Could there be an alternative to Riemannian geometry?
alen
this is really a pure math topic, so I am posting
it here as well.
I have never liked the complexity of Riemannian
geometry since the first time I encountered it, and
always thought 'there must be an easier way than
this!'
In order to see if this is possible, the basic question
I ask is, if we have a situation of 'flatlanders', living
within the 2 dimensional curved surface of their 2
dimensional world, and who have initially no geometry,
no formal concept of 'distance', 'length', or 'direction',
will they eventually be able to determine the geometry
of their curved surface?
Achieving a positive answer to this question results
in a principle of the geodesic as the fundamental,
immediate basis of all surfaces, or spaces, flat or
curved. This means that the geodesic of any space
can be simply assumed, and does not have to be
derived, and does not have to have an 'equation'.
We do not derive the geodesic of a euclidian space
(the straight line). We begin with it as the experimental
basis of all euclidian geometry. In fact, the same applies
also to curved spaces. That is, anything confined to a
curved space or surface can be used to experimentally
discover its geodesics. One can even argue that a
space fundamentally IS the set of its geodesics.
So the analysis of curved spaces does not require
and affine connection, or a derivation of its geodesics,
etc. The argument therefore is that it is the geodesic
that is the fundamental, immediate, underived connection
in any space.
If you want to follow this argument, check out this
page:
Tim Little
> Achieving a positive answer to this question results in a principle
> of the geodesic as the fundamental, immediate basis of all surfaces,
> or spaces, flat or curved. This means that the geodesic of any space
> can be simply assumed, and does not have to be derived, and does not
> have to have an 'equation'.
It is useful to have an equation if you want to do anything useful
like determine whether two geodesics intersect.
> We do not derive the geodesic of a euclidian space (the straight
> line).
You can use properties of the space of geodesics to establish a system
of coordinates. The systems you derive in Euclidean space have very
nice symmetry properties. Easy!
More general geodesics have much less symmetry, making the practical
math extremely messy.
> So the analysis of curved spaces does not require and affine
> connection, or a derivation of its geodesics, etc.
It does if you intend to use a model to make testable predictions. In
particular, if you have a model that you wish to compare with how the
geodesics behave in reality, you need some way of deriving predictions
for the behaviour of the geodesics from the model in a quantitative
manner under various conditions.
> The argument therefore is that it is the geodesic that is the
> fundamental, immediate, underived connection in any space.
You can assume that if you like. This is math; you can make any
assumptions you like and see what conclusions you can deduce from
them.
> If you want to follow this argument, check out this page:
>
> http://www.alenspage.net/Geodesic.htm
I am reminded of the thought experiments that underlie a part of Greg
Egan's science fiction novel Incandescence:
"Incandescence grew out of the notion that the theory of general
relativity - widely regarded as one of the pinnacles of human
intellectual achievement - could be discovered by a pre-industrial
civilisation with no steam engines, no electric lights, no radio
transmitters, and absolutely no tradition of astronomy."
http://gregegan.customer.netspace.net.au/INCANDESCENCE/Idea/Idea.html
--
Tim
alen
> On 2011-01-13, alen <al...@westserv.net.au> wrote:
>
> > Achieving a positive answer to this question results in a principle
> > of the geodesic as the fundamental, immediate basis of all surfaces,
> > or spaces, flat or curved. This means that the geodesic of any space
> > can be simply assumed, and does not have to be derived, and does not
> > have to have an 'equation'.
>
> It is useful to have an equation if you want to do anything useful
> like determine whether two geodesics intersect.
>
> > We do not derive the geodesic of a euclidian space (the straight
> > line).
>
> You can use properties of the space of geodesics to establish a system
> of coordinates. The systems you derive in Euclidean space have very
> nice symmetry properties. Easy!
>
> More general geodesics have much less symmetry, making the practical
> math extremely messy.
Yes - but is it not helpful if you can start by
simply drawing a geodesic coordinate system,
at least locally, on a curved surface, rather than
having to derive the coordinate axes themselves?
One doesn't derive the initial coordinate axes in a
euclidian surface, but simply draws them, so why
not do so on a curved surface, if it can be shown
that they, also, are immediate realities? Once you
have the geodesics you can use normal
spherical geometry, at least locally.
> > So the analysis of curved spaces does not require and affine
> > connection, or a derivation of its geodesics, etc.
>
> It does if you intend to use a model to make testable predictions. In
> particular, if you have a model that you wish to compare with how the
> geodesics behave in reality, you need some way of deriving predictions
> for the behaviour of the geodesics from the model in a quantitative
> manner under various conditions.
Why use an affine connection if the inhabitants of a curved
surface can identify and use an embedding space in its
entirety? The affine connection supposes an intrinsic
perspective in which one doesn't have access to a
(theoretical at least) embedding space?!
Alen
Tim Little
> Why use an affine connection if the inhabitants of a curved surface
> can identify and use an embedding space in its entirety? The affine
> connection supposes an intrinsic perspective in which one doesn't
> have access to a (theoretical at least) embedding space?!
In many cases of interest, a manifold cannot be embedded in a
Euclidean space.
Besides, if the behaviour of the geodesics is the only observable
aspect in the reality, an embedding space is an extraneous hypothesis
with no predictive value and should therefore be discarded.
--
Tim
alen
> On 2011-01-13, alen <al...@westserv.net.au> wrote:
>
> > Why use an affine connection if the inhabitants of a curved surface
> > can identify and use an embedding space in its entirety? The affine
> > connection supposes an intrinsic perspective in which one doesn't
> > have access to a (theoretical at least) embedding space?!
>
> In many cases of interest, a manifold cannot be embedded in a
> Euclidean space.
I don't believe that - give us an example!!
>
> Besides, if the behaviour of the geodesics is the only observable
> aspect in the reality, an embedding space is an extraneous hypothesis
> with no predictive value and should therefore be discarded.
>
> --
> Tim
I am not clear about the point you are making.
What do you mean by 'predictive value' - i.e., what
predictions do you have in mind?
All I am really saying is that you can immediately
set up a geodesic coordinate system locally in a
curved space, rather than having to derive it
mathematically.
Alen
Tim Little
> On Jan 14, 2:23 pm, Tim Little <t...@little-possums.net> wrote:
>> In many cases of interest, a manifold cannot be embedded in a
>> Euclidean space.
>
> I don't believe that - give us an example!!
My apologies, I misremembered a result and so contradicted Nash's
Embedding Theorem.
>> Besides, if the behaviour of the geodesics is the only observable
>> aspect in the reality, an embedding space is an extraneous hypothesis
>> with no predictive value and should therefore be discarded.
>
> I am not clear about the point you are making. What do you mean by
> 'predictive value' - i.e., what predictions do you have in mind?
You started by talking about scientific models, so I was talking about
the scientific predictions that are the primary purpose of those models.
> All I am really saying is that you can immediately set up a geodesic
> coordinate system locally in a curved space, rather than having to
> derive it mathematically.
Sure, that's one of the first principles introduced when beginning any
presentation of GR, though you're glossing over the usual caveats.
The messy math details are for deriving theoretical predictions that
can be compared with experimental results.
--
Tim
alen
I haven't come across that myself (if I am understanding
you correctly).
A GR presentation, as I have found them, will start by saying
that vectors at two points P and P', within a curved surface,
cannot be compared, because their tangent planes are in
different directions to one another. It will then go on to derive
the usual elaborate vector differentiations, using the embedding
space to compare vectors at the two points, to get expressions
for a geodesic, and for the curvature of the surface (the
curvature tensor).
What I have tried to show, however, is that vector directions
at points P and P' can be immediately compared via a
geodesic. What, in essence is the significance of 'direction'?
I think that the answer is that it indicates something about
moving objects. Especially, in euclidian space, it predicts how
an object will move through space under its own momentum,
i.e. along a straight line, or geodesic of the space. So vectors
along this geodesic have the 'same' direction. Since it can
be shown that objects, confined to a curved surface, and
moving freely under their own momentum (within the surface),
must follow the geodesic of the surface, it follows that a
geodesic of a curved surface specifies direction no less
than does a geodesic in euclidian space. That means that
a vector pointing along the geodesic at point P, within
the surface, is in the same direction, within the surface,
as a vector pointing along the same geodesic at point P'.
Also, a set of basis vectors at P, parallel transported
along the geodesic to P', will all have the same directions
at P', irrespective of changes of direction within an
embedding space. These changes need not be involved
at all, since they are irrelevant to objects confined to the
surface. That, at least, is my argument, which I think
should be able to simplify the whole of the mathematics
of curved spaces!?
Alen
Tim Little
> What, in essence is the significance of 'direction'? I think that
> the answer is that it indicates something about moving
> objects. Especially, in euclidian space, it predicts how an object
> will move through space under its own momentum, i.e. along a
> straight line, or geodesic of the space. So vectors along this
> geodesic have the 'same' direction.
You then have the messy situation that vector U has the same direction
as V, and V has the same direction as W, but U does not have the same
direction as W. So in a curved space, the concept of "same direction"
is not an equivalence class of vectors at points.
> That, at least, is my argument, which I think should be able to
> simplify the whole of the mathematics of curved spaces!?
No, it does not simplify it at all.
--
Tim
alen
> On 2011-01-15, alen <al...@westserv.net.au> wrote:
>
> > What, in essence is the significance of 'direction'? I think that
> > the answer is that it indicates something about moving
> > objects. Especially, in euclidian space, it predicts how an object
> > will move through space under its own momentum, i.e. along a
> > straight line, or geodesic of the space. So vectors along this
> > geodesic have the 'same' direction.
>
> You then have the messy situation that vector U has the same direction
> as V, and V has the same direction as W, but U does not have the same
> direction as W. So in a curved space, the concept of "same direction"
> is not an equivalence class of vectors at points.
I am not clear about what the vectors U, V, and W are.
Perhaps you are referring to different geodesics?. If U
is a vector at P and V is a vector at P', along geodesic 1,
from P, and W is a vector at P'', along geodesic 2 from
P, and V and W are parallel transported versions of U,
then it is true that U and V have the same direction along
geodesic 1, and U and W have the same direction along
geodesic 2, but V and W do not have the same direction
along a geodesic connecting P' and P''. That is quite true
in a curved surface. A 'direction' cannot be broadcast
across a curved surface as across a flat surface.
But I think that the concept of direction on a curved
surface has meaning as long as it is referred to a particular
geodesic. That is, every vector parallel transported along
a geodesic has the same direction along that geodesic.
If you parallel transport W, which is on geodesic 2, along a
geodesic from P'' to P' you can measure the change of direction
of W at P', on geodesic 1. This is the same as the change in
direction of transporting vector W, in a closed loop, from P to
P' to P'', and back to P.
So it is possible to make sense of 'direction changes' of
vectors parallel transported along more than one geodesic
of a curved surface, on the basis that 'direction' is always
referred to a vectors either at a point or parallel transported
along one geodesic only from that point. This should make
the calculation of direction changes mathematically simpler,
I would say.
I guess that, as you say, this is 'messy', but also you can
make SOME sense out of direction across a curved surface!??
Alen
Tim Little
> That is quite true in a curved surface. A 'direction' cannot be
> broadcast across a curved surface as across a flat surface.
Yes, and that's exactly why flat geometry is fairly trivial, and
almost any curved geometry is not.
> But I think that the concept of direction on a curved surface has
> meaning as long as it is referred to a particular geodesic.
Geodesics are not special in that regard. Transport along any path
makes sense equally well.
> This should make the calculation of direction changes mathematically
> simpler, I would say.
Simpler than what? It seems no less complex than the standard
derivations. You also require the restriction that only one geodesic
exists between two points, and will have to rule out large classes of
surfaces (including all the ones useful in GR) as a result.
> I guess that, as you say, this is 'messy', but also you can make
> SOME sense out of direction across a curved surface!??
Sure, in the standard treatments you have the usual parallel transport
along paths, with geodesics being just a particular case of paths.
--
Tim
alen
> On 2011-01-17, alen <al...@westserv.net.au> wrote:
>
> > That is quite true in a curved surface. A 'direction' cannot be
> > broadcast across a curved surface as across a flat surface.
>
> Yes, and that's exactly why flat geometry is fairly trivial, and
> almost any curved geometry is not.
>
> > But I think that the concept of direction on a curved surface has
> > meaning as long as it is referred to a particular geodesic.
>
> Geodesics are not special in that regard. Transport along any path
> makes sense equally well.
Transport along a non-geodesic path makes sense as 'transport', but
does not preserve 'direction'. The only fundamental definition of
direction
there can be is a geodesic. A non-geodesic path is really an arbitrary
path made of multiple different geodesic segments. An arbitrary path
cannot define direction, because there is nothing special about one
arbitrary path compared to another. There are an infinite number of
arbitrary paths connecting one point with another.
Only transport of a vector along a single geodesic path can preserve
its 'direction' within the surface (not the embedding space)
> > This should make the calculation of direction changes mathematically
> > simpler, I would say.
>
> Simpler than what? It seems no less complex than the standard
> derivations. You also require the restriction that only one geodesic
> exists between two points, and will have to rule out large classes of
> surfaces (including all the ones useful in GR) as a result.
Using geodesics creates the possibility of using simple spherical
geometry to calculate changes in the direction of a vector transported
within the surface. I think that this could enable a great
simplification
of the mathematics!?
Yes, before any geometry exists, the geodesic can be defined
within a surface only where only one geodesic connects one point
with another. Once the properties of a geodesic are defined, however,
it should be possible to apply it in a case where multiple geodesics
exist, although such cases cannot be used to initially define what
a geodesic is.
Alen
Tim Little
> Transport along a non-geodesic path makes sense as 'transport', but
> does not preserve 'direction'. The only fundamental definition of
> direction there can be is a geodesic.
Rubbish.
> A non-geodesic path is really an arbitrary path made of multiple
> different geodesic segments.
More rubbish.
> Using geodesics creates the possibility of using simple spherical
> geometry to calculate changes in the direction of a vector
> transported within the surface.
No, it cannot. At best you get only an approximation that way, at
worst there is no spherical geometry that even approximates the
behaviour of the geodesics. For example, the scalar curvature of a
spherical surface is everywhere positive. That is not true of
arbitrary manifolds.
As one example, there is no spherical geometry that models the
behaviour of free-fall paths in vacuum.
--
Tim
alen
> On 2011-01-17, alen <al...@westserv.net.au> wrote:
>
> > Transport along a non-geodesic path makes sense as 'transport', but
> > does not preserve 'direction'. The only fundamental definition of
> > direction there can be is a geodesic.
>
> Rubbish.
No - without a geodesic there is no possibility
of creating an ORIGINAL definition of 'direction'.
Without the euclidian geodesic, a straight line,
you cannot draw a 'vector', or even define it as
a concept.
> > A non-geodesic path is really an arbitrary path made of multiple
> > different geodesic segments.
>
> More rubbish.
Again, I disagree.
> > Using geodesics creates the possibility of using simple spherical
> > geometry to calculate changes in the direction of a vector
> > transported within the surface.
>
> No, it cannot. At best you get only an approximation that way, at
> worst there is no spherical geometry that even approximates the
> behaviour of the geodesics. For example, the scalar curvature of a
> spherical surface is everywhere positive. That is not true of
> arbitrary manifolds.
Yes, it can apply only to positive curvature. But that is still
something, at least.
Locally, a curved surface around a point approximates
to its tangent plane, and its geodesics are straight lines.
In a slightly larger region it can be approximated by a spherical
surface segment, and its geodesics are arcs of circles.
> As one example, there is no spherical geometry that models the
> behaviour of free-fall paths in vacuum.
A direct free-fall path is a straight line in space
and curved in the direction of time. So it has a
'cylindrical' type of geometry.
Alen
>
> --
> Tim
Tim Little
A direct free-fall path is not curved at all, by your own definition:
it is geodesic. Furthermore, a cylindrical geometry is intrinsically
identical to a flat one.
I think you need to clear up some misconceptions before attempting to
revolutionize the mathematics of curved spaces.
--
Tim
alen
> On 2011-01-17, alen <al...@westserv.net.au> wrote:
>
> > On Jan 17, 11:11 pm, Tim Little <t...@little-possums.net> wrote:
> >> As one example, there is no spherical geometry that models the
> >> behaviour of free-fall paths in vacuum.
>
> > A direct free-fall path is a straight line in space and curved in
> > the direction of time. So it has a 'cylindrical' type of geometry.
>
> A direct free-fall path is not curved at all, by your own definition:
> it is geodesic. Furthermore, a cylindrical geometry is intrinsically
> identical to a flat one.
I don't dispute that.
> I think you need to clear up some misconceptions before attempting to
> revolutionize the mathematics of curved spaces.
>
> --
> Tim
I think you shouldn't assume that I am a typical
competitive, aggressive male, who is aiming to gain
some kind of victory, or major prize etc., as the tone
of your remark would seem to indicate.
Even modest truths are worth expressing, for their
own sake, imo.
Alen