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An erroneous claim about spacetime geodesics

"Certain types of world lines (called geodesics of the spacetime) are the shortest paths between any two events, with distance being defined in terms of spacetime intervals." This claim is false. I've altered it twice, but both times the edit has been reverted. In Euclidean (and Riemannian) spaces, paths of (locally) stationary length are (locally) shortest paths. But the same is not true in Minkowski spacetime. Timelike geodesics are paths of longest temporal length. This is why, in the context of the "twin paradox" scenario, the twin whose motion is geodesic (unaccelerated) is older when they meet again than the twin whose motion is non-geodesic (accelerated). In Minkowski spacetime the length of spacelike geodesics is stationary, but they are neither longest nor shortest paths (e.g., they are shortest wrt variations confined to a spacelike hyperplane containing them, but longest wrt variations in a plane with timelike tangent vectors). This is all completely standard, but since it appears to be controversial, here are some references.

"A curve on $M$ that minimises its length $\int ds$... between two fixed points is actually a geodesic of for the metric g. \ldots When g is not positive definite, the argument is basically the same, but now geodesics do not minimize $\int ds$, the integral being what is called `stationary' for a geodesic" (Penrose, Road to Reality, pp 318-9)

"In the case of positive definite metrics, i.e. ones with signature of form (n, 0), we know, geodesics are *locally shortest* curves. The corresponding result for Lorentz metrics is that timelike geodesics are *locally longest* curves." (David Malament, [1], p. 8.)

The latest edit by [2] is still incorrect. Under no sensible construal of "shortest" are geodesics in Lorentzian spacetime shortest. In particular, spacelike geodesics are stationary but not extremal (and so not shortest -- in fact, for any two spacelike separated points in Minkowski spacetime, and for any positive number e, there is an everywhere spacelike curve connecting the points the length of which is less that e).— Preceding unsigned comment added by 86.9.64.18 (talk • contribs) 12:11, 6 March 2014‎ (UTC)

Please put new talk page messages at the bottom of talk pages and sign your messages with four tildes (~~~~). Thanks.

My edit does not say that geodesics in Lorentzian spacetime are shortest. Please feel free to improve the wording, but make sure it reflects the cited source. - DVdm (talk) 12:21, 6 March 2014 (UTC)

This is a tad confusing, not too clear in the article. I agree that it says "longest", but this is correct only for purely timelike worldlines. However, the reference uses the term "worldline" essentially to mean an arbitrary path in spacetime, not restricted to being timelike. And as the OP points out, the spacelike case seems to be considerably more complicated. We can either qualify the worldline as timelike and gloss over the rest, or we can sensitize the reader to the quirks of non-timelike geodesics. I favour something more sophisticated than a gloss. —Quondum 05:19, 7 March 2014 (UTC)

Clarity of Expression

I find the following statement to be unclear. In particular, I find the phrase "to a single abstract universe" to be hard to decode. Can it be clarified, please?

In cosmology, the concept of spacetime combines space and time to a single abstract universe.

Expos4ever (talk) 14:37, 22 December 2014 (UTC)

Portal links

To editor Tetra quark: Curious as to why the link to the Space portal was deleted? – Paine Ellsworth^CLIMAX! 13:16, 11 January 2015 (UTC)

I was adding portals on many pages yesterday so I don't remember exactly. I guess that its because spacetime is everywhere, and not only in outer space. It is more related to physics and (maybe)cosmology than anything else. Feel free to add it back anyway Tetra quark ^{(don't be shy)} 14:32, 11 January 2015 (UTC)

Yes, I agree that spacetime is found both in "outer space" and in everything within all the planets, moons and stars down to atomic levels and smaller. For example, what is between the tiny nucleus of an atom and its electrons can't be air, because air molecules won't fit. However in my opinion, and I may be wrong, the vast measure of spacetime must comprise a great deal more "outer space", than anything else. Thank you for your neutral "Feel free..."! Joys! – Paine  17:28, 11 January 2015 (UTC)

Dimensional analysis

Is spacetime a physical quantity? If yes, what is its dimension? L^3 T, namely volume times time?--188.26.22.131 (talk) 11:21, 19 January 2015 (UTC)

For the first Q, definitely yes, unless one has some particular (and somewhat peculiar) definition of what a physical "quantity" is. It may be that physical quantity of a physical "system" is required to be a number, like mass in kg or charge in Coulombs. But it is certainly a physical entity (or a physical thing if you want to) having an interpretation of being a background in which everything conceivable takes place. We spend our lives in it (for good and for bad). (Quotation marks are placeholders for a more rigorous definitions than the mere words suggest.) Spacetime is most certainly a physical thing, while mathematically equivalent things (like momentum space) do not have the same status of actually surely existing as a matter of physical reality.

For the second Q, yes again assuming how one should express a chunk of spacetime. But then there are different sets of units, e.g. natural units, that assigns different dimensions, but this is not really important. These are just mathematical rewrites of the same thing. YohanN7 (talk) 14:03, 19 January 2015 (UTC)

Indeed spacetime is qualitatively a background/frame for phenomena that unfold or a condition for the existence of substances and fields.

It seems that natural units have no physical dimensions. (It should not forgotten that a physical quantity is expressed as (symbolic) product between a numerical value and a unit of mesure (Maxwell's axiom).) I have just wondered on talk:speed of light if the speed of light can be expressed dimensionlessly and how this dimensionless numerical value is related to a unit of spacetime.--188.26.22.131 (talk) 15:36, 19 January 2015 (UTC)

It can be, but I'm unable to explain exactly how, I haven't tried to investigate, I just trust the words. This issue about units is decidedly non-trivial, and every physics textbook there is treats it as if it was (they all suck in this respect, as well as in many other respects) obvious. It isn't obvious. There is a reasonable treatment of natural units (with references) in J. D. Jackson's Classical electrodynamics. YohanN7 (talk) 16:22, 19 January 2015 (UTC)

Spacetime interval

In this case, we need to be careful of introducing new notation, partly for historical reasons. The angle-bracket notation is not defined nor otherwise used in this article, and most references (rather sloppily) actually do use the notation s² for this, even though they seem to be less consistent about the meaning of the term "spacetime interval". So, I'd avoid the angle brackets if possible (lots of books do) – we really need a symbol here, not an expression; as I've notes in the footnote, several authors seem to stick with a symbol that looks like an expression. History unfortunately carries with it a lot of inertia.

We also should point out that in its quadratic use, "spacetime interval" is a misnomer, but also we should find where it is used differently, e.g. to mean the Minkowski norm. A bit of checking references (e.g. Google books) should suffice. —Quondum 03:00, 28 February 2015 (UTC)

Added footnote

A recently added footnote: "More generally the spacetime interval can be written as $ds^{2}=g_{\alpha \beta }dx^{\alpha }dx^{\beta }$ with metric tensor g."

Surely this is not accurate? It could be thought of as a sort of infinitesimal interval, but I guess it might still be called a spacetime interval (yet another misnomer use for this term). My observation is that it is not correct to call it a generalization of s². In Minkowski space, the closest one would come to a true generalization is $\Delta s^{2}=g_{\alpha \beta }\Delta x^{\alpha }\Delta x^{\beta }$ (but this would not make sense in a general pseudo-Riemannian manifold). And if the notation of differential forms is intended, it is not consistent. —Quondum 17:21, 3 March 2015 (UTC)

It is from Line_element#General_spacetime. One generalisation is that in curved space we have to use differentials (or "approximately equal"), the other is that we can use other coordinates, and then have a more general symmetric matrix than a diagonal matrix with only 1 and -1. - Patrick (talk) 22:30, 3 March 2015 (UTC)

Yes, I understand what is happening; I'm just querying the use of the term "spacetime interval" (without any qualification to say that is the square of a differential), and of the description of it as a "generalization"; it seems there should be another word added or something. Another WP article is hardly suitable to use as a reference for usage of terms; we'd have bad terminology permeating related articles faster than we could clean it up. Actually, I'm wondering about the value of the footnote: it does not say whether it is dealing with straight, nonorthogonal coordinates, curvilieanr coordinates in Minkowski space, or general relativity. Whichever, this section is about basic concepts, and any generalization should be discussed later in the article. BTW, there is a difference between proper time τ (for which the zero point is arbitrary, like an integration constant, and for which an orientation must be assigned) and proper time interval |Δτ| (for which these considerations do not hold). —Quondum 23:27, 3 March 2015 (UTC)

I added "flat space" to the section title, and restricted the remark to that case. - Patrick (talk) 14:05, 4 March 2015 (UTC)

Did you notice the comment about a link to that section? It means that the link was broken when the section header was changed. {{Anchor}} is your friend. – Paine Ellsworth^CLIMAX! 02:19, 5 March 2015 (UTC)

Wrong definition

The definitions presented here are not applicable to general relativity, but only to special relativity and Newtonian physics. Definitely spacetime is NOT a "mathematical model that combines space and time into a single interwoven continuum", and it is NOT obtained "By combining space and time into a single manifold". The spacetime coordinates ("space" and "time") in GR have NO physical meaning. Certainly they do not have the meaning that is assigned to "space" and "time". The definitions given here are (at best) highly misleading and, if one wants to be rigorous, they are wrong. Probably a more rigorous definition would be something along the lines: "Spacetime in GR is an abstract manifold that identifies events through four numbers (coordinates). These four numbers can be related to the (proper) time and/or (proper) space of one or more observers (e.g. the GPS coordinates). The spacetime coordinates space and time do not have the conventional meaning of space and time (e.g. they are not measured through rods and clocks, except in special situations such as in special relativistic spacetime)."

A good reference for this is Rovelli's book (http://www.cpt.univ-mrs.fr/~rovelli/book.pdf), chapters 2.3 and 2.4. Rovelli is a leading researcher in general relativity.

Quantization of Spacetime

For the authors, if this article is of interest: https://www.quantamagazine.org/20150424-wormholes-entanglement-firewalls-er-epr The Quantum Fabric of Space-Time Jcardazzi (talk) 20:31, 25 April 2015 (UTC)jcardazzi

Spacetime vs. Space-time

Because of its unified nature, it should always be spelled spacetime without a hyphen. 69.180.104.60 (talk) 14:03, 4 January 2016 (UTC)

The literature seems to disagree: see Google Scholar and Google Books. - DVdm (talk) 14:10, 4 January 2016 (UTC)

3D + t = 4D

The three dimensions of regular space plus time equalling the four common dimensions is expressed through 3D + t = 4D. 69.180.104.60 (talk) 14:06, 4 January 2016 (UTC)

I have never seen that in the literature, so I have reverted. Please find a source for that. - DVdm (talk) 14:10, 4 January 2016 (UTC)

Its more usual to say "3+1" spacetime for 3 space + 1 time dimension. In general D + N, where the first number D is the number of space dimensions, and the second N the number of time dimensions. The sum is the number of spacetime dimensions. M∧Ŝc²ħεИ_τlk 15:00, 4 January 2016 (UTC)

Sure, but my point is that this content is already in the article, and there is no need to translate the fact from words into a nonsensical crippled pseudo-formula. - DVdm (talk) 15:09, 4 January 2016 (UTC)

By the way: [3]. - DVdm (talk) 17:10, 4 January 2016 (UTC)

Sorry, my comment was meant for the IP, shouldn't have indented it a space too far, but if blocked no matter anyway. Anyway thanks for the reply, M∧Ŝc²ħεИ_τlk 20:35, 4 January 2016 (UTC)

Ok, no problem, and happy 2016! - DVdm (talk) 20:36, 4 January 2016 (UTC)