Space/Spacetime Metrics: Comparing Routes in PHY 101

[robphy]

[Moderator's note: Spin off from previous thread due to more advanced level of discussion.]

In relativity, it turns out that the elapsed time measured by your clock is a measure of "distance" traveled through spacetime. Just like with distance between points in space, the "distance" between two events in spacetime depends on the route taken.
The twins took different routes between the departure and return events, ...
(snip...)

The above is true even on an ordinary position-vs-time graph in PHY 101,
when "distance" is appropriately defined.

(...snip)
so have different elapsed times.

OOPS (Thanks @PeterDonis ) This turns out to be true only in the Galilean/Newtonian case,

CORRECTED: ...in general.
They are only the same in the Galilean/Newtonian case

an exceptional case in this Euclidean-Galilean-Minkowskian viewpoint of geometry.

 

[PeterDonis]

The above is true even on an ordinary position-vs-time graph in PHY 101,
when "distance" is appropriately defined.

There is no "distance through spacetime" in Newtonian physics, because there is no spacetime metric. @Ibix is not talking about ordinary distance through space.

This turns out to be true only in the Galilean/Newtonian case,

Isn't this backwards? In Newtonian physics, if two twins separate and then meet up again, the elapsed time on both their clocks will always be the same.

 

[robphy]

There is no "distance through spacetime" in Newtonian physics, because there is no spacetime metric. @Ibix is not talking about ordinary distance through space.

There is a metric (the degenerate Galilean metric in Cayley-Klein Geometry) that can assign a distance to timelike curves:
$$ds^2 =dt^2$$,
which is a special case of
$$ds^2 =dt^2 - \frac{c^2}{c_{max}^2}\frac{dx^2}{c^2}$$.

Those interested can revisit this discussion
https://www.physicsforums.com/threads/going-from-galilei-to-minkowski.930035/post-5871574 (especially here https://www.physicsforums.com/threads/going-from-galilei-to-minkowski.930035/post-5872323 )

By the way, I first learned of this notion from a chapter in Emch's book (see reference below),​

then found that it appears in the relativity literature (as given in links in the previous discussion).​

​

Mathematical and Conceptual Foundations of 20th-Century Physics​

By G.G. Emch (2000)​

https://www.google.com/books/edition/Mathematical_and_Conceptual_Foundations/eYQHIjkaEroC?hl=en&gbpv=​

look up "Cayley" and advance to page 94;

Isn't this backwards? In Newtonian physics, if two twins separate and then meet up again, the elapsed time on both their clocks will always be the same.

Oops... yes, that's backwards. I've corrected the post. Thanks.

 

[PeterDonis]

There is a metric (the degenerate Galilean metric in Cayley-Klein Geometry) that can assign a distance to timelike curves

But this "distance" has nothing to do with the actual path of the timelike curve; in a "twin paradox" scenario, for example, this "distance" would be the same for both twins even though they are following very different timelike curves. What @Ibix was saying, in the post you originally responded to, was that in relativity, the "distance" along a timelike curve does have everything to do with the actual path of the curve, and depends on the route taken; in Newtonian physics, it doesn't.

 

[vanhees71]

Within Newtonian dynamics there is no time dilation nor a twin paradox. By assumption time is absolute in Newtonian mechanics!

 

[robphy]

But this "distance" has nothing to do with the actual path of the timelike curve; in a "twin paradox" scenario, for example, this "distance" would be the same for both twins even though they are following very different timelike curves. What @Ibix was saying, in the post you originally responded to, was that in relativity, the "distance" along a timelike curve does have everything to do with the actual path of the curve, and depends on the route taken; in Newtonian physics, it doesn't.

The general idea is that,
in a general space with a metric,
given two points and a path joining them, a "distance" can be assigned along the path.

In general (like in the Euclidean and Minkowski case),
the "distance" is generally different between the different paths between fixed points [fixed events].
(In spacetime, all observers carry clocks (Bondi's private time) and these clocks define the "distance-in-spacetime", as an odometer defines distances in Euclidean space.)

In the exception Galilean/Newtonian case,
the "distance" is the same between the different paths between fixed points [fixed events].
(This is related to what we call Absolute Time.
In Galilean spacetime, all observers carry clocks (Bondi's private time) and these clocks define the "distance-in-Galilean spacetime". In the Galilean case, these elapsed times are the same along different worldlines between two fixed events. This is essentially what the section from Emch is saying.)

The "Route-Dependence" of Time

...
The crucial discovery of relativity was the route-dependence of time which, having previously been considered a public universal quantity, was naturally thought to be route-independent. Of course, it is only the fact that time has become private, rather than public, that allows it to be route-dependent at all, but the important point to emerge in the next chapter is that we cannot escape within the Principle of Relativity from the notion of route-dependence of time.

The reason for bringing this is up is
that we aren't surprised in the familiar Euclidean case.
In this framework,
we should learn not to be surprised in the Minkowski case
because it is the Galilean case that is the exceptional (very special case).

 

[robphy]

Within Newtonian dynamics there is no time dilation nor a twin paradox. By assumption time is absolute in Newtonian mechanics!

Yes, but not because of the "geometrical construction" (the figure on the diagram)...
but because of the metric (even if degenerate) on the space.

 

[PeterDonis]

in a general space with a metric

But spacetime in the Newtonian/Galilean case does not have a metric. There is a spatial metric, and a "temporal metric" (if you can call it that), but no spacetime metric. So Galilean spacetime is not a "space with a metric" and the "distance through time" cannot be thought of as "distance through spacetime along a timelike curve".

it is the Galilean case that is the exceptional

Yes, agreed, but the reason is is exceptional is that, as above, Galilean spacetime does not have a metric, whereas Euclidean space and Minkowski spacetime do.

In the exception Galilean/Newtonian case,
the "distance" is the same between the different paths between fixed points [fixed events].

But this "distance" is not even a "distance between points in spacetime" to begin with. See above. That seems to me to be a much simpler and cleaner way to describe why the Galilean/Newtonian case is exceptional, than to try to talk about this "distance" as though it were the same basic kind of thing as "distances" in Euclidean space or Minkowski spacetime, just with one special difference. It's really a different kind of thing altogether.

 

[PeterDonis]

because of the metric (even if degenerate) on the space.

There is no spacetime metric in Newtonian mechanics, not even a degenerate one. There are two separate, disconnected "metrics"--one on space (the usual Euclidean metric on 3-space) and one on time (which isn't even really a "metric", just a labeling of each "space" in an infinite stack of "spaces" by an absolute time parameter).

 

[robphy]

There is no spacetime metric in Newtonian mechanics, not even a degenerate one. There are two separate, disconnected "metrics"--one on space (the usual Euclidean metric on 3-space) and one on time (which isn't even really a "metric", just a labeling of each "space" in an infinite stack of "spaces" by an absolute time parameter).

Yes, there are two pieces, each degenerate.
And while there might be an issue of "terminology" (akin to relaxing "metric" to allow for indefinite signatures without using an explicit "pseudo-riemannian" adjective each time),
this does not take away from the construction of assigning a real-number to a path that corresponds to the elapsed time along the worldline (as Emch does above).

Call it whatever you want... the construction and calculation can be done.

 

[PeterDonis]

there are two pieces, each degenerate.

And I would say that that is why this...

the construction of assigning a real-number to a path that corresponds to the elapsed time along the worldline

...is not route dependent in the Galilean case, whereas it is in the Euclidean and Minkowskian cases because there is just one non-degenerate piece, not two degenerate ones.

 

[vanhees71]

Galiean space time is a fiber bundle, not an affine pseudo-Euclidean space as the Minkowski spacetime of special relativity or a pseudo-Riemannian manifold as in general relativity. At each point along the time axis you have a Euclidean space. This of course reflects Newton's idea of a fixed spacetime where time is absolute and independent of any physical actions. That's why a 4d pseudometric constuct as in relativity makes no sense in Newtonian mechanics.

 

[robphy]

Galiean space time is a fiber bundle, not an affine pseudo-Euclidean space as the Minkowski spacetime of special relativity or a pseudo-Riemannian manifold as in general relativity. At each point along the time axis you have a Euclidean space. This of course reflects Newton's idea of a fixed spacetime where time is absolute and independent of any physical actions. That's why a 4d pseudometric constuct as in relativity makes no sense in Newtonian mechanics.

Having the fiber-bundle structure does not prevent it from being an affine pseudo-Euclidean space.

all bolding below is mine

• https://en.wikipedia.org/wiki/Newton–Cartan_theory#Classical_spacetimes
  which is well-represented in the relativity literature.
  
  The Galilean/Newtonian case is the zero-curvature version of it.
• Andrzej Trautman's "Comparison of Newtonian and Relativistic Theories of SpaceTime": http://www.fuw.edu.pl/~amt/CompofNewt.pdf
  

  The metric structure of space-time according to the Newtonian theory is rather different from that in relativity. The Newtonian metric is degenerate; clearly, it is the limit as $c\rightarrow\infty$, of the relativistic metric.
  ...
  Clearly, if we agree to consider the world-lines of free motions as geodesics, the Newtonian space-time becomes endowed with an integrable affine connection. In other words, in the absence of gravitation, the first law says the following: the Newtonian space-time is an affine space whose straight lines correspond to free motions.

  ( http://trautman.fuw.edu.pl/publications/scientific-articles.html has many articles by Trautman )
  
• Trautman does also make reference to fiber bundles in
  http://trautman.fuw.edu.pl/publications/Papers-in-pdf/25.pdf
  (I have no trouble with this. My point is that this is not incompatible with the degenerate-metrics viewpoint.)
  
• In this degenerate-metrics spirit, there is also
  Jurgen Ehlers' frame theory in "Examples of Newtonian limits of relativistic spacetimes": http://pubman.mpdl.mpg.de/pubman/item/escidoc:153004:1/component/escidoc:153003/328699.pdf
• Topics in the Foundations of General Relativity and Newtonian Gravitation Theory
  David Malament
  https://www.socsci.uci.edu/~dmalamen/bio/GR.pdf#page=260
  https://press.uchicago.edu/ucp/books/book/chicago/T/bo12893557.html
  

  We take a classical spacetime to be a structure $(M, t_{ab}, h^{ab}, \nabla)$ where (i) M is a smooth, connected, four-dimensional manifold; (ii) $t_{ab}$ is a smooth, symmetric field on M of signature (1, 0, 0, 0); (iii) $h^{ab}$ is a smooth, symmetric field on M of signature (0, 1, 1, 1); (iv) $\nabla$ is a derivative operator on M; and (v) the following two conditions hold:
  (4.1.1) $h^{ab} t_{bc} = 0$ and (4.1.2) $\nabla_a t_{bc} = 0 \mbox{ and } \nabla_a h^{bc} = 0$. We refer to them, respectively, as the “orthogonality” and “compatibility” conditions. M is interpreted as the manifold of point events (as before). Collectively, the objects $t_{ab}, h^{ab}, \mbox{ and } \nabla$ on M represent the spacetime structure presupposed by classical Galilean relativistic dynamics. It will soon emerge how they do so
  ...
  (p.251)
  

  

  (This shows a construction, which begins generally, but reduces in the Galilean-case to the familiar path-independence of the elapsed time between two events.)
  -- I audited a class by Malament when I was in grad school
• The Galilean geometry can be viewed in the Cayley-Klein framework (from a projective geometry point of view) in
  Richter-Gebert's Perspectives on Projective Geometry: A Guided Tour through Real and Complex Geometry (a draft from his website)
  https://www-m10.ma.tum.de/foswiki/pub/Lehre/WS0910/ProjektiveGeometrieWS0910/GeomBook.pdf
  https://www.amazon.com/dp/3642172857/?tag=pfamazon01-20
  
  This projective point of view (which has Euclidean, Galilean, Minkowski as affine spaces
  and elliptic and hyperbolic as riemannian-signature spaces
  and (1+1)-deSitter and (1+1)-anti-deSitter as lorentz-signature spaces,
  these are 7 of the so-called nine Cayley-Klein geometries in 2 dimensions.
  (The remaining two are the Galilean-limits of the deSitter and anti-deSitter cases,
  analogous to the hyperbolic and elliptic cases.)
  
  Non-euclidean geometries: the Cayley-Klein approach
  Horst Struve & Rolf Struve - Journal of Geometry volume 98, pages151–170 (2010)
  https://link.springer.com/article/10.1007/s00022-010-0053-z
  
  A Simple Non-Euclidean Geometry and Its Physical Basis: An Elementary Account of Galilean Geometry and the Galilean Principle of Relativity
  I.M. Yaglom
  https://link.springer.com/book/10.1007/978-1-4612-6135-3
  
  Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry
  Francisco J. Herranz, Ramon Ortega, Mariano Santander
  J. Phys. A 33 (2000) 4525-4551
  https://arxiv.org/abs/math-ph/9910041
  
  and from G.G. Emch's Mathematical and Conceptual Foundations of 20th-Century Physics
  
  
  
  
• and these are the basis of this old poster of mine
  New Ideas for Teaching Relativity: Space-Time Trigonometry
  https://www.aapt.org/doorway/Posters/SalgadoPoster/Salgado-GRposter.pdf

So, while one may have issues with terminology,
the above references in the relativity literature
show how one can try to put such "geometries" (in the sense of Felix Klein)
under an umbrella where the constructions and structures are similar,
but can have differences depending on the
specifics of the metrical structure...however they are specified or approximately-specified...
by distance functions, angle functions, dot-products, metrics, matrices, quadratic forms, cross-ratios, etc...

 

[robphy]

For completeness, I provide the following unified construction
to assign an "arc-length" to timelike paths [here, composed of displacements in a vector space],
whose numerical value is determined by a real parameter $E$,
which can take values

• (-1 ) for Euclidean geometry
• ( 0 ) for Galilean spacetime geometry, and
• (+1) for Minkowski spacetime geometry

The construction essentially counts the number of "unit vectors" along each displacement,
where the "unit vector" is determined by the "slope" [of the displacement] (an affine concept)
and the value of $E$ in that matrix.

I'm not going to worry about terminology or generality (e.g. applying to other curves).
I'm just doing what I've claimed is possible:
using metrical information (provided somehow) to assign an arc-length to a
timelike curve between two fixed point-events,
which suggests that the notion of proper time (wristwatch time) that Minkowski developed
when he formulated the "spacetime viewpoint"
applies to the Galilean spacetime in the analogous way given here.

For the Euclidean case, this is like an odometer carried by the surveyor reading the distance traveled along that path.
For the Galilean and Minkowski cases, this is like a wristwatch carried by the astronaut reading the proper-time elapsed along that worldline.

[I have omitted the calculation for OT=5 and TZ=5, using the given grid and the affine structure.]
The key point is that without further metrical structure (provided somehow),
there is no specific assignment of the magnitudes of the vectors $\vec{OS}$ and $\vec{SZ}$ that are not along the given grid.

So, we have, for this triangle with legs 5 and 3 on the given grid,

• for $E=-1$, we have $OS=\sqrt{34}$ and $SZ=\sqrt{34}$,
  essentially leading to the triangle-inequality for Euclidean space
• for $E=0$, we have $OS=5$ and $SZ=5$,
  essentially leading to the "no clock-effect" ... absolute time for Galilean spacetime geometry
• for $E=+1$, we have $OS=4$ and $SZ=4$,
  essentially leading to the reverse triangle-inequality ... the clock effect for Minkowski spacetime geometry

Borrowing from an old diagram in an old post,
these results can be visualized as

...where the "diamonds with vertical diagonals along the verticals" are analogous to the usual rectangular grid boxes
and the "diamonds with diagonals along the OS and SZ displacements" are the results of a Euclidean rotation or Galilean or Lorentz boost of the original diamonds-with-vertical-diagonals, corresponding to the value of E.

These diagrams are just a representation of the calculation.
The key point is that
the magnitudes of the vectors OS and SZ (not parallel to the given grid lines)
are determined by the specification of the value of E, as given above.
(This is in the spirit of the Cayley-Klein geometry framework referenced earlier.)

(To play with this E=-1,0,1 parameter, visit my spacetime diagrammer: https://www.desmos.com/calculator/emqe6uyzha )