Re: Curvature of spacetime

Subject	Re: Curvature of spacetime
Newsgroups	sci.physics.relativity
References	(6 earlier) <189fd0fe0d6aa5ae$7$157150$c2565adb@news.newsdemon.com> <10q0cke$1r43v$1@dont-email.me> <10q0tmh$lciq$1@gwaiyur.mb-net.net> <10q2684$mvcb$1@gwaiyur.mb-net.net> <IKWVyWcCsR7ckRyt5-ImR7NptmA@jntp>
From	Ross Finlayson <ross.a.finlayson@gmail.com>
Date	2026-03-25 21:22 -0700
Message-ID	<vimdncPyeo8IKVn0nZ2dnZfqnPWdnZ2d@giganews.com> (permalink)

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On 03/25/2026 09:11 PM, Python wrote:
> Le 26/03/2026 à 03:39, Thomas 'PointedEars' Lahn a écrit :
>> Thomas 'PointedEars' Lahn wrote:
>>> The "additional effect" that is observed irrespective of relative
>>> motion can
>>> be explained by the curvature of spacetime: if spacetime is curved
>>> *and* the
>>> local speed of light is the same in all frames of reference, then this
>>> effect would be observed.
>>>
>>> For example, with the Schwarzschild metric, a static solution of the
>>> Einstein Field Equations for a spherically-symmetric uniform
>>> distribution of
>>> the total mass M (m := G M/c^2)
>>> [
>>>          ds^2 = c^2 (d tau)^2
>>>               = (1 - 2m/r) c^2 dt^2 - (1 - 2m/r)^-1 dr^2 - r^2 (d
>>> Omega)^2,
>>>               = c^2 dt^2 [1 - 2m/r - (1 - 2m/r)^-1 (dr/dt)^2/c^2
>>>                           - r^2 (d Omega/dt)^2/c^2],
>>>
>>>   (d Omega)^2 = (d theta)^2 + sin^2(theta) (d phi)^2,
>>> ]
>>> it is clear that as/if c is the same in all frames of reference, then
>>> even
>>> at relative rest, (dr/dt)^2, (d Omega/dt)^2 = 0, the elapsed proper time
>>> (Delta tau = int_W d tau, W the section of a corresponding worldline)
>>> depends on the distance r from the center of mass.
>>> [...]
>>
>> Another thing that confuses a lot of (lay)people (which included me
>> before
>> my studies) is the concept that I have mentioned here: "the curvature of
>> spacetime".  How can spacetime be "curved"?  Is space "curved", is time
>> "curved", and if yes, what does that mean?  And what is this "spacetime"
>> anyway?  Pictures of bent rubber sheets certainly do not do this concept
>> justice, and without proper explanation confuse more than they help.
>>
>> Why do we even call this "curvature"?  This had confused me, too, until I
>> studied general relativity.  It is actually a term taken over from
>> analyzing
>> three-dimensional surfaces that we (humanoids) can still intuitively
>> understand as we can see and touch them.  If such a surface is curved,
>> i.e.
>> not flat like a sheet, and you put a coordinate grid on it where grid
>> lines
>> would have been evenly spaced if it were flat, then it depends on
>> where you
>> are on the surface what the distance between points on it along it is
>> if you
>> use those coordinates.  For example, the distance between two lines of
>> longitude that differ by 1°, i.e. between two points on the surface whose
>> geocoordinates only differ in longitude and only by that much, is much
>> larger at the equator [ca. (40000/360) km = 111 km] than at the poles
>> [all
>> lines of longitude meet there, so the distance between them there is 0
>> km].
>>
>> How do we measure the distance between points on such a surface
>> *along* that
>> surface?  It is curved, so if the triangle becomes large enough,
>> Euclidean
>> geometry and thus the Pythagorean theorem do not work anymore.  (They
>> never
>> work then, but for small areas the error is too small and can be
>> neglected.
>> For example, you can have a large triangle on this surface along the
>> equator, to one of the poles, and back to the equator at a different
>> longitude, that has an inner angle sum that is greater than 180°.)
>>
>> But it turns out that this theorem is just a special case: We choose a
>> path
>> that connects the points and measure/calculate its length.  The path
>> between
>> two points can in general be arbitrarily long (we can make as many
>> detours
>> from point A to point B as we want), so for defining the distance between
>> the two points, we choose a special path between them: the locally
>> shortest
>> path, called a *geodesic path* or simply a *geodesic* (which comes from
>> geodesy/geodetics, the science of measuring and representing the
>> surface of
>> Earth; literally: "dividing Earth").  Its length is determined by a
>> function
>> into which we basically put the pairwise differences between the
>> coordinates
>> of the points: the *metric*.
>>
>> The metric is of the general form
>>
>>   ds^2 = g_ab dx^a dx^b,
>>
>> where Einstein notation has been used (a and b are placeholders for the
>> labels of the coordinates, and one sums over indices that appear as
>> subscript and as superscript).  g_ab are components of the *metric
>> tensor*.
>>
>> I claimed before that the Pythagorean theorem is just a special case of
>> this, and here is the proof: If you use as coordinates just x and y,
>> label
>> them x^1 and x^2 instead, and define that g_11 := 1, g_12 = g_21 := 0,
>> and
>> g_22 := 1, i.e.
>>
>>   [g]_ab = (1 0),
>>            (0 1)
>>
>> and you get
>>
>>   ds^2 = dx^1 dx^1 + dx^2 dx^2 = (dx^1)^2 + (dx^2)^2 = dx^2 + dy^2.
>>
>> If you set c := ds, a := dx, and b := dy, there you have it:
>>
>>   c^2 = a^2 + b^2  or  a^2 + b^2 = c^2.
>>
>> So the metric tensor tells us how the surface (in general, a manifold)
>> differs from a flat one, and as the components of the metric tensor
>> appear
>> in the metric, the metric does, too.  One can use the metric (tensor) to
>> calculate various objects, some of them just numbers (scalars), that
>> provide
>> information about the curvature.
>>
>> Finally, this concept of curvature can be extended to an arbitrary
>> number of
>> dimensions, and we (Minkowski, Einstein, and others) found out that
>> reality
>> can be best described using an (at least) (1+3)-dimensional manifold
>> that we
>> (Minkowski) call(ed) "spacetime".
>>
>> Sean Carroll gives a very good introduction to this that is (AFAICS)
>> correct, easy to understand for laypeople (he gives this public
>> lecture at
>> The Royal Institution), and humorous (i.e. not boring), which IMO (having
>> studied GR at university now) is quite a feat:
>>
>> The Royal Institution/YouTube: The secrets of Einstein's unknown
>> equation –
>> with Sean Carroll
>>
>> <https://www.youtube.com/watch?v=BRudidBcfXk&list=PL41EYJuJ5YuDn3d13ryZwpzGBXewXa9AH&index=9>
>>
>>
>> Einstein has also written about it for the 1922 edition of the
>> Encyclopædia
>> Britannica, which after the primers above should be easier to understand:
>>
>> <https://www.britannica.com/topic/Albert-Einstein-on-Space-Time-1987141>
>
> Very wise post.
>
> Unfortunately (for this group) : Wozniak will, again, scream for ages....
>
>> I claimed before that the Pythagorean theorem is just a special case of
>> this,
>
> Thanks for that :'-(
>
>

Oftentimes there will be Cauchy-Schwarz a.k.a. triangle inequality
instead of the Pythagorean theorem, for the metric and norm its
establishment, since otherwise in the "curving" of the space-time
there would never be a planar triangle, so instead it's the
triangle inequality used in the formalism, later for things
like Hoelder inequality.

Tulli Levi-Civita in the "Absolute Differential Calculus" gives
a section on "the indefiniteness of ds^2" i.e. the metric, then
for something like "normed rings" about the norm, since, for
example, the pieces that detect the Euclidean geometry of the
Galilean (it's straight) with the Riemannian geometry of a
well of gravity (it's down) is quite entirely arbitrary.

So, it seems you have both rubber sheet and chalkboard at
the same time, still.

"Einstein, is gravity down?"
"Yeah, straight down."


Otherwise that's still mostly just giving an account of the Galilean.


"A first order approximation classical in the limit, ..."
is not the same as "assume classicality".

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