"James Stokes" <dkjk@bigpond.net.au> wrote in message
news:a3ea3c76.0411292221.6b2a277a@posting.google.com...

"Dirk Van de moortel" <dirkvandemoortel@ThankS-NO-SperM.hotmail.com>

in message news:<fHIqd.4484$EP.243780@phobos.telenet-ops.be>...

"James Stokes" <dkjk@bigpond.net.au> wrote in message
news:a3ea3c76.0411290807.afc9528@posting.google.com...

Hi group,

Well, I've finally completed my HSC (Australian equivalent of the
SATs) and have just returned from two weeks in China to celebrate for
schoolies. Anyway, now that I have an abundance of free time on my
hands, nerdy as always, I've decided to explore the intricacies of
spacetime using Taylor and Wheeler's Spacetime Physics. However,
throughout the course of my readings, several questions have arisen
about the nature of spacetime and events. As I understand it, the
spacetime of relativity is Minkowski space R^{3,1} comprised of the
set of quadruples of real numbers R^4 endowed with a form g :

R that defines the squared distance between two points in R^{3,1}.
Objects in spacetime are regarded as timelike subsets of R^{3,1}

as worldlines so that the tangent vector v to any point in the
worldline obeys g(v,v) < 0.

Since they define the interval as the time separation squared
minus the space separation squared, that should be g(v,v) > 0.

When we transform from one inertial
observer to another, the components of the points comprising the
worldline change so as to maintain the value of g(v,v). Although the
worldline of an object changes from one observer to observer, it

to me that the object will pass through the same physical events
regardless of the observer.

Actually, the *equation* of the worldline changes from one
observer to another. The worldline itself is merely a collection
of events and does not change. See below...

So this is basically akin to saying that the subset of points of
R^{3,1} changes, however, the set of events defining the worldline
remains unchanged.

However, events in spacetime are defined
only in terms of points that are readily transformed into other

in R^{3,1}.

And events are defined as independent of any coordinate system, so
they are not defined as elements of R^{3,1} or as elements of R^4.
That should take away the objection you make below...

I defined Minkowski space earlier as the set of quadruples of reals
plus a form that defines that spacetime interval. How do we
mathematically define an event, and where do events exist in relation
to (R^4,g)?

An (ideal) event is a possible occurrence in the physical world that
lasts for an infinitesimally short duration of time and takes place in
an infinitesimally small region of space. For example, consider a very
small firecracker whose explosion take place very rapidly.

Now, I'm going to introduce some mathematics at the level of a first
course in linear algebra.

Choose one event as the event with respect to which all other events are
to be referenced. Then, spacetime is the collection of all events, and,
in special relativity, spacetime is modeled by a 4-dimensional abstract
vector space V with the reference event represented by the zero vector
for V. Spacetime is absolute, but, as will be seen below, views of
spacetime are relative.

Also included in the model for spacetime is a mapping g that maps pairs
of vectors in V into the set of real numbers. So, g(v,w) is a real
number for any vectors v,w in V.

Call v in V:

1) timelike if g(v,v) < 0;
2) lightlike if if v =/= 0 and g(v,v) = 0;
3) spacelike if g(v,v)>0.

V and g have the following properties:

4) g linear in each argument;
5) g(v,w) = g(w,v) for all v and w (symmetric);
6) timelike vectors exist;
7) v timelike, w =/=0, and g(v,w) =0 implies that w is spacelike.

Roughly, 7) says that since spacetime has, for any observer, one
dimension of time and 3 dimensions of space, a vector orthogonal to a
timelike vector has nowhere to go but into space.

A worldline is curve in V whose tangent vector is always timelike, so
you're right - worldlines are absolute. Being a bit more precise, a
worldline is a mapping, h say, that takes an interval I of the reals
into a subset of V.

h: I -> V.

h models the path in spacetime taken by an object. Often I is taken to
be the proper time values for the object, so h associates events in
spacetime with readings on the object's wristwatch.

Where does R^4 fit into all of this? Model an inertial reference frame
for spacetime by an orthonormal basis {v0, v1, v2, v3} for spacetime
with v0 pointing along the time axis of the spacetime and the other v's
point along the space axes of the frame. Then any v in V can be uniquely
expressed as

v = A*v0 + B*v1 + C*v2 + D*v2,

where A, B, C, and D are all real numbers, i.e., (A,B,C,D) is an element
of R^4.

(A,B,C,D) gives the coordinates of v with respect to the reference frame
{v0, v1, v2, v3}. Choosing a different reference frame results in a
different set of coordinates for v, i.e., in a different element of R^4.

Events in spacetime are absolute, but coordinates of events are not!

One final math point: at the start of all this I singled out an event
as the reference event. This *seems* to make one event special (but
actually doesn't), and can be avoided by modeling spacetime with an
affine space instead of a vector space.

My advice: study Taylor and Wheeler closely without worrying too much
about the abstract mathematics right now - there will plenty of time for
that later. While in high school, I learned special relativity from
Taylor and Wheeler.

Regards,
George

"James Stokes" <dkjk@bigpond.net.au> wrote in message

I defined Minkowski space earlier as the set of quadruples of reals
plus a form that defines that spacetime interval. How do we
mathematically define an event, and where do events exist in relation
to (R^4,g)?

An (ideal) event is a possible occurrence in the physical world that
lasts for an infinitesimally short duration of time and takes place in
an infinitesimally small region of space. For example, consider a very
small firecracker whose explosion take place very rapidly.

Now, I'm going to introduce some mathematics at the level of a first
course in linear algebra.

Choose one event as the event with respect to which all other events are
to be referenced. Then, spacetime is the collection of all events, and,
in special relativity, spacetime is modeled by a 4-dimensional abstract
vector space V with the reference event represented by the zero vector
for V. Spacetime is absolute, but, as will be seen below, views of
spacetime are relative.

Also included in the model for spacetime is a mapping g that maps pairs
of vectors in V into the set of real numbers. So, g(v,w) is a real
number for any vectors v,w in V.

Call v in V:

1) timelike if g(v,v) < 0;
2) lightlike if if v =/= 0 and g(v,v) = 0;
3) spacelike if g(v,v)>0.

V and g have the following properties:

4) g linear in each argument;
5) g(v,w) = g(w,v) for all v and w (symmetric);
6) timelike vectors exist;
7) v timelike, w =/=0, and g(v,w) =0 implies that w is spacelike.

Roughly, 7) says that since spacetime has, for any observer, one
dimension of time and 3 dimensions of space, a vector orthogonal to a
timelike vector has nowhere to go but into space.

A worldline is curve in V whose tangent vector is always timelike, so
you're right - worldlines are absolute. Being a bit more precise, a
worldline is a mapping, h say, that takes an interval I of the reals
into a subset of V.

h: I -> V.

h models the path in spacetime taken by an object. Often I is taken to
be the proper time values for the object, so h associates events in
spacetime with readings on the object's wristwatch.

Where does R^4 fit into all of this? Model an inertial reference frame
for spacetime by an orthonormal basis {v0, v1, v2, v3} for spacetime
with v0 pointing along the time axis of the spacetime and the other v's
point along the space axes of the frame. Then any v in V can be uniquely
expressed as

v = A*v0 + B*v1 + C*v2 + D*v2,

where A, B, C, and D are all real numbers, i.e., (A,B,C,D) is an element
of R^4.

(A,B,C,D) gives the coordinates of v with respect to the reference frame
{v0, v1, v2, v3}. Choosing a different reference frame results in a
different set of coordinates for v, i.e., in a different element of R^4.

Events in spacetime are absolute, but coordinates of events are not!

One final math point: at the start of all this I singled out an event
as the reference event. This *seems* to make one event special (but
actually doesn't), and can be avoided by modeling spacetime with an
affine space instead of a vector space.

My advice: study Taylor and Wheeler closely without worrying too much
about the abstract mathematics right now - there will plenty of time for
that later. While in high school, I learned special relativity from
Taylor and Wheeler.

Regards,
George