In article <uloseh$3dbv3$
1...@dont-email.me>, Luigi Fortunati wrote:
> In the Wikipedia entry "Spacetime" there is the figure of the
> space-time sheet that curves *downward* when we place a mass on it.
>
> If there are many directions in the GR, why does the elastic sheet
> sink *always and only* downwards, i.e. towards a single direction?
As you point out, the rubber-sheet analogy has some serious conceptual
problems. My personal opinion is that this analogy is actively misleading,
and doesn't contrube to any useful understanding of general relativity
(see also <
http://xkcd.com/895/>).
Ideally, forget you ever saw the rubber-sheet analogy. :)
<<following text copied from a 2019 posting of mine in this newsgroup>>
In my opinion a better analogy is to consider horizontal motion on the
Earth's surface (which we can treat as being spherical for present
purposes).
Suppose you have two nearby ships on the Earth's (ocean) surface, whose
courses are initially parallel (i.e., d/dt of the distance between them
is zero), and that both ships move along "locally straight" lines, i.e.,
symmetrical hulls and rudders not deflected left or right. Then each
ship will travel along a great circle on the Earth's surface. This
implies that the ships paths will *converge* (i.e., d^2/dt^2 of the
distance between them must be negative).
You could attribute this convergence-of-paths to a mysterious
"Newtonian gravity" ship-to-ship attraction, but it's informative to
instead regard it as a manifestation of the Earth's surface having
"intrinsic curvature", i.e., *not* satisfying the axioms of Euclidean
geometry.
[Here the adjective "intrinsic" means that we're treating
the curvature as an attribute of the 2-dimensional surface
itself; not as an artifact of an embedding in a 3-dimensional
world.]
As supporting evidence for this interpretation, note that the
ship-to-ship acceleration (i.e., d^2/dt^2 of the distance between
nearby ships at the moment they're some standard distance apart) is
universal, independent of the properties of this ship. [This is
analogous to the universality (at a given place and time) of the
free-fall acceleration of test masses in Newtonian gravitation,
regardless of the properties of the free-falling test mass.]
Moreover, we observe that a "locally straight" route [= that taken
by ships & aircraft] from (say) Paris (France) to Vancouver (Canada)
curves far to the North of the origin & destination cities, typically
passing over central to northern Greenland. If we didn't know about
the curvature of the Earth's surface, we might explain this by saying
that there is an equator-attracting "action at a distance" force on
the Earth's surfce, causing ship and aircraft trajectories to be
concave towards the equator for the same reason that a thrown ball's
path is concave towards the ground.
>From a "Newtonian" perspective, this equator-attracting force can
acclerate ships and airplanes "down" towards the equator, and thus
is a real physical force (it can do work in the Newtonian sense).
>From a curved-space perspective, we instead see that these ship and
aircraft simply follow great circles (geodesics).
Ok, that's enough of the analogy. I can think of 2 major ways in
which this analogy fails:
1. Gravitation is actually a phenomenon of curved *spacetime*, not just
of curved *space*. It's this that allows a very weak gravitational
field like that of the Earth, to curve a thrown ball's path so
strongly (from 45-degrees up to 45-degrees down in just a second
or so).
[One of Wheeler's books has a nice sketch showing a
ball's path in 3-D, with axes x, y, and c*time, making
it clear that in a 1-second flight, the thrown ball
has travelled some small number of meters in x and y,
but also 300,000 km in c*time, so the actual *curvature*
of its path in spacetime is very small.]
2. In our analogy, the Earth's surface has *extrinsic* curvature: it's
curved (does not satisfy Euclid's axioms of plane geometry) due to
the way it is embedded as a 2-dimensional surface in a 3-dimensional
world. In contrast, we do not believe that the curvature of
4-dimensional spacetime is due to it's actually being embedded in
some 5-dimensional "hyper-spacetime"; rather, spacetime curvature
is seen as *intrinsic* to spacetime itself.
--
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