Motivation for the introduction of spacetime

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What exactly are the theoretical motivations for considering space and time as a four dimensional continuum? Is it a natural consequence of requiring that the speed of light is independent of the frame of reference that it is measured in, since this implies that time and time are not absolute, but depend on the frame of reference that one measures them in, i.e. they are relative quantities. Also, requiring the constancy of the speed of light and that the laws of physics are independent any particular inertial frame, requires that spatial and temporal coordinates transform via the Lorentz transformations from one inertial reference frame to another. Such transformations explicitly "mix-up" spatial and temporal coordinates, such that the temporal coordinate of the "new" inertial frame is dependent on the spatial and temporal coordinates of the "old" inertial frame (that one has transformed from), and likewise for the spatial coordinates.
Finally, unlike in classical mechanics where the spatial line element is frame independent, it is a combination of temporal a spatial coordinates that form a frame independent quantity, suggesting that the "natural" geometry is four dimensional, forming a continuum called "spacetime", with a length in this four dimensional spacetime being determined by this new combination of spatial and temporal coordinates.

Is this in any way a correct motivation for considering spacetime instead of space and time as separate entities?
 

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PeterDonis
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What exactly are the theoretical motivations for considering space and time as a four dimensional continuum?
The four dimensional part is easy: three dimensions of space and one of time.

The reason it can't be separated into two independent continua, 3-d space and time, is, as you say, because "space" and "time" are frame-dependent; the invariant, frame-independent quantities that appear in the laws of physics are spacetime intervals, i.e., "lengths" in the 4-d continuum.

Actually, the part that still might need revision is the "continuum" part; some speculative theories of quantum gravity lead to spacetime being quantized, i.e., not a continuum. But that's still speculation at this point; we don't have any way of testing such hypotheses. As far as we can tell, spacetime is a continuum at the smallest scales we can probe; but the speculations about quantum gravity only predict anything different on scales about twenty orders of magnitude smaller.
 
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The reason it can't be separated into two independent continua, 3-d space and time, is, as you say, because "space" and "time" are frame-dependent; the invariant, frame-independent quantities that appear in the laws of physics are spacetime intervals, i.e., "lengths" in the 4-d continuum.
So would the reasons a gave be a satisfactory argument for why we must consider space and time as a single entity be correct then? They are frame independent in the sense that the coordinates that we use to label a spatial and a temporal point in one frame will necessarily both contain a mixture of temporal and spatial coordinates when labelled in another frame, and hence it doesn't make sense to treat the two independently, since they are frame dependent quantities?!
 
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PeterDonis
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So would the reasons a gave be a satisfactory argument for why we must consider space and time as a single entity be correct then?
They are the reasons standardly given in relativity theory. Whether they are "satisfactory" depends on who you are trying to satisfy. :wink: But I'm not aware of any serious scientific challenges to them.
 
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They are the reasons standardly given in relativity theory. Whether they are "satisfactory" depends on who you are trying to satisfy. :wink: But I'm not aware of any serious scientific challenges to them.
Ok, cool. Thanks for your help. I think the motivation is clearer for me now.
 
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pervect
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What exactly are the theoretical motivations for considering space and time as a four dimensional continuum? Is it a natural consequence of requiring that the speed of light is independent of the frame of reference that it is measured in, since this implies that time and time are not absolute, but depend on the frame of reference that one measures them in, i.e. they are relative quantities. Also, requiring the constancy of the speed of light and that the laws of physics are independent any particular inertial frame, requires that spatial and temporal coordinates transform via the Lorentz transformations from one inertial reference frame to another. Such transformations explicitly "mix-up" spatial and temporal coordinates, such that the temporal coordinate of the "new" inertial frame is dependent on the spatial and temporal coordinates of the "old" inertial frame (that one has transformed from), and likewise for the spatial coordinates.
Finally, unlike in classical mechanics where the spatial line element is frame independent, it is a combination of temporal a spatial coordinates that form a frame independent quantity, suggesting that the "natural" geometry is four dimensional, forming a continuum called "spacetime", with a length in this four dimensional spacetime being determined by this new combination of spatial and temporal coordinates.

Is this in any way a correct motivation for considering spacetime instead of space and time as separate entities?
I would suggest reading "The Parable of the Surveyor" from "Spacetime Physics", the first chapter of the 1960 version is downloadable from the internet on the author's website at http://www.eftaylor.com/special.html. There is an updated 1992 version of the text (but it's not available for free). The text is a fairly standard and popular textbook.

To very briefly recap the argument, one considers the question: Why do we regard north/south and east/west as being part of a unified entity, space, rather than regarding North/south and East/west as two different things?

The authors address this question in more depth, but as a short summary I'd give the following interpreation. The existence of distance as an invariant equal to (nort-south-distance)^2 + (east-west-distance)^2 justifies regarding them as part of something larger. The name for that "something larger" is the rotation group. We note that if we rotate our maps, by using true north vs. magnetic north, one sort of distance turns into the other. Similarly, the existence of the Lorentz interval (space-distance)^2 - (time-interval)^2, where the time interval is suitably scaled, justifies the unification of space-time as something larger, which is the Lorentz group. Additionally, the Lorentz boost shows a similar property - purely spatial intervals in one frame of reference have components in both the space and time dimensions in a boosted frame of reference - "boost" is just a technical term for a frame of reference that's moving relative to the first.
 
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robphy
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What exactly are the theoretical motivations for considering space and time as a four dimensional continuum? Is it a natural consequence of requiring that the speed of light is independent of the frame of reference that it is measured in, since this implies that time and time are not absolute, but depend on the frame of reference that one measures them in, i.e. they are relative quantities.
Technically speaking, the 3D-position-vs-time graph in Galilean/Newtonian physics is also a four-dimensional continuum.
Admittedly, unlike in special relativity, one can get (and has gotten) away with not looking as it as such [or treating it geometrically].
The spacetime geometries of special relativity and of Galilean relativity are both noneuclidean geometries
built upon on the same [topological] four-dimensional continuum.
 
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PeterDonis
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Technically speaking, the 3D-position-vs-time graph in Galilean/Newtonian physics is also a four-dimensional continuum.
Sort of. There is no "spacetime distance" in this case. See below.

The spacetime geometries of special relativity and of Galilean relativity are both noneuclidean geometries built upon on the same [topological] four-dimensional continuum.
No, the "geometry" of Galilean relativity is not a "spacetime geometry", because there is no invariant 4-dimensional "distance". The 4-dimensional spacetime "distance" between a given pair of events is frame-dependent.
 
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robphy
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Sort of. There is no "spacetime distance" in this case. See below.

[snip]

No, the "geometry" of Galilean relativity is not a "spacetime geometry", because there is no invariant 4-dimensional "distance". The 4-dimensional spacetime "distance" between a given pair of events is frame-dependent.

The Galilean / Newtonian Spacetime is well-studied in the literature. (references below)
Although its metric structures are degenerate, it is still considered a geometry.
[In an analogous way, Lorentz-signature geometries were not considered geometries early on because their distance function didn't satisfy the triangle-inequality or it wasn't positive-definite. That viewpoint has been relaxed somewhat these days.]
(In projective geometry [where Euclidean, Galilean, and Minkowski (as well as spherical, hyperbolic, and deSitter) geometries have a common foundation], the Euclidean structure has some degenerate structures. So, degeneracy is not as bad as one might think.)

references:
http://www.fuw.edu.pl/~amt/CompofNewt.pdf (Trautman)
https://www.amazon.com/dp/0226288641/?tag=pfamazon01-20&tag=pfamazon01-20 (Geroch)
http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf (Malament)
https://eudml.org/doc/75762 (Kunzle)
https://en.wikipedia.org/wiki/Newton–Cartan_theory (Galilean spacetime is a limiting case of Newton-Cartan)
https://en.wikipedia.org/wiki/Jürgen_Ehlers#Frame_theory_and_Newtonian_gravity (Ehlers)
https://www.amazon.com/dp/3642172857/?tag=pfamazon01-20&tag=pfamazon01-20 (Richter-Gebert)
https://www.amazon.com/dp/0387903321/?tag=pfamazon01-20&tag=pfamazon01-20 (Yaglom)
 
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PeterDonis
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Although its metric structures are degenerate, it is still considered a geometry.
I think "its metric structures are degenerate" understates the issue. The issue is that there is no "spacetime metric" at all. There are other mathematical objects in the theory, but no spacetime metric ##g_{ab}##. There can't be, because any such metric would have to give a "spacetime distance" between two events, along a particular curve, that was invariant under Galilean transformations of coordinates, and that is impossible.

It looks like the term "geometry" is still used to describe this thing, so I'll withdraw my flat statement that it isn't a geometry; that's a matter of terminology conventions, not physics. But I would still say that it lacks a key property, a spacetime metric, that "spacetime geometry" is usually associated with in discussions of GR.
 
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robphy
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But I would still say that it lacks a key property, a spacetime metric, that "spacetime geometry" is usually associated with in discussions of GR.
Yes, it lacks that property... and other structures have to be developed in order to describe the physics of a Galilean/Newtonian system.
The value of such an enterprise is to appreciate what physics can be modeled by the available structures... and in trying to understand the [unified?] structure of theories and their "classical limits"... rather than saying "with this kind of physics, you should think like this" and "with that kind, you should think like that"... probably because of historical precendent.

While the spacetime metric has been the central structure (and quite convenient) in most discussions of GR (and in Einstein's formulation [historically]), it may not be the only starting point.
In some formulations, the connection is central: https://en.wikipedia.org/wiki/Palatini_variation .
In attempts to generalize GR [for, say, quantum gravity],
one often starts varying (or weakening) aspects of the theory [with the "correspondence principle" constraint that one should recover GR in the appropriate limit]... asking questions like:
Is the metric "truly" fundamental? Or is it the connection? Curvature?
Dimensionality? Signature? Smooth Manifold?
Why [pseudo-]Riemannian and not Finslerian? (Ehlers http://projecteuclid.org/download/pdf_1/euclid.cmp/1103859104 )
Causal Structure? (Penrose Causal Spaces http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2063916)
Algebraic Structure? (Geroch Einstein Algebras http://projecteuclid.org/download/pdf_1/euclid.cmp/1103858122 )

There is some similarity to the idea of "not being too rigid" about Euclid's postulates by giving up Euclid's Parallel Postulate,
then realizing that hyperbolic and spherical geometries are possible and might actually be useful for something.

[added references to Geroch "Einstein Algebra" and Ehlers continuation of "Ehlers-Pirani-Schild"]
 
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stevendaryl
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I think "its metric structures are degenerate" understates the issue. The issue is that there is no "spacetime metric" at all. There are other mathematical objects in the theory, but no spacetime metric ##g_{ab}##. There can't be, because any such metric would have to give a "spacetime distance" between two events, along a particular curve, that was invariant under Galilean transformations of coordinates, and that is impossible.

It looks like the term "geometry" is still used to describe this thing, so I'll withdraw my flat statement that it isn't a geometry; that's a matter of terminology conventions, not physics. But I would still say that it lacks a key property, a spacetime metric, that "spacetime geometry" is usually associated with in discussions of GR.
I can't seem to find a reference online, but I read one treatment of Galilean spacetime that did use something like a metric, but unlike with the case for GR, this tensor (not a real metric) was not invertable. So there were two different metric-like tensors, [itex]g_{\alpha \beta}[/itex] and [itex]h^{\alpha \beta}[/itex], and they were not inverses. I don't remember the details.
 
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robphy
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I can't seem to find a reference online, but I read one treatment of Galilean spacetime that did use something like a metric, but unlike with the case for GR, this tensor (not a real metric) was not invertable. So there were two different metric-like tensors, gαβg_{\alpha \beta} and hαβh^{\alpha \beta}, and they were not inverses. I don't remember the details.
From my URLs above,
http://www.fuw.edu.pl/~amt/CompofNewt.pdf#page=3 (Trautman page "415")
http://www.socsci.uci.edu/~dmalamen/bio/GR.pdf#page=261 (Malament page "249")
http://archive.numdam.org/ARCHIVE/A..._17_4_337_0/AIHPA_1972__17_4_337_0.pdf#page=8 (Kunzle page "343")

Here's someone's dissertation ("The Newtonian Limit of General Relativity" by Reimold) that uses Ehlers' Frame Theory
https://www.math.uni-tuebingen.de/user/loose/studium/Diplomarbeiten/Diss.Reimold.pdf#page=21
 

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