Morphisms or Morphine?
the softrat
In my study of GR, I get confused and sleepy with the plethora of
xxx-morphisms. Aren't they all 'functions', just defined on different,
but very similar, kinds of spaces? Or mebbe all 'mappings'? (God, this
is like learning Greek all over again!)
For that matter, are there different kinds of 'spaces' or is it all
short-hand for 'vector space'?
And aren't vectors fundamentally all ordered n-tuples? (*I* can depict
an n-tuple without a coordinate system.)
"I'm s-o-o-o-o-o-o-o confused!" --- Jahn Travolta.
I've now backed up to restudying linear algebra from Hadley.
George D. Freeman IV
the softrat
Curmudgeon-at-Large
mailto:sof...@pobox.com
--
If you find yourself in a hole, the first thing to do is stop
diggin'.
Zig
the softrat wrote:
> In my study of GR, I get confused and sleepy with the plethora of
> xxx-morphisms. Aren't they all 'functions', just defined on different,
> but very similar, kinds of spaces? Or mebbe all 'mappings'? (God, this
> is like learning Greek all over again!)
yes, they are all functions or mappings between different kinds of spaces.
>
> For that matter, are there different kinds of 'spaces' or is it all
> short-hand for 'vector space'?
there are definitely different kinds of spaces, vector spaces being one
of the easiest kinds of spaces.
>
> And aren't vectors fundamentally all ordered n-tuples? (*I* can depict
> an n-tuple without a coordinate system.)
>
yes, vectors are all essentially n-tuples.
here is a short list of mathematical sets or spaces that i can think of
off the type of my head. perhaps others can add to it. each one is a
set with objects in it, plus an additional structure or relation among
the points in the set. each kind of mathematical structure has its
mappings.
topological space is a set of points together with a notion of closeness
among the points. one to one onto mappings that preserve this notion of
closeness between topological spaces are called homeomorphisms.
a topological space that also has enough smoothness that you can also do
calculus on it is called a smooth manifold. one to one onto mappings
with preserve this smoothness are called diffeomorphisms.
a smooth manifold that also has a notion of precise distance, in
addition to closeness and smoothness, is called a Riemannian manifold.
in GR, you study pseudo Riemannian manifolds (which is just a slight
variation of a regular Riemannian manifold). one to one onto mappings
which preserve the notion of distance on a manifold are called isometries.
most of the rest of the *-morphisms live in algebra: homomorphisms,
monomorphisms, epimorphisms, endomorphisms, isomorphisms, and
automorphisms are all mappings between sets that have some algebraic
structure on them, like rules for multiplication or addition. if a
mapping between two algebraic sets preserves the algebraic structure, it
is a homomorphism. if it is also one to one, then it is a monomorphism,
if it is onto, then it is an epimorphism. if it is both one to one and
onto, then it is an isomorphism. if the two sets are the same, then it
is an endomorphism. if the two sets are the same, and the mapping is
one to one and onto, then it is an automorphism.
a vector space is really an algebraic set, even though we often have
geometric pictures of them in our head. a homomorphism on a vector
space is just a linear mapping. an isomorphism is an invertible linear
mapping.
often, you can associate an algebraic structure to a topological or
geometric space. then you can translate between *-morphisms between the
topological spaces and *-morphisms between the associated algebraic
structures. a natural language for describing these translations is
category theory, and there we drop the prefix, and just use the word
"morphism" to encompass all of the above.
Philip Charlton
In article <q1v0qv4vpasdjf4u1...@4ax.com>, the softrat wrote:
>
> In my study of GR, I get confused and sleepy with the plethora of
> xxx-morphisms. Aren't they all 'functions', just defined on different,
> but very similar, kinds of spaces? Or mebbe all 'mappings'? (God, this
> is like learning Greek all over again!)
No doubt others will answer this question better but here goes...
It's my understanding that "morphism" is shorthand for a class of functions
which are also structure-preserving in some sense, where the structure that
is preserved is dependent on the spaces being acted upon. For example,
there are morphisms that map topological spaces to topological
spaces and preserve topology, morphisms that act on linear spaces and preserve
linearity, morphisms that act on groups or algebras and preserve alegebraic
structures, morphims that act on metric spaces and preserve metric properties
(ie. geometry) and so on.
> For that matter, are there different kinds of 'spaces' or is it all
> short-hand for 'vector space'?
there are lots of things called spaces - the simplest is the topological space,
ie. a set plus a local topology (I've never heard a plain old set called a
space but I suppose one could...). Topological space is the basis for
all other spaces (AFAIK) eg. a linear space is a topological space where
there is an "addition" operation on elements of the sets.
> And aren't vectors fundamentally all ordered n-tuples? (*I* can depict
> an n-tuple without a coordinate system.)
Technically, all n-dimensional vector spaces are isomorphic, but not
canonically. So there are lots of different isomorphisms that map one
n-dimensional vector space to another, but usually none of them are singled
out as being "natural" or "canonical" eg. the space of polynomials of
degree 2 spanned by { 1, x, x^2 } is isomorphic to R^3, but not canonically.
The act of choosing a basis for a vector space is equivalent to constructing
an isomorphism between it and the space of n-tuples R^n.
But note that "almost all" vector spaces are infinite-dimensional and
are not isomorphic to one another :) spaces of functions for example.
I think it's fair to say that the majority of pure mathematics research is
concerned with the study of infinite-dimensional spaces these days.
Cheers,
Philip
Lubos Motl
On Thu, 30 Oct 2003, the softrat wrote:
> In my study of GR, I get confused and sleepy with the plethora of
> xxx-morphisms. Aren't they all 'functions', just defined on different,
Yes, they're all functions of some sort - functions that must preserve
some structure i.e. some relations between the elements of the set.
The word "morphosis" means "form" or "shape" in Greek while "homo" is
"alike" (like in "homosexual").
An example what structure must be preserved: if a morphism "f" acts on a
group where elements a,b,c satisfy a.b=c, then you must also get
f(a).f(b)=f(c), otherwise it is not a morphism.
Essentially every function may be called homomorphism or morphism. The
word "homomorphism" is supposed to be the most general one.
If the function is *onto*, i.e. a *surjection* (all the elements of the
image set are realized), you can also call the map an *epimorphism*. If
different elements always map to different images, the function is
*injection* and the morphism can be called *monomorphism*.
A morphism from a set X to the same set X - not necessarily *onto* - is
called *endomorphism* (an internal one).
A homomorphism that is completely reversible, i.e. a morphism that is both
a *monomorphism* as well as *epimomorphism*, is called *isomorphism* - it
is a one-to-one map between two sets - and if it is also endomorphism at
the same moment, such an isomorphism can be called *automorphism*.
An automorphism is a one-to-one relabeling of the elements of the set that
preserves all of the required relations between the elements. For
geometrical objects, the automorphism can be thought of as an *isometry*,
for example.
Diffeomorphism is a map from one manifold to another manifold that must be
smooth enough - usually we require it to be differentiable (therefore
"diffeo-").
> For that matter, are there different kinds of 'spaces' or is it all
> short-hand for 'vector space'?
The sets that participate in xxx-morphisms can be very different sets. The
word "space" usually means a "vector space" or a "linear space", but more
generally it can represent any manifold. For example, "the moduli spaces"
are rarely linear spaces - they are rather curved shapes, i.e. general
manifolds.
> And aren't vectors fundamentally all ordered n-tuples? (*I* can depict
> an n-tuple without a coordinate system.)
Yes, they are. Vectors can always be thought of as ordered n-tuples of
numbers (not necessarily real numbers). Just keep in mind that "n" may be
infinite, and that the precise choice of the numbers depends on your
choice of the basis - and many people with a very formal and less physical
approach to these concepts prefer to avoid coordinates completely.
Cheers,
Lubos
______________________________________________________________________________
E-mail: lu...@matfyz.cz fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
phone: work: +1-617/496-8199 home: +1-617/868-4487
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.
Stein A. Stromme
| A homomorphism that is completely reversible, i.e. a morphism that
| is both a *monomorphism* as well as *epimomorphism*, is called
| *isomorphism* - it is a one-to-one map between two sets
Two small points to note here:=20
1. the word homomorphism and its friends belong to a pretty generic
terminology, used also in contexts where they do not rest on some
underlying maps of sets, and
2. even when the morphisms actually are maps of sets with some
structure-preserving property, it is not always the case that a
homomorphism which happens to be a bijection of the underlying sets is
automatically an isomorphism. To have an isomorphism, one has to add
the requirement that also the inverse map be a homomorphism. =20
Example: the map f:R->R given by f(x)=3Dx^3 is a bijective morphism from
the set R of real numbers to itself in the category of differentiable
manifolds (f is differentiable and bijective), but it is not an
isomorphism in this category since the inverse map is not
differentiable at 0. Another example: map the half-open interval
[0,1) continuously and bijectively onto the unit circle; it is still
not a homeomorphism (isomorphism in the category of topological spaces).
So "completely reversible" is correct, but be careful about the nature
of the inverse.
--=20
Stein Arild Str=F8mme +47 55584825, +47 95801887
Universitetet i Bergen Fax: +47 55589672 =20
Matematisk institutt www.mi.uib.no/stromme/ =20
Johs Brunsg 12, N-5008 BERGEN str...@mi.uib.no
Philip Charlton
>
>> And aren't vectors fundamentally all ordered n-tuples? (*I* can depict
>> an n-tuple without a coordinate system.)
>
> Yes, they are. Vectors can always be thought of as ordered n-tuples of
> numbers (not necessarily real numbers). Just keep in mind that "n" may be
> infinite, and that the precise choice of the numbers depends on your
> choice of the basis - and many people with a very formal and less physical
> approach to these concepts prefer to avoid coordinates completely.
>
It's not really the case that "all vectors are n-tuples"...
It's true that all n-dimensional vector spaces are isomorphic to the
space of n-tuples R^n. This is true even if n is "infinite", in which case
the vector space is usually identified with the space of infinite sequences.
However, there are lots of vector spaces with *uncountably* infinite
dimensions (such as function spaces) and these aren't isomorphic
to any space of n-tuples, infinite or not.
Cheers,
Philip
Jason
Lubos Motl wrote:
> and many people with a very formal and less physical
> approach to these concepts prefer to avoid coordinates completely.
There's nothing less physical about avoiding coordinates! In fact, since
coordinates are unphysical, using coordinates is LESS, not MORE
physical. Avoiding coordinates isn't formal either. Formal is when you
manipulate the series obtained by summing Feynman diagrams, which are
really asymptotic series, or using dimension regularization or
manipulating functional integrals formally because you don't know how to
do them rigorously! Formal is when you toss about Grassmann numbers
everywhere but not treat them as actual values. That's formal...
Arnold Neumaier
Lubos Motl wrote:
>
> Diffeomorphism is a map from one manifold to another manifold that must be
> smooth enough - usually we require it to be differentiable (therefore
> "diffeo-").
Not quite.
A diffeomorphism is a map between manifolds which is
differentiable and has a differentiable inverse.
Arnold Neumaier
Alfred Einstead
> And aren't vectors fundamentally all ordered n-tuples?
No. Not fundamentally. See the thread "Coordinates without
coordinates". A vector is no more "fundamentally" an n-tuple
than an ordinary number is. It just can be seen that way,
as well as other, such as an ordinary 'decimal' number, as in
the thread above, or an element of an algebra such as the
one given by the axioms:
Operation: [,,]: V x F x V -> V; F a field (other than {0,1}):
Axiom 1: [x,0,y] = x
Axiom 2: [x,0,y] = y
Axiom 3: [x,rt(1-t),[y,s,z]] = [[x,rt(1-s),y],t,[x,rs(1-t),z]]
which defines affine geometry; and also vector spaces, modulo
the selection of a 0 vector.
Imam Tashdid ul Alam
> On Thu, 30 Oct 2003, the softrat wrote:
>
> > In my study of GR, I get confused and sleepy with the plethora of
> > xxx-morphisms. Aren't they all 'functions', just defined on different,
>
> Yes, they're all functions of some sort - functions that must preserve
> some structure i.e. some relations between the elements of the set.
> The word "morphosis" means "form" or "shape" in Greek while "homo" is
> "alike" (like in "homosexual").
Disturbing similarity. So if homomorphism is the most general kind of
morphisms, why not call it just morphism? I know people do that, but
why not popularize it to a level that these confusions (thinking of
homomorphism as a kind of morphism) does not arise?
They are functions. There are so many names for functions. I would
prefer to say that they are mappings from one algebraic structure to
another that preserve the algebra. Mapping seems to be a slightly
better word in this context. Of course, mathematically a mapping is
the same as a function. Psychologically a little distinction helps.
>
> An example what structure must be preserved: if a morphism "f" acts on a
> group where elements a,b,c satisfy a.b=c, then you must also get
> f(a).f(b)=f(c), otherwise it is not a morphism.
What Lubos is saying, I would like to say it again. Morphisms preserve
structure. If a morphism is from a group to another group that
preserves the group operation, as Lubos showed, then it's a group
morphism. If a morphism from a ring to a ring preserves *both*
multiplication and addition [if c = a + b and d = ab then eta(c) =
eta(a) + eta(b) and etc(d) = eta(a) eta(b)], it is a ring morphism.
There might very well be a morphism from a ring to another right that
only preserves the addition as a group operation, needless to say, it
is a group morphism, treating both of the rings as groups under
addition.
>
> Essentially every function may be called homomorphism or morphism. The
> word "homomorphism" is supposed to be the most general one.
>
Every function? Or you just meant, every function that is consistent
with the previous paragraph? That's what I was saying...how are we
supposed to know that homomorphism is the most general one? The most
general bird, as opposed to crow or sparrow, is called a bird, no one
called it a homobird.
> If the function is *onto*, i.e. a *surjection* (all the elements of the
> image set are realized), you can also call the map an *epimorphism*. If
> different elements always map to different images, the function is
> *injection* and the morphism can be called *monomorphism*.
... and these two words are used only by the nerds. A 'surjective
morphism' or an 'injective morphism' defined the situation completely
and descriptively.
>
> A morphism from a set X to the same set X - not necessarily *onto* - is
> called *endomorphism* (an internal one).
This one is more common. Think of a finite dimensional linear
transformations from a linear space to itself. Linear transformations
preserve linearity, and thus are morphisms of linear spaces. Someday,
people in elementary linear algebra *will* call it a morphism. So a
linear transformation to itself is an endomorphism. Generally, the
endomorphisms will map the space to a subspace of its, having
dimension less than the original one. But sometimes, they will be
isomorphisms also, being invertible, placing a one-to-one and onto
correspondence. That's when they are called automorphisms.
Isomorphism means it's basically the same thing. It's sort of a
relabelling. Changing basis is establishing an isomorphism.
Endomorphism (from X to X) + Isomorphism (relabelling) = Automorphism
(symmetry)
Symmetry in the sense that if the object has a large number of
automorphisms, it is very symmetric. Think about it a little bit, the
words are deciving.
I think I should elaborate on the matter. Take 3d space. Now replace
x, y, z with y, z, x. The math remains the same, although the vectors
don't. But when you are only interested in calculating, it doesn't
matter. This one was an automorphism (auto means own, doesn't it?)
Lookup dictionary:
Homo => similar, that's the whole point of having a morphism, better
drop it
Iso => same, the math is exactly the same, like a cyclic group and
Z_n
Auto => own, a morphism within itself, relabelling, like iso
Endo => internal within itself, this time allows something to be
missing
learn greek and latin, it helps ;)
>
> A homomorphism that is completely reversible, i.e. a morphism that is both
> a *monomorphism* as well as *epimomorphism*, is called *isomorphism* - it
> is a one-to-one map between two sets - and if it is also endomorphism at
> the same moment, such an isomorphism can be called *automorphism*.
That's what I said. Lubos is a pro.
>
> An automorphism is a one-to-one relabeling of the elements of the set that
> preserves all of the required relations between the elements. For
> geometrical objects, the automorphism can be thought of as an *isometry*,
> for example.
Cool. Almost exactly what I said. That makes me think, may be I missed
my target of being helpful.
>
> Diffeomorphism is a map from one manifold to another manifold that must be
> smooth enough - usually we require it to be differentiable (therefore
> "diffeo-").
For completeness, I would like to mention homeomorphism also, which
preserves continuity. Homeomorphic objects are topologically
equivalent. Nonsense. Topology studies only those properties that
remain after a successful homeomorphism. Sounds better.
>
> > For that matter, are there different kinds of 'spaces' or is it all
> > short-hand for 'vector space'?
>
> The sets that participate in xxx-morphisms can be very different sets. The
> word "space" usually means a "vector space" or a "linear space", but more
> generally it can represent any manifold. For example, "the moduli spaces"
> are rarely linear spaces - they are rather curved shapes, i.e. general
> manifolds.
Yeah. He is a pro.
Space means something you can have a vague visualization of. Not
function space though. The formal definition of the word space in
maths, I suspect is that of a set.
>
> > And aren't vectors fundamentally all ordered n-tuples? (*I* can depict
> > an n-tuple without a coordinate system.)
>
> Yes, they are. Vectors can always be thought of as ordered n-tuples of
> numbers (not necessarily real numbers). Just keep in mind that "n" may be
> infinite, and that the precise choice of the numbers depends on your
> choice of the basis - and many people with a very formal and less physical
> approach to these concepts prefer to avoid coordinates completely.
>
Just to illustrate what we are talking about....every finite linear
space is *isomorphic* to F^n, where F is the underlying field. Choose
a basis { v_1, v_2, ... , v_n } in the original linear space. Define a
morphism from your space to F^n as eta(v_i) = (0 0 0 .. in the i-th
position 1 .. 0 0 ) transpose. There you go. Notice that the original
space might very well be a function space, or matrices. But for all of
them you can essentially calculate addition and scalar multiplication
and linear transformations in F^n. After you are done, send them back
using the inverse of the isomorphism. Isomorphism means the same
shape.
> Cheers,
> Lubos
>
Cheers
Lubos Motl
> There's nothing less physical about avoiding coordinates! In fact, since
> coordinates are unphysical, using coordinates is LESS, not MORE
> physical.
A particular choice of coordinates is unphysical because various
symmetries imply that different choices of the coordinates can be
physically equivalent. But *a* choice of coordinates is necessary to
describe a particular process in the laboratory or elsewhere. Coordinates
- or at least the information about the distances between various pieces
of the experiment, which is essentially equivalent - is necessary to know
what the configuration looks like and to calculate anything specific.
It's true that a person who has never calculated a particular physics
problem and never plans to do so does not need coordinates. But such a
person is very formal, and his or her methods are formal, too.
> really asymptotic series, or using dimension regularization or
> manipulating functional integrals formally because you don't know how to
> do them rigorously!
Nope, this is not the right explanation of the word. "Formally" is not the
opposite of "rigorously". On the contrary, the "formal" people are usually
more rigorous in their treatment of mathematics. The difference is that
they don't care too much about the explicit results and their practical
consequences. They prefer to play with the *formalism*, and this makes the
people as well as their calculations formal. XY is formal if XY is based
on formalism (and the visual analogy between various more or less abstract
equations), without the ability to imagine that the variables in the
expression can take particular numerical or other values that can be
compared with reality. Formal is rather the antonymum (the opposite word)
of the word "practical", not of the word "rigorous".
Jason
> On Sun, 2 Nov 2003, Jason wrote:
> But *a* choice of coordinates is necessary to
> describe a particular process in the laboratory or elsewhere.
Nope, it's just a description with coordinates is often much simpler
than a coordinate-free description. But it's certainly possible to
describe everything in terms of coordinate-free variables, just like in
gauge theories, where we could use Wilson loops. It's just much more
cumbersome, but it definitely is possible!
> Coordinates
> - or at least the information about the distances between various pieces
> of the experiment, which is essentially equivalent
The distance between two objects along a path is a
coordinate-independent quantity!
> Nope, this is not the right explanation of the word. "Formally" is not the
> opposite of "rigorously".
You misunderstood me. By formal, I didn't mean not rigorous. Instead, I
simply mean the standard usage of "formal" as meaning playing about with
symbols and equations without giving an actual semantic meaning to them.
I brought up these examples not because I think they're not rigorous,
but because I see them as merely playing about with symbols.
> Formal is rather the antonymum (the opposite word)
> of the word "practical", not of the word "rigorous".
As I've said, you misunderstood me, as usual!
Dieter Menszner
the softrat wrote:
...
> And aren't vectors fundamentally all ordered n-tuples? (*I* can depict
> an n-tuple without a coordinate system.)
>
Elements of vector spaces are called vectors :-)
The vectors of finite dimensional vector spaces can
be considered n-tuples because they always can be
represented as a linear combination of a fixed
set of n 'basis' vectors.
But this is only a special case. Standard example:
the vector space C[a,b] of continuous functions f:[a,b] -> R
A function of this space is a 'vector' but is
not a n-tuple.
--
Dieter
matt grime
> Lubos Motl <mo...@feynman.harvard.edu> wrote in message news:<Pine.LNX.4.31.031031...@feynman.harvard.edu>...
> > On Thu, 30 Oct 2003, the softrat wrote:
> >
> > > In my study of GR, I get confused and sleepy with the plethora of
> > > xxx-morphisms. Aren't they all 'functions', just defined on different,
> >
<edit>
> > If the function is *onto*, i.e. a *surjection* (all the elements of the
> > image set are realized), you can also call the map an *epimorphism*. If
> > different elements always map to different images, the function is
> > *injection* and the morphism can be called *monomorphism*.
>
> ... and these two words are used only by the nerds. A 'surjective
> morphism' or an 'injective morphism' defined the situation completely
> and descriptively.
Be careful:
an epimorphism is not a surjection (although a surjection is an
epimorphism), and similarly for monomorphism and injection.
e is an epimorphism, if whenever ge=fe, we have g=f,
m is a monomorphism if mf=mg => f=g
<edit>
Kevin A. Scaldeferri
In article <bo3rdm$g5k$1...@news.udel.edu>, Jason <pri...@excite.com>
wrote:
>Lubos Motl wrote:
>> But *a* choice of coordinates is necessary to
>> describe a particular process in the laboratory or elsewhere.
>Nope, it's just a description with coordinates is often much simpler
>than a coordinate-free description. But it's certainly possible to
>describe everything in terms of coordinate-free variables, just like in
>gauge theories, where we could use Wilson loops. It's just much more
>cumbersome, but it definitely is possible!
Lubos is correct here. To make predictions about real experiments in
GR (or any other covariant theory) it is essential to carefully define
the measurements, which is going to involve fixing the laboratory
frame of reference. It is extremely easy (and common among those
without experience in the subject) to come to completely wrong
conclusions about GR if you ignore this issue.
>> Coordinates
>> - or at least the information about the distances between various pieces
>> of the experiment, which is essentially equivalent
>The distance between two objects along a path is a
>coordinate-independent quantity!
The interval between to events is a coordinate independent quantity.
The (spatial) distance between two points is not, nor is the temporal
interval between events.
--
======================================================================
Kevin Scaldeferri Calif. Institute of Technology
The INTJ's Prayer:
Lord keep me open to others' ideas, WRONG though they may be.
Jason
Kevin A. Scaldeferri wrote:
> In article <bo3rdm$g5k$1...@news.udel.edu>, Jason <pri...@excite.com>
> wrote:
>>Nope, it's just a description with coordinates is often much simpler
>>than a coordinate-free description. But it's certainly possible to
>>describe everything in terms of coordinate-free variables, just like in
>>gauge theories, where we could use Wilson loops. It's just much more
>>cumbersome, but it definitely is possible!
> Lubos is correct here. To make predictions about real experiments in
> GR (or any other covariant theory) it is essential to carefully define
> the measurements, which is going to involve fixing the laboratory
> frame of reference. It is extremely easy (and common among those
> without experience in the subject) to come to completely wrong
> conclusions about GR if you ignore this issue.
The raw data of experiments and observations are coordinate-independent.
They go like something like this: relative to the direction on Earth
defined by the north pole from the center of the Earth, with respect to
the Greenwich meridian, which runs through London, such-and-such an
object, as determined from electromagnetic radiation reaching Earth
arrived in the direction at a certain angle (as induced from the local
metric) from the North pole direction and the Greenwich meridian at a
time, which, as determined by clocks synchronized to sunrises and
sunsets as observed from Earth, coincides with a certain numerical
reading on a clock, and as deduced from its
intensity/redshift/interferometry/speculation, is at such-and-such a
distance away from Earth.
Or maybe, with respect to the distances from the walls of a room (as
induced by a metric whose curvature is negligible with respect to the
dimensions of the room), and coinciding with a certain numerical display
on a clock synchronized to sunrises and sunsets, such-and-such an object
did such-and-such.
>>The distance between two objects along a path is a
>>coordinate-independent quantity!
>
> The interval between to events is a coordinate independent quantity.
> The (spatial) distance between two points is not, nor is the temporal
> interval between events.
I wrote "along a path"! Maybe I should be more careful and say something
like given a path determined by physical means, like let's say an actual
worldline of a particle, the length as induced by the dynamical metric
along the path between any two points on it specified physically, like
let's say the creation and decay of the particle, is coordinate-independent.
Jason
> than a coordinate-free description. But it's certainly possible to
> describe everything in terms of coordinate-free variables, just like in
> gauge theories, where we could use Wilson loops. It's just much more
> cumbersome, but it definitely is possible!
Much, much more cumbersome, actually. For gauge theories, Wilson loops,
and if you have charged fields, Wilson lines intertwined with the
charged fields at both ends would do the job (or more generally,
gauge-invariant variables evaluated over spin networks). For
diffeomorphic covariant theories, though, what simple variables would do?
Kevin A. Scaldeferri
>
>Kevin A. Scaldeferri wrote:
>
>> To make predictions about real experiments in
>> GR (or any other covariant theory) it is essential to carefully define
>> the measurements, which is going to involve fixing the laboratory
>> frame of reference. It is extremely easy (and common among those
>> without experience in the subject) to come to completely wrong
>> conclusions about GR if you ignore this issue.
>
>The raw data of experiments and observations are coordinate-independent.
No, the raw data are generally coordinate-dependent.
The construction you describe:
>They go like something like this: relative to the direction on Earth
>defined by the north pole from the center of the Earth ...
doesn't really carry any content. In the sense you have described,
you can make any quantity "coordinate-independent" by contracting it
against some arbitrary quantity with the appropriate tensor
signature. However, this requires an unnatural choice (I'm using
"unnatural" in a technical sense here, by the way), so it really
shouldn't be considered coordinate-independent in a real sense.
>Or maybe, with respect to the distances from the walls of a room (as
>induced by a metric whose curvature is negligible with respect to the
>dimensions of the room), and coinciding with a certain numerical display
>on a clock synchronized to sunrises and sunsets, such-and-such an object
>did such-and-such.
Right, but you just defined a coordinate system.
Jason
> In article <boaubr$rq1$1...@news.udel.edu>, Jason <pri...@excite.com> wrote:
>>The raw data of experiments and observations are coordinate-independent.
> No, the raw data are generally coordinate-dependent.
>
> The construction you describe:
>>They go like something like this: relative to the direction on Earth
>>defined by the north pole from the center of the Earth ...
> doesn't really carry any content. In the sense you have described,
> you can make any quantity "coordinate-independent" by contracting it
> against some arbitrary quantity with the appropriate tensor
> signature. However, this requires an unnatural choice (I'm using
> "unnatural" in a technical sense here, by the way), so it really
> shouldn't be considered coordinate-independent in a real sense.
OK, let me be more precise: There are unphysical coordinate systems not
defined with respect to anything else, in other words, a coordinate
system which exists independently of the configuration of
fields/particles/whatever over spacetime (this is usually the kind of
coordinate system used in theoretical calculations as opposed to
experimental ones), and for these coordinate systems, we have (active)
diffeomorphism covariance, and there are relational coordinate systems
defined with respect to the configuration of fields/particles over
spacetime and such coordinate systems don't have (active) diffeomorphism
covariance. All the examples I gave are coordinate systems of the second
kind, and from the point of view of someone who considers only
coordinate systems of the first kind to be "true coordinates", such
coordinates are coordinate-independent.
the softrat
(Kevin A. Scaldeferri) wrote:
>Some poor uncited soul wrote:
>>The raw data of experiments and observations are coordinate-independent.
>No, the raw data are generally coordinate-dependent.
Yes! Without measurement, you do not have Physics, you have Scholastic
Speculative Philosophy. One needs coordinates in order to measure real
phenomena. The fact that the mathematics used to describe physical
phenomena may be formulated largely without coordinates is interesting
metaphysically and epistemologically, but NOT Physics! Mathematicians
who deny the 'reality' of coordinates and coordinate systems are not
in tune with the history of their own subject. Mathematics is derived
historically from attempts to measure things, like land and money.
Measurement requires coordinates.
Jason
the softrat wrote:
> Yes! Without measurement, you do not have Physics, you have Scholastic
> Speculative Philosophy. One needs coordinates in order to measure real
> phenomena.
>
> Measurement requires coordinates.
Not really. In 2D Euclidean geometry, for example, the statement that
point A is at (0,0), point B at (3,0) and point C at (0,4) contains
exactly the same physical content as AB=3, AC=4 and BC=5.
Kevin A. Scaldeferri
Actually, it doesn't. Clearly you need at least one more piece of
information.
Kevin A. Scaldeferri
>OK, let me be more precise: There are unphysical coordinate systems not
>defined with respect to anything else, in other words, a coordinate
>system which exists independently of the configuration of
>fields/particles/whatever over spacetime (this is usually the kind of
>coordinate system used in theoretical calculations as opposed to
>experimental ones), and for these coordinate systems, we have (active)
>diffeomorphism covariance, and there are relational coordinate systems
>defined with respect to the configuration of fields/particles over
>spacetime and such coordinate systems don't have (active) diffeomorphism
>covariance. All the examples I gave are coordinate systems of the second
>kind, and from the point of view of someone who considers only
>coordinate systems of the first kind to be "true coordinates", such
>coordinates are coordinate-independent.
The earth provides a preferred reference frame to the extent that the
earth is relevant to the problem at hand. However, most GR
observations don't care about the earth, so using it to define a
frame-of-reference is just as arbitrary as any other choice.