Sunday, September 21, 2008

100 Years of Space-Time

Die Anschauungen über Raum und Zeit, die ich Ihnen entwickeln möchte, sind auf experimentell-physikalischem Boden erwachsen. Darin liegt ihre Stärke. Ihre Tendenz ist eine radikale. Von Stund' an sollen Raum für sich und Zeit für sich völlig zu Schatten herabsinken und nur noch eine Art Union der beiden soll Selbständigkeit bewahren.

Hermann Minkowski, opening words of his talk "Raum und Zeit" at the 80. Meeting of Natural Scientists and Physicians, Cologne 1908 (English translation see footnote).

Hermann Minkowski, 1864-1909 (from MacTutor History of Mathematics Archive)
September 21, 1908, was a wonderful and sunny late-summer Monday in Cologne, Germany, where scientists from all over the country had come together for the 80th General Meeting of the Society of Natural Scientists and Physicians.

On that day, Hermann Minkowski, a well-known mathematician from Göttingen, gave a talk with the title "Raum und Zeit" – "Space and Time". In this now famous talk, Minkowski proposed a new formulation of the special theory of relativity. His formulation implied a unification of the notions of space and time, which traditionally have been seen as completely independent, to a four-dimensional entity dubbed "space-time".

Points in this "space-time" correspond to "events", e.g. things happening at a certain time and at a certain point in space, and Minkowski proposed to define a distance between events x (at time t and location x, y, z) and x' (at time t' and location x', y', z') by

distance(x, x') = c²(tt')² − (xx')² − (yy')² − (zz')²,

where c is the speed of light. The distance between two events defined in this way is, according to the special theory of relativity, the same for all observers in uniform relative motion, or, using the technical jargon, does not change under Lorentz transformations. This definition is a generalization of the Euclidean distance between two points in space, which does not change ("is invariant") under rotations, and the corresponding four-dimensional space-time is now called "Minkowski space".

All the concepts we now use to describe the kinematics of special relativity – events, worldlines, light cones – were presented in front of a large public audience for the first time one hundered years ago today, in Minkowski's lecture.

Future ("Nachkegel") and past ("Vorkegel") light cones, and timelike ("zeitartiger") and spacelike ("raumartiger") vectors in the writeup of Minkowski's talk (page 82 of Raum und Zeit, Jahresbericht der Deutschen Mathematiker-Vereinigung 18, 1909).

Worldline ("Weltlinie") of a particle in Minkowski spacetime (page 86 of Raum und Zeit).

Hermann Minkowski was born in Lithuania, and had studied mathematics at the University of Königsberg. His contributions to number theory, complex analysis and algebra had made him quite renowned at a young age, and he held positions as professor of mathematics at the universities of Bonn, Zürich (the ETH), and Göttingen. At Göttingen, he shared the interest of Hilbert in the problems of the theory of the electron and special relativity.

Curiously, his worldline hat crossed that of Albert Einstein before: Einstein, as a student of physics at Zürich, had been taught mathematics by Minkowski. But it seems that Minkowski didn't have a very good impression of his student. On the other hand, Einstein had some difficulties to make sense of the reformulation of his theory by his former teacher. Arnold Sommerfeld quotes Einstein as having said in reaction to Minkowski's work that "since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore."

But it's clear that Minkowski's four-dimensional world was an essential conceptual step in the understanding of relativity, and indispensable for the later formulation of general relativity. Unfortunately, Minkowski didn't live to see or even foster these developments. His lecture on "Space and Time" was his last scientific work – he died from a ruptured appendix in January 1909, at the age of 44.

The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

Translation by W. Perrett and G.B. Jeffery, taken from Hermann Minkowski, "Space and Time", in Hendrik A. Lorentz, Albert Einstein, Hermann Minkowski, and Hermann Weyl, The Principle of Relativity: A Collection of Original Memoirs on the Special and General Theory of Relativity (Dover, New York, 1952), pages 75-91.

For more about Minkowski and his role in special relativity, see e.g. L Corry: Hermann Minkowski and the postulate of relativity, Arch. Hist. Exact Sci. 51 (1997), 273-314 (a free preprint as PDF is here), and Scott Walter: Hermann Minkowski’s approach to physics, Math Semesterber. 55 (2008) 213–235 (preprint as PDF), and Minkowski, Mathematicians, and the Mathematical Theory of Relativity, in H. Goenner, J. Renn, J. Ritter, T. Sauer (eds.): The ExpandingWorlds of General Relativity (Einstein Studies 7), Boston/Basel: Birkhäuser 1999, 45–86 (preprint as PDF).


  1. Let's get the party started!!!

    It's already the 22nd where I live. :(

  2. "Imagination is more important than knowledge."
    -- Einstein

    I do remember reading about how a mathematician (Polish name) helped Einstein with the mathematics of his Relativity theory. Didn't know it was his teacher! (my knowledge of Physics is superficial, my background is in Electrical Eng).

    I like the idea of 2-man teams. A HS friend of mine told me of a couple of Stanford research profs in Geophysics. One guy is a thinker ("Einstein" concept), the other one is a doer ("Minkowski" engineering the mathematics into the Concept).

    Every researcher should find a "partner" (or partners). So,'re a mathematical physicist, are you a Concept-eer or Mathematical Engineer? I think that interesting "satellite R&D Institute" that's been in development (Kea's Category Theory Inst, G. Lisi's "science hostels", my idea of "satellite R&D think-tanks" as part of a network, etc) should incorporate some novel tactics/strategies for effective R&D. Just look to the past ("retro"), & you can find jewels like this Einstein/Minkowski collaboration. Every Dept (not only Physics) should have a mathematician ON-STAFF, to check mathematical rigor & work hand-in-hand with scientists (who are the Concept-eers).

  3. What experimental evidence do we have that Godel's solutions to the field equations are not physical? Possible future topic?

  4. Minkowski deserves to be called the co-discoverer of relativity. If he could see the way SR is usually taught today, 100 years later, he would be shocked and depressed. *Still* with all that crap about "postulates" of relativity, still we have MIT undergraduates working out elaborate trivialities about long cars being parked in short garages....when are we going to dispense with all that nonsense and announce to our students: "Spacetime has a semi-riemannian structure. The rest is details." No, we are condemned to an eternity of lightning striking trains; those trains will still be being struck by lightning 500 years hence when everyone else has forgotten what a train *is*. End result: we *still* have ignoramuses like LM declaring that general relativity is "more general" than SR because it allows us to make acceleration "relative". What a mess!

  5. Hi Stefan,

    A nice piece reminding us of Monkowski’s key role in the development of the concept of space-time. As you point out it would have been interesting to see what he would have thought of Einstein’s extension and generalization of the concept. It is also interesting what Einstein said about mathematicians entering physics in regards to comprehension of theory, as it seems today that with string theory and other concepts it has in the main been turned over more to the mathematicians or rather the mathematical physicists.

    As such I’ve often wondered if perhaps mathematical and physical possibility is too often seen as being the same, giving rise to such pursuits in the first place. One could argue that anthropic concepts also have similarly been so inspired, where instead of asking what math helps us describe our world we often ask which world the math describes. I’ve often questioned if this is a reasonable approach in attempting to explain reality?



  6. Yes thanks Stefan.

    The value of non-Euclidean geometry lies in its ability to liberate us from preconceived ideas in preparation for the time when exploration of physical laws might demand some geometry other than the Euclidean. Bernhard Riemann

    Your blog post has produced some thoughts for me about what already existed at the time. In terms of non-euclidean geometries and people leading up to it. The mathematical work to resolving the fifth postulate. This was then awork in progress by the mathematicians, and was current at the time. This preceding the formulation of how we see the dynamical views of spacetime?

    Some additional historical perspective here.

    Giovanni Girolamo Saccheri

    Saccheri entered the Jesuit order in 1685, and was ordained as a priest in 1694. He taught philosophy at Turin from 1694 to 1697, and philosophy, theology, and mathematics at Pavia from 1697 until his death. He was a protege of the mathematician Tommaso Ceva and published several works including Quaesita geometrica (1693), Logica demonstrativa (1697), and Neo-statica (1708).

    What causes some thinking here is Grossman's introduction to Einstein on this subject.Minkowski was part of that group to deliver on the heritage of that move from non-euclidean.


    I think you have the right idea just that it's much more in depth then what you are saying in term of the relationship between two individuals.

    When one is doing mathematical work, there are essentially two different ways of thinking about the subject: the algebraic way, and the geometric way. With the algebraic way, one is all the time writing down equations and following rules of deduction, and interpreting these equations to get more equations. With the geometric way, one is thinking in terms of pictures; pictures which one imagines in space in some way, and one just tries to get a feeling for the relationships between the quantities occurring in those pictures. Now, a good mathematician has to be a master of both ways of those ways of thinking, but even so, he will have a preference for one or the other; I don't think he can avoid it. In my own case, my own preference is especially for the geometrical way. Paul Dirac and Geometrical Thinking


  7. There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.Nikolai Lobachevsky

  8. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

    So he turned out to be wrong about this. Physics still has time. FQXi's essay contest about nature of time suggests that we actually know time less precisely than spacetime but spacetime did not obviate time.

  9. Minkowski space-time concept has a deep meaning in concept of particle environement. By Aether theory time is formed by compacted space dimension (i.e. gradient of Aether density, similar to water surface) - therefore it's nothing strange, the matter spreading through space-time follows the Hamiltonian flow like during light spreading through water surface, including refraction phenomena.

    If you don't believe, the concept of particle environment is compatible with relativity, try to answer a few easy questions for yourself:

    Which particle environment isn't compatible with relativity exactly? The environment, where the energy spreading occurs via longitudinal waves (i.e. through gas), transversal waves (like during spreading of waves through foam) - or both?

    The spreading of water waves along water surface violates the relativity, if observed by light waves (a pair of waves involved in experiment). How such spreading would appear, if it would be observed by using of surface waves - i.e. exactly by the same way, like we are observing the energy spreading in vacuum (one kind of waves involved in experiment)?

    Can be (reference frame/motion of) some environment ever observed by its own waves? Can be particles of environment observed as an obstacles by using of wave, if they're serving as an environment for this wave already? Can some object serve as its own mean of observation, i.e. can be some object observed by itself?

    Did we ever think about these questions seriously? If not, why not?

  10. Hi Phil,

    at least in his later work, searching for a "unified theory", Einstein used quite sophisticated math.

    perhaps mathematical and physical possibility is too often seen as being the same

    maybe.. tell that Max Tegmark ;-)

    Hi zeynel,

    So he turned out to be wrong about this. Physics still has time.

    I don't remember who has pointed that out, but in fact Minkowski is inconsistent in his very first sentences, as he talks about "henceforth" and uses the future tense, "will survive", thus maintaining the special role of time.

    Best, Stefan

  11. Hi Stefan,

    “at least in his later work, searching for a "unified theory", Einstein used quite sophisticated math.”

    Yes that’s very true, yet at the same time sophisticated math attempting to describe a physical model that he thought might hold true. Also, in the end he was confident that he had the correct physical insight and concept yet complained that perhaps the math still need to be discovered. He leaves us with what that physical concept might be when in the last added appendix to his book on relativity he states as an overview the following.

    “In this edition I have added, as a fifth appendix, a presentation of my views on the problem of space in general and the gradual modifications of our ideas on space resulting from the influence of the relativistic viewpoint. I wished to show that space-time is not necessarily to which one subscribes a separate existence, independently of the actual objects of physical reality. Physical objects are not in space, but these objects are spatially extended. In this way the concept of “empty space” loses its meaning.”

    -Albert Einstein- June 9th, 1952
    - Relativity “The Special and the General Theory”-Note to the Fifth edition

    Interestingly enough it’s concepts such as Smolin’s Loop Quantum Gravity which is more or less the continuance of this approach, with its background independence and indeed there are new math’s such as Penrose’s spin network method that form to be just one of them. Never the less they only serve to describe the physical reality, not replace or diminish it.

    “maybe.. tell that to Max Tegmark ;-)”

    Actually from my way of looking at it this goes further to support my position rather then negate it:-)



  12. Great post Stefan, Minkowski is so often overlooked.
    On a trivial note, is it known if he ever regretted/retracted his 'lazy dog' comment on Einstein as a student? he must have got a shock when the 1905 paper first emerged..

  13. Hi Stefan,

    Thanks for the effort. Interesting post. I didn't know that Einstein's quotation about Minkowski. The photo is kind of intimidating though:-).

    Unification of space and time was a revolution; no doubt about it. Time is just another dimension. After 100 years, has the time come to abandon the notion of space-time as a whole? The fields do not propagate on the 4D space-time manifold. They just interact (and self interact) with the "surrounding" i.e. with other fields. In Nature there are only fields and their interactions, nothing else. No evolution in time, no moving in space and no space-time background.


  14. I didn't know he died so early on... very sad.

    The big question on everyone's mind is: in his talk, did he use the (+---) signature or the (-+++) signature?

  15. Hi quantum,

    The big question on everyone's mind ...

    Now, that's very important ;-), but easy to answer - you can check it out in the writeup of the talk:

    Minkowski uses coordinates x, y, z, t (page 76), writes the line element as cdt² − dx² − dy² − dz² (page 79), and later introduces, but more lightheartedly, the notation s = it, hence dτ² = − dx² − dy² − dz² − ds² (page 86).

    Cheers, Stefan

  16. Hi Cormac,

    I do not know the source or circumstances of the "lazy dog" remark, maybe someone can give a pointer?

    Anyway, in the summer of 1905, while Einstein was writing his paper on relativity as a patent clerk in Bern, there was a seminar in Göttingen on electron theory and the electrodynamics of moving bodies, run by Hilbert and Minkowski, and with Max Born participating as a graduate student, among others. It seems that Einstein was never mentioned in this seminar (he was unknown, and his paper not yet out), but that later, Minkowski commented to Born about Einstein: "Oh that Einstein, always missing lectures - I really would not have believed him capable of it" (quoted literally, in German, in Reids biography of Hilbert, but without given source). So it seems to me he was really surprised by his former student, but I guess he did stick to that opinion.

    Cheers, Stefan

  17. (Based on a comment at Uncertain Principles, in "Everything is Relative")

    The idea of "space-time" led, in General Relativity, to the idea of space as being like a "rubber sheet" than could be curved as the basis of gravity. So people asked, what does "space" curve through unless there's more space (hyperspace) around it. That led to many diagrams showing a surface in emptiness, and the notion we can generalize to a three-surface curved into a four-space (sic) etc.

    But here's a fundamental problem I have with thinking of "space" (in effect, the constraint domain of the movement of matter and radiation, right?) as being like a "sheet" (rubber or otherwise) that can be inside another "space" with more dimensions. Here I mean a macro space with more dimensions to "hold" a space with fewer, and not confusing with time either (so: a four-space dimensional manifold to "hold" a curved 3-D space within it.) Sure, in math I can just specify a manifold, a surface or space as part of another space by stating the rule for the locus of points. I can say, "the locus of points equidistant from a given point" which creates by semantic fiat a spherical shell in any space. The shell is literally curved (showing intrinsic non-Euclidean geometry), and has dimensionality one less than the parent space.

    Some physicists and philosophers of science say, there really isn't (or "doesn't need to be," seen as the same point to the empirical minded) some hyperspace that our space has to be "curved into", it's just a way of talking about what happens here. Hence you can imagine that space doesn't really pucker around a mass, but rather that rulers shrink in the radial direction. Some writers literally phrase it that way. That effect would effectively seem to be space curvature (e.g., more rulers can be placed = more distance to travel, when going through the pucker versus around it, etc.) But even if curvature can be "simulated" by distorted rulers "on a flat space,": don't you need "real curvature" and not just the equivalent distortion on a flat surface, in order to get closed, finite volumes of space? (Otherwise, the mapping doesn't work out does it?)

    So as far as physics goes, in what sense is the "space" that holds how matter can move distinguished from the equally empty "space" that holds the first space? I know, there are quantum issues and maybe gravitons work differently, but we still can't merely draw pictures of surfaces inside another "space" and think we've explained anything. Like I said, with mathematics you get to specify loci literally by saying so, but in a natural world there's "something" that has to keep objects held inside a locus of points defined within a more bountiful (in whatever sense) "space". IOW, if there's no difference in "kind" about space and its containing "hyperspace" then it's just like trying to have water surfaces distinguished inside of water etc. - what makes the difference between them (unless you follow Tegmark's modal realism.) So, what does that job? What keeps particles etc. penned in when there's "more room" available in principle? (Please, no circular arguments or semantic tricks. And it isn't just "metaphysics" to ask this, if your theory employs "spaces" as a physical constituent.)

  18. Thanks Stefan, very interesting. Other people that also contributed to what was to become relativity were Lorentz and Poincare. The genius of Einstein was to 'compile' all of this, and add some insights of his own into his special and general theories of relativity. Absolutely in no way do these things dimish Einsteins' genius.

  19. That we live in space-time is very consequential; that we live in Lorentzian spacetime is seemingly much less consequential, because our theories seem to work quite well with analytic continuation to imaginary time. Yet, we believe we are not living in this analytically continued space; to me it means the euclidean rotation fails to capture something essential. But I can't put my finger on what precisely it is.


  21. I know this is not a "ask-the-experts" forum, but still, here goes - in Lorentzian space-time how do I define the neighborhood of an event?

    Thanks in advance!

  22. Well one can parametrize curves through other variables than their proper length. The existence of local neighborhoods is a consequence of our space-time manifold typically assumed to be a Hausdorff space, it has a priori nothing to do with the metric over the space. You however need the metric if you want to define 'sizes' of neighborhoods. Does that help?

  23. It is just that the coordinate description of a neighborhood is funny. Nor is it clear to me what a neighborhood means physically.

  24. Hi Stefan,
    I'm still trying to find an original source for the'lazy dog' comment.
    It is widely quoted by reputable people like John Gribbin (or see NS at (
    and refers to the fact that E. rarely attended lectures. I'm sure it's in the PAIS biography, I seem to have lent it to someone...Cormac

  25. P.S. Do you know a good source for a timeline of physics?
    I keep just missing great anniveraries like the Minkowski one

  26. Hi Cormac,

    thanks for getting back about the "lazy dog". I couldn't find the quote in Pais, or in other sources, and I'm quite convinced now that the phrasing "lazy dog" is not authentic. Gribbin and the NS are a bit too journalistic to be trusted for exact phrasing, when the made-up formulation is more colourful.

    My guess is that the "lazy dog" goes back to the quote cited in the Hilbert biography by Constance Reid I have mentioned, which unfortunately doesn't give sources. But the German "schwänzen" is how you typically call it when lazy students skip classes ;-)

    Do you know a good source for a timeline of physics?

    No, unfortunately not...

    Best regards, Stefan

  27. The pocked diary for physicists from the PDG lists birthdays of (deceased) physics VIPs.

  28. Hi Bee,
    The venerable Choquet-Bruhat DeWitt-Morette Analysis, Manifolds and Physics tells me that I need very little extra to give me a locally Lorentzian space. But it gave me another headache.

    Let me quote a little: (section V.3)

    A line element (direction) at x ∈ X is a 1-dimensional vector subspace of T_x.

    Theorem: On a paracompact C^1 manifold X the existence of a continuous line element field is equivalent to the existence of a hyperbolic riemannian structure on X.

    Theorem: (1) Any non-compact paracompact C^1 manifold can be given a hyperbolic riemannian structure.

    (2) A connected compact C^1 manifold can be given a hyperbolic riemannian structure if its Euler-Poincare characteristic is zero.

    (3) If a compact connected orientable manifold can be given a hyperbolic riemannian structure then its Euler-Poincare characteristic is zero.

    -- And I'm thinking I'm understanding something, when I realize that no, I don't - what are higher genus string world sheets? Is this theorem restricted to 4 dimensions? or are worldsheets not compact? not connected? or without hyperbolic riemannian structure?

  29. You can ask questions! I have no clue, I don't even know the book. I'd guess if it's compact and the EPC is nonzero the riemannian structure is not hyperbolic?

  30. Hi Bee,
    This is the book CIP determinedly tried to read but never got past the first 15 pages.

    Quote: "Baez once advised every aspiring mathematical physicist to buy the two volumes of Analysis, Manifolds, and Physics by Yvonne Choquet-Bruhat and Cecile Dewitt-Morette, and keep it at his/her bedside until the entire contents were mastered. I only have volume one by my bedside, and somehow, every time I get to page 15, I feel my eyes closing, so clearly I wasn't meant to be a mathematical physicist."

    I'm in the same boat. :(

    Since you don't have the book, let me quote from a little before where I quoted previously:

    "We now restrict our attention to the case of fundamental interest in physics, the 4-dimensional pseudo-riemannaian manifold of index 1 {i.e., one timelike direction }. Such a manifold is called a hyperbolic manifold. The metric is called lorentzian".

    Then there is a whole bunch of definitions and theorems and then the stuff I quoted.

    I thought the string worldsheet action was written in terms of a lorentzian metric on the worldsheet; is that theorem i quoted telling me that such a metric doesn't exist except on the torus?

  31. Ah. Well, without the book I'm still somewhat poking in the dark, so 'hyperbolic riemannian structure' means just it's a manifold with lorentzian metric on which one can define a riemann tensor etc, or is there more to it?

    Well, the string worldsheet isn't 4-dim, so yes, possible it's a matter of dimension? I don't usually worry about compact 4-dim spaces with holes admittedly, so the interesting part of the theorem for me seems to be (1), which is good to know. Best,



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