Sorry for the late reply. I've been very busy.

Spoonfed wrote:
I count TWO notions of time in relativity, and thus two notions of
distant simultaneity. You and Friedmann have been using "proper time"
and I've been using "coordinate time"

No, the t in a(t) is a coordinate time; they're just different coordinates.

I hope you are exaggerating when you say "there is no notion of distant
simultaneity at all" because I can see no way to have any discussion of
this topic at all without some notion of simultaneity.

There's no physically meaningful notion of distant simultaneity. If you
choose a coordinate system, you get a notion of distant simultaneity (equal
coordinate time), but it's physically meaningless.

The Lorentz transformation represents the only possibility that
maintains all lightcones, and all collisions while allowing changes in
velocity. And it does this very, very elegantly, taking care of all of
your "complicated business" of drawing conclusions about the origin of
those photons.

You simply must unlearn this stuff about the Lorentz transformation in order
to understand general relativity. It is because you have this idea of the
Lorentz transformation operating globally on the universe that you're having
trouble with GR, and with abandoning distant simultaneity. See below about
Alice and Bob.

I don't understand this idea of equivalence. It seems to me redshift
must either be explained by the SR formula or the change in the scale
factor. If putting the redshift effect into the scale factor makes the
math easier, this should be described explicitly as a mathematical
shortcut for calculation purposes.

That would be similar to describing the primed coordinates in the Lorentz
transformation as just a mathematical shortcut to make the calculation more
convenient, which is what Lorentz did. It's better to treat the unprimed and
primed coordinates as equally valid. That's also true of other coordinate
systems that aren't related by a global Lorentz transformation.

What phenomenon are you expecting to make distant objects seem larger?

See this message:

http://groups.google.com/group/sci.p...2984b2c8511d91

This is for a(t)=t and k=0, but I think it applies to your model also. (But
keep in mind that I'm less sure of this than of the other stuff in this
discussion.)

The simplest difference I know of is that I predict that a 600km/second
change in velocity would not significantly effect a measurement of the
CMBR dipole. This is very much at odds with the explanation for the
dipole given by NASA.

But that's not even consistent with SR, let alone GR or the big bang theory.
A 600km/sec boost leads to Doppler shift and aberration *of your visual
field* which is completely independent of where that light originally came
from. The effect of a 600km/sec boost on the CMBR dipole is independent of
any cosmological assumptions. It only depends on local Lorentz symmetry.

Redshift is not independent of the speed of its source.

Indeed not, but what I said is correct. If Alice and Bob are at the same
place at a particular moment, both moving inertially, and you know what
Alice sees, and Alice and Bob's relative velocities, that is enough to
determine what Bob sees. Information about the objects that originally
produced the light is unnecessary. This is true in SR -- nothing to do with
GR -- but it remains true in GR, and I think that once you understand it you
will understand GR better.

Restricting ourselves to SR, consider the following two ways of working out
what Bob sees, in terms of what Alice sees:

* In terms of Bob's rest frame: the whole universe is
Lorentz-transformed from Alice's rest frame; distances, times,
redshifts, etc. change, so Bob sees different things.

* In terms of Alice's rest frame: the universe is not
Lorentz-transformed, but Bob is Lorentz-transformed from his
own rest frame: his eyes are distorted, causing him to see
different things than Alice sees.

You should be able to see that these two approaches yield equivalent
predictions: they are, after all, Lorentz transformations of each other.

Most SR courses only teach the first (global) approach. But only the second
(local) approach works in GR.

-- Ben

Ben Rudiak-Gould wrote:
Sorry for the late reply. I've been very busy.

Spoonfed wrote:
I count TWO notions of time in relativity, and thus two notions of
distant simultaneity. You and Friedmann have been using "proper time"
and I've been using "coordinate time"

No, the t in a(t) is a coordinate time; they're just different coordinates.


First off, I will need to know where to find a derivation of this:
http://en.wikipedia.org/wiki/Friedmann_equation

Second,
If you have sound and a good bandwidth on your computer, have a look at
this
http://www.spoonfedrelativity.com/movies/Plot6to9.htm

If you mapped a region of constant "space-time interval" as I call it
in the demo, or constant "proper time" as I've been calling it here,
you would get a hyperbola. If you mapped a region of constant
coordinate time, you would get a straight line.

I used the terms "proper time" vs. "coordinate time" as defined in
Lewis Carroll Epstein's Relativity Visualized. Maybe they are not in
common usage.

Coordinate time goes along with coordinate space, with all the
(x,y,z,t) defined events that go along with them. Proper time goes
along with individual particles describing how much they have aged.

If you looked at the universe from the perspective of a given particle
at a given age, it should look the same as the universe from the
prespective of any other particle at the SAME age.

Specifically, each particle should observe itself to be at the center
of a sphere that looks (in cross-section) like this:
http://www.spoonfedrelativity.com/files/250%20plus.JPG

The circle should have a radius proportional to the age of the particle
(since the initial event), and all of the particles should be moving
outward at a speed of v=H*d, where H=1/s where s is the age of the
particle, AKA the proper time of the particle, AKA the spacetime
interval.

By this logic, the universe looks the same FROM all places in the
universe. Expanding at an equal speed. But we don't look at the
universe FROM all places, we look at the universe from HERE. And from
HERE, the universe appears to increase in density towards infinity
toward the edges. Just as it does FROM everywhere else.

Anyway, I just want to see the derivation of the Freidmann equation,
and I thought you might have a good reference.

Spoonfed wrote:
First off, I will need to know where to find a derivation of this:
http://en.wikipedia.org/wiki/Friedmann_equation

I imagine it's in most textbooks. I'm not sure about online. I can outline
it here.

The assumptions leading to the Friedmann equations a

* Spacetime is described by the FRW metric.

* The stress-energy tensor is that of a homogeneous isotropic perfect
fluid, which I seem to recall is given in FRW coordinates by

/rho 0 0 0\
| 0 p 0 0|
| 0 0 p 0|
\ 0 0 0 p/

where rho is the energy density and p is the pressure. (Both are
functions of the FRW coordinate t.)

* The GR field equations hold.

The Friedmann equations are the extra constraints needed to make all of
these assumptions consistent. You get them by working out the Einstein
tensor G_uv from the FRW metric (which is straightforward but tedious --
just tons of differentiation), and setting it equal to 8 pi G times the
stress-energy tensor given above.

Second, If you have sound and a good bandwidth on your computer, have a
look at this
http://www.spoonfedrelativity.com/movies/Plot6to9.htm

Yes, I think it's neat. Keep up the good work. But in order to understand GR
you have to realize that the Lorentz transformation is valid only locally.

I used the terms "proper time" vs. "coordinate time" as defined in
Lewis Carroll Epstein's Relativity Visualized. Maybe they are not in
common usage.

Well, his approach to special relativity is quite different from everyone
else's, as I'm sure you know. I'm pretty sure his definition of proper time
is the same as mine. When I say "coordinate time" I simply mean the value of
the time coordinate of some event, with respect to some agreed-upon
coordinate system.

Coordinate time goes along with coordinate space, with all the
(x,y,z,t) defined events that go along with them. Proper time goes
along with individual particles describing how much they have aged.

Right.

If you looked at the universe from the perspective of a given particle
at a given age, it should look the same as the universe from the
prespective of any other particle at the SAME age.

Under certain symmetry assumptions, yes.

Specifically, each particle should observe itself to be at the center
of a sphere that looks (in cross-section) like this:
http://www.spoonfedrelativity.com/files/250%20plus.JPG

I understand where this picture comes from, mathematically. But you are
attaching to it far more physical significance than it deserves.

(Pseudo-)Riemannian manifolds are not like (pseudo-)metric spaces. Metric
spaces are nonlocal: you plug any two points in the space into the distance
function, and wham, you get a distance between them. Riemannian manifolds
are local: From a point, you can only figure out the distance to nearby
points. You can't jump from here to there; you have to move continuously
from here to there. Your notion of spacetime interval is s^2 = t^2 - x^2,
but there's no such thing on a Riemannian manifold. There's only ds^2 = dt^2
- dx^2.

-- Ben

Ben Rudiak-Gould wrote:
Spoonfed wrote:
First off, I will need to know where to find a derivation of this:
http://en.wikipedia.org/wiki/Friedmann_equation

I imagine it's in most textbooks. I'm not sure about online. I can outline
it here.

The assumptions leading to the Friedmann equations a

* Spacetime is described by the FRW metric.

* The stress-energy tensor is that of a homogeneous isotropic perfect
fluid, which I seem to recall is given in FRW coordinates by

/rho 0 0 0\
| 0 p 0 0|
| 0 0 p 0|
\ 0 0 0 p/

where rho is the energy density and p is the pressure. (Both are
functions of the FRW coordinate t.)

* The GR field equations hold.

The Friedmann equations are the extra constraints needed to make all of
these assumptions consistent. You get them by working out the Einstein
tensor G_uv from the FRW metric (which is straightforward but tedious --
just tons of differentiation), and setting it equal to 8 pi G times the
stress-energy tensor given above.


It seems like there was some derivation along these lines in Schuetz's
book. I will try to go back to it again to look. He presented a model
of "dust" representing a infinite, static and homogeneous distribution
of particles throughout space, but I couldn't remember him actually
making the assumption that it represented the universe. To me, it
seemed like one possible distribution, and I skimmed through the rest
of the book looking around for a more believable non-static
distribution, but it seemed like he wasn't going to get around to
presenting another example.

Perhaps I'll have a chance to go review it again, now that I have some
better idea how the static homogeneous distribution in the FRW metric
with k=-1, a(t)=t is the same as a relativistically expanding
lobachevskian distribution in Euclidian space.

(Maybe it will be back at the library next time I check.)

Second, If you have sound and a good bandwidth on your computer, have a
look at this
http://www.spoonfedrelativity.com/movies/Plot6to9.htm

Yes, I think it's neat. Keep up the good work. But in order to understand GR
you have to realize that the Lorentz transformation is valid only locally.

I used the terms "proper time" vs. "coordinate time" as defined in
Lewis Carroll Epstein's Relativity Visualized. Maybe they are not in
common usage.

Well, his approach to special relativity is quite different from everyone
else's, as I'm sure you know. I'm pretty sure his definition of proper time
is the same as mine. When I say "coordinate time" I simply mean the value of
the time coordinate of some event, with respect to some agreed-upon
coordinate system.


That's the same as my definition.

I think Epstien gets into trouble for calling the business about curved
space "hocus-pocus." To me, his exile from the ranks of respected
authors on Relativity appears to be more a matter of politics than
substance. Here are a couple quotes from Relativity Visualized.

"Proper means the measure of a thing as perceived by an agent not in
motion relative to the thing being measured. If you ride on a ship,
you measure its proper length. If you measure the ship's length as it
flies past you, you don't measure it's proper length."

And another quote from Epstein:
"The spacetime diagram in this book represents the speed of light as a
horizontal line. The spacetime diagram in many other books represents
the speed of light as a sloped 45 degree line. How come? Because the
diagram in this book plots proper time against space. The diagram in
the other books plots coordinate time against space. Which is right?
Both are. They are different views of the same thing"

Coordinate time goes along with coordinate space, with all the
(x,y,z,t) defined events that go along with them. Proper time goes
along with individual particles describing how much they have aged.

Right.

If you looked at the universe from the perspective of a given particle
at a given age, it should look the same as the universe from the
prespective of any other particle at the SAME age.

Under certain symmetry assumptions, yes.


Good point.

Specifically, each particle should observe itself to be at the center
of a sphere that looks (in cross-section) like this:
http://www.spoonfedrelativity.com/files/250%20plus.JPG

I understand where this picture comes from, mathematically. But you are
attaching to it far more physical significance than it deserves.

(Pseudo-)Riemannian manifolds are not like (pseudo-)metric spaces. Metric
spaces are nonlocal: you plug any two points in the space into the distance
function, and wham, you get a distance between them. Riemannian manifolds
are local: From a point, you can only figure out the distance to nearby
points. You can't jump from here to there; you have to move continuously
from here to there. Your notion of spacetime interval is s^2 = t^2 - x^2,
but there's no such thing on a Riemannian manifold. There's only ds^2 = dt^2
- dx^2.

-- Ben

Also a good point; I am considering events which are considerably more
than differentially separated. I am assuming that for a first order
approximation, gravitational effects can be neglected, and that we have
several choices in how to perform a path integral to calculate s.

You can choose to take the path integral either along hyperbolic-curved
lines, between differentially separated events of the same proper-time,
which is what you and Baez appear (to me) to be doing. Epstein does
this at well, but he makes it a lot more explicit.

.... or you can take the path integral along horizontal straight lines,
between differentially separated events of the same coordinate time
(for a particular observer), which is what I have been suggesting.

A third, most appropriate option, given our observational limitations,
would be to take the path integral along 45-degree-downward straight
lines representing differentially separated events along the
past-light-cone. This would represent the locus of events in the
universe which we currently see, because the light would be currently
reaching us.