermann Minkowski (one of Einstein’s professors) was the one who came up with the idea of a unified 4-dimensional space-time continuum. Einstein himself hadn’t thought of it that way (according to a history of relativity that I read somewhere) until sometime later. Einstein didn’t like the idea at first.
Some chronology:
[ul][li]1905: Einstein’s Special Theory of Relativity.[/li][li]1907: Minkowski develops 4-dimensional unified space-time geometry, suggests that Einstein’s Relativity can be modelled that way.[/li][li]1908: Einstein rejects that idea.[/li][li]1912: Einstein decides that 4-D space time is the way to go after all.[/ul][/li]
Citations:
Wikipedia:
– Biography of Minkowski, with brief paragraph on his space-time idea.
– History of Relativity, with a little more detail on Minkowski’s space-time development, including the time-line mentioned above.
Though too late to be considered the first, William Rowan Hamilton advocated the use of his algebra of quaternions – four-dimensional generalizations of complex numbers – to represent points in space-time (see the second quote in particular). He even wrote a poem about it:
THE TETRACTYS
Or high Mathesis, with her charm severe,
Of line and number, was our theme; and we
Sought to behold her unborn progeny,
And thrones reserved in Truth’s celestial sphere:
While views, before attained, became more clear;
And how the One of Time, of Space the Three,
Might, in the Chain of Symbol, girdled be:
And when my eager and reverted ear
Caught some faint echoes of an ancient strain,
Some shadowy outlines of old thoughts sublime,
Gently he smiled to see, revived again,
In later age, and occidental clime,
A dimly traced Pythagorean lore,
A westward floating, mystic dream of FOUR.
However, his view lost out against the vector-based proposals of Gibbs and others. What’s interesting however is that there is a direct connection between quaternions and special relativity: space and time are treated in the same asymmetric, but equivalent way as Indistinguishable describes with respect to Minkowsky space.
There are many stories about moving through time before Wells. Time was a major interest not just in scientific publications but in philosophical ones. The Industrial Revolution brought the metaphors of machines and clockwork into the public consciousness. Time was something that clocks measured, but anything that could be measured could be controlled. From that came stories in which clocks run backwards and move people back in time. At the moment I don’t have access to my copy of Edward F. Bleiler’s indispensable Science Fiction: The Early Years, in which he recounts the plot of every science fiction story before Gernsback. I know that he describes these stories there.
It’s difficult to say that they thought of time as the fourth dimension, but they did see it as something travel-able. Space-time as we know it emerged from 20th century science, not 19th century storytelling, though.
Oh, don’t get me started… Division algebras in physics are kind of a hobby, and yes, octonions are used in string theory, but they’re fascinating way beyond that! This would take us a bit too far afield, though.
Hm… there seems to me to be rather a bit more asymmetry/inequivalence in the quaternions, in that “time” is singled out as the multiplicative identity (and, as a corollary, “time” is singled out as squaring specifically to +1; the sign convention for whether “space” or “time” goes with +1 is no longer arbitrary in this context).
But perhaps the connection you are noting is something further than just that there is 1 dimension which squares to positives and 3 dimensions which square to negatives in the quaternions? Is there actually a way to usefully apply quaternions to studying Minkowski space? [I would be pleasantly surprised to learn how; I always think of quaternions as rather Euclidean objects, in that, just as complex numbers are scalings-and-rotations in 2d Euclidean space, quaternions are scalings-and-(isoclinic rotations of a designated handedness) in 4d Euclidean space]
For what it’s worth, appealing to google’s scanned book database turns up:
Inquiries into human faculty and its development - Sir Francis Galton (1883)
“It is difficult to withstand a suspicion that the three dimensions of space and a fourth dimension of time may be four independent variables…”
Four-Dimensional Space, Nature March 26, 1885
“…in our fourth dimension of time.”
I’ve also pondered some about modeling Minkowski space using quaternions, but haven’t really been able to find any practical implications of it. The bit about three of them having opposite squaring-sign from the fourth fits well enough, but the part about i*j = k etc. doesn’t really seem to correspond to anything in physics. In other words, you could do it, but it wouldn’t help any more than any other way of treating Minkowski space.
Oh, and I don’t know about any practical uses of octonions, but I can say that they don’t have as interesting a structure as the quaternions. The quaternions form a group, but you can’t set up the octonions in such a way as to form an equivalent group (I think you end up losing associativity, but I’m not certain on that).
Quaternions don’t quite form a group, except under addition; they’ve got 0, after all. But they do form a skew field, and so the non-zero quaternions do form a group.
Yes, you lose associativity of multiplication when moving to octonions, which as a category theorist causes me to eye them with suspicion. It means octonions can’t possibly be represented as some ring of linear operators on some space (since they aren’t a ring!). But I suppose they still have their uses, all the same.
Mendel Sachs has done some interesting stuff with quaternions in GR (see here, for example), claiming to derive QM using GR’s quaternionic reformulation. He is generally considered a crackpot (he uses his formulation, for example, to prove that there is no asymmetric aging in the twin paradox). But in all honesty I haven’t seen his ideas discussed very much at all by mainstream physicists. It would be nice to see a good solid debunking from someone.
A word of warning: if you’re not interested in division algebras in physics, it’s probably best to skip this post… I don’t want to end up killing the thread with this diversion.
Well, it’s not an entirely straightforwards story (or at least, to my knowledge, it hasn’t been really straightened out), but there’s an argument to be made, I think, that the connection goes a little further. For instance, the received wisdom is that you need biquaternions (quaternions with complex coefficients) to represent Lorentz transformations; a variation on that story (which doesn’t mention or use biquaternions, but essentially just transfers the complex part to the rotation angle) is told here (pdf link). There’s also a way to make do ‘only’ with real quaternions, essentially by noting that a quaternion can be written as a certain 2 x 2 complex matrix, and relating that to SL(2, C), which is isomorphic to SO(3, 1), the Lorentz group; but the construction is not very natural (see here (pdf)).
Well, the most obvious place for octonions in physics is indeed in string theory: they’re the reason it has to be ten dimensional. Basically, one can only have supersymmetry in certain dimensions, because it hinges on a certain algebraic relation holding in division algebras. Those have the dimensions 1, 2, 4, and 8, so a (classical) superstring Lagrangian only exists in the dimensions 3, 4, 6, and 10 (the extra two come from the string’s two-dimensional worldsheet). This continues: a classical supermembrane Lagrangian exists correspondingly in dimensions 4, 5, 7, and 11. Quantization considerations pick out the cases of 10 resp. 11 dimensions for string and M theory. So if string theory turns out to be right (big if, I know), then octonions are actually absolutely fundamental to physics!
But they pop up in other places, too. For instance, in the classification of entangled qubit states: basically, the cases of one, two, and three qubits can be related to the three parallelizable spheres (via Hopf fibration), S1, S3, and S7 – the spaces of unit complex numbers, quaternions, and octonions, respectively. (Essentially, the state space of a single qubit is S3; this can be represented as the usual Bloch sphere, S2, via modding out the S1 global phase. The S3 can thus be seen as a S2 base manifold with an S1 fiber. This pattern continues: the 2-qubit state space S7 – the unit quaternions – can be viewed as an S3 fiber over S4; the 3-qubit S15 as a S7 fiber over S8. Thus, the fiber of one always gives you the state space of the next lower level!)
Crucially, the unit quaternions form the group SU(2), relating them to the Pauli matrices and the mathematics of spin (that SU(2) is the double cover of SO(3) is why they’re so good for representing rotations in 3-space). This is similar to the unit complex numbers forming the group U(1). These are two of the three groups making up the standard model symmetry, U(1) x SU(2) x SU(3). What about the third? Well, it’s not quite as easy as the unit octonions forming SU(3) – as obviously, the octonions don’t form a group because of their non-associativity. But still, one can relate the imaginary octonion units to the Gell-Mann matrices, the generators of SU(3), in a way similar to how the quaternions relate to the Pauli matrices. More technically, the automorphism group of the quaternions is G2, which, upon ‘picking out’ a special imaginary octonion direction to be left invariant, breaks down to – SU(3)! This might seem arbitrary, but a second thing happens: the ‘Lorentz group’ associated with the octonions, SO(9, 1), breaks down to SO(3, 1); we get a mechanism of dimensional reduction leaving us with a U(1) x SU(2) x SU(3) symmetric theory in 3 + 1 dimensions!
Now, this may be nothing more than a mathematical coincidence, but it’s something that is seriously pursued, if a little off the beaten path. And actually, a kind of ‘weak theory’ (i.e. a theory of the weak force, not a theory that’s not very good) was historically found using quaternions, before the U(1) x SU(2) modern electroweak theory. I always thought the fact that the division algebras so (almost) uniquely pick out these of all the possible Lie groups to be quite intriguing…
A clearer telling of the story of quaternions, biquaternions, and special relativity can be found here, if anyone’s interested (this also contains an introduction to quaternions and calculus done using them).
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