What Are the Key Questions About Supernova Data and the Modified Milne Model?

[Chalnoth]

CMB was discovered in 1965. The big discovery of COBE was that the cosmic background radiation was not constant.

Also that the frequency spectrum of the CMB is extraordinarily close to a perfect black body spectrum.

 

[Chalnoth]

Also, I think you can get the Milne model as a subset of the standard model. If you set all of the densities to zero in the standard cosmology, what you get is the Milne model.

Yes, this is correct, which is why we can say it is definitively ruled out by current observations. We don't need to bother to modify anything to examine the Milne model: we merely compare our parameter estimates and see if the "total density = 0" case is ruled out. And it is to around 100 standard deviations with a combination of baryon acoustic oscillation and WMAP data (with this combination of data, $\Omega = 1$ to within about a percent at one standard deviation).

You can attempt to get around this by proposing that the CMB doesn't come from the phase transition in the early universe from a plasma to a gas, but then this presents two problems:
1. Why is the frequency spectrum of the CMB nearly a perfect black body? A large collection of distant stars, for instance, will not produce a black body spectrum.
2. Why is the baryon acoustic oscillation effect visible at all? This effect is a correlation of an angular scale on the CMB with the typical separation between galaxies in the nearby universe. It isn't a trivial correlation, but instead a correlation that relies upon the physics that would have been active when our universe was still a plasma.

If you want to get around saying that there was an early hot plasma state, you need to present a new model that predicts both of these effects (as well as others, such as the primordial helium abundance). If you don't, then the Milne cosmology is ruled out to hundreds of standard deviations, not even counting the obvious fact that there is matter in the universe, while the Milne cosmology assumes none.

 

[twofish-quant]

You can attempt to get around this by proposing that the CMB doesn't come from the phase transition in the early universe from a plasma to a gas

You still have big bang nucleosynthesis to worry about, and then there is galaxy count information. There are a lot of charts in the internet that describe the experiment limits on parameters and where they come from.

You could get really crazy and then reject GR and redshifts altogether, but at that point you've blown away the Milne model in addition to standard cosmologies. If you don't think that redshifts are caused by Hubble flow, that's interesting, but then you are rejecting all of the supernova and CMB data, at which point I don't see how looking at the details are going to be useful.

It's generally considered rude to pick and choose what data you think is valid and which ones you think aren't, but the fun thing about this discussion is that even if you cherry pick data, you are going to find it difficult to get anywhere close to the Milne model.

 

[Chalnoth]

You still have big bang nucleosynthesis to worry about, and then there is galaxy count information.

Indeed, mentioned the primordial helium abundance :)

You could get really crazy and then reject GR and redshifts altogether, but at that point you've blown away the Milne model in addition to standard cosmologies. If you don't think that redshifts are caused by Hubble flow, that's interesting, but then you are rejecting all of the supernova and CMB data, at which point I don't see how looking at the details are going to be useful.

It's generally considered rude to pick and choose what data you think is valid and which ones you think aren't, but the fun thing about this discussion is that even if you cherry pick data, you are going to find it difficult to get anywhere close to the Milne model.

Yup. Well, there's a large degeneracy in the supernova data with regard to the total energy density of the universe. It measures reasonably accurately the ratio of matter (normal + dark) to dark energy, but it doesn't do so well at measuring the total. Because of this, early supernova experiments taken alone were quite consistent with a Milne cosmology (except for the whole issue of the universe not being empty).

But this degeneracy isn't perfect, and more recent supernova experiments which have much more data rule out the Milne cosmology rather well. Of course, once you combine them with something like WMAP or BAO, the Milne cosmology becomes so far out of bounds of observation that the whole idea becomes patently absurd.

 

[JDoolin]

Milne model, if given in spherical coordinates, should have the metric:

$$ds^2 = (cdt)^2 - dr^2$$​

...which is the same metric that results from having no matter present.

In addition to the typo that I already corrected, I made a rather more boneheaded conceptual error. As penance, I went through the entire derivation this morning four times until I finally got it right, and did my best to get the results in LaTeX.

The goal here is to find (cdt)² - dx² - dy² - dz^2 in spherical coordinates. We must make the coordinate transformations of

$$\begin{matrix} z=r cos(\theta)\\ x=r sin(\theta) cos(\phi)\\ y= r sin(\theta) sin(\phi)\\ \\ dz = cos(\theta) dr - r sin(\theta)d(\theta)\\ dx = sin(\theta)cos(\phi)dr + r cos(\theta)sin(\phi)d\theta - rsin(\theta)sin(\phi)d\phi\\ dy = sin(\theta)sin(\phi)dr + r cos(\theta)sin(\phi)d\theta + r sin(\theta)cos(\phi)d\phi \end{matrix}$$​

I was thinking the calculation of dx²+dy²+dz² was trivial and could be equated to dr². However, it is not so trivial as that. We have to multiply term-by-term, and combine like terms.

$$\begin{tabular} {|c|ccc|c|} \hline term & dz & dx & dy & total \\ \hline dr^2 & cos^2(\theta)& sin^2(\theta)cos^2(\theta) & sin^2(\theta)sin^2(\phi) & dr^2 \\ d\theta^2 & r^2 sin^2(\theta) & r^2 cos^2(\theta)cos^2(\phi) & r^2 cos^2(\theta)sin^2(\phi) & r^2 d\theta^2\\ d\phi^2 & 0 & r^2 sin^2(\theta)sin^2(\phi) & r^2 sin^2(\theta)cos^2(\phi)& r^2 sin^2(\theta)d\phi^2\\ dr d\theta & -2r sin(\theta)cos(\theta) & 2 r sin(\theta)cos(\theta)cos^2(\phi)&2rsin(\theta)cos(\theta)sin^2(\theta)&0 \\ dr d\phi & 0 & -2rsin^2(\theta)sin(\phi)cos(\phi)&2rsin^2(\theta)sin(\phi)cos(\phi) & 0\\ d\theta d\phi & 0 & -2r^2 sin(\theta)cos(\theta)sin(\phi)cos(\phi) & 2 r^2 sin(\theta)cos(\theta)sin(\phi)cos(\phi)&0\\ \hline \end{tabular}$$​

Hence, the metric for the Milne model in spherical coordinates is

$$ds^2=(c dt)^2 - dr^2 - r^2(d\theta^2 +sin^2(\theta)d\phi^2)$$​

The equation given by the Wikipedia article for the Milne model has a hyperbolic sine function in it, and is clearly not appropriate for empty space or a gravity-free region.

 

[JDoolin]

You are in the wrong game then. I can make any model work if I don't care about the details. It's getting the details right that's a pain.

You understanding is wrong. The way that the standard model handles electron recombination is exactly the same as how the Milne model does.

Right.

So can you with the Milne model calculate the exact spectrum of the lumpiness. With lamba-CDM, I can put in some parameters and get a fit to get the lumpiness.

Can you do that?

One should look for the devil in his details, but should not look to the devil for his details.

Yes, it is a pain to get the details right. And yes, the devil is in the details, because the details are hard to get right. The worst details are the ambiguous details; the "not even wrong" variety.

For instance, you claim that the Milne model and the standard model handle electron recombination in the same way. Of course they do! However, why is the frequency of the light redshifted? Why is it that this light can still be seen? In the standard model, the the events happened long ago, (the plane of last scattering is no longer happening anywhere in the universe) but the light is just now reaching us.

But in the Milne model the redshift is entirely due to the recession velocity of the plane of last-scattering. And the fact that this plane can still be seen is due to time-dilation. The events are still happening; the plane of last-scattering is still there, moving away from us at nearly the speed of light.

On your other detail, you are asking "Can the Milne model predict the lumpiness." Of course not! We can see it, and perhaps hope to find an explanation for it, but the idea that you could or should predict the lumpiness of the CMBR from a metric is mere conceit. It is analogous to asking a man to "predict" the shape of the mountains and valleys on an unseen planet, using only the Pythagorean theorem.

What you can do, though, if you have a map , then you can begin forming real theories. In the case of a planetary map, you can develop theories about tectonic plate motion, volcanic activity, sedimentary action, etc.

If you have a map, you can develop a real theory: With WMAP and COBE, for instance, they could see that the light was a thermal spectrum, and then they were able to come up with a phenomenon that caused it. Other than using the idea of redshift, the standard model neither helped nor hindered them in figuring out that the process must be caused by recombination.

Jonathan

 

[JDoolin]

With lamba-CDM, I can put in some parameters and get a fit to get the lumpiness.

By the way, you can generate any function with a Fourier Series expansion. With only a few parameters, generally, you can get a pretty good fit. Your ability to put in parameters to an equation and get an approximate fit uses a similar kind of math using spherical Bessel functions or some related idea.

Jonathan

 

[Chalnoth]

Hence, the metric for the Milne model in spherical coordinates is

$$ds^2=(c dt)^2 - dr^2 - r^2(d\theta^2 +sin^2(\theta)d\phi^2)$$​

The equation given by the Wikipedia article for the Milne model has a hyperbolic sine function in it, and is clearly not appropriate for empty space or a gravity-free region.

This is the metric for Minkowski space-time. The Milne metric is a simple coordinate change. Simply take the above metric, and replace:

$$r \to t \sinh r$$
$$t \to t \cosh r$$

...and you will have the Milne metric.

 

[Chalnoth]

On your other detail, you are asking "Can the Milne model predict the lumpiness." Of course not! We can see it, and perhaps hope to find an explanation for it, but the idea that you could or should predict the lumpiness of the CMBR from a metric is mere conceit. It is analogous to asking a man to "predict" the shape of the mountains and valleys on an unseen planet, using only the Pythagorean theorem.

Then why should we pay it any attention? We have a perfect explanation for the lumpiness we do see, an explanation that is not only very specific and detailed, but also accords with other observations. And that explanation precludes the possibility of a Milne cosmology. Unless you can present a model that has at least as much predictive power as the current cosmology, nobody is going to care.

If you have a map, you can develop a real theory: With WMAP and COBE, for instance, they could see that the light was a thermal spectrum, and then they were able to come up with a phenomenon that caused it. Other than using the idea of redshift, the standard model neither helped nor hindered them in figuring out that the process must be caused by recombination.

This isn't in any way the case. The big bang theory predicted the thermal spectrum. This was expected back when the CMB was first predicted.

And as for the lumpiness, at the time COBE was launched, there were two major competing theories that predicted its statistical properties: cosmic strings, and inflation. Inflation won out. This was apparent with COBE plus balloon data, but became blatantly and wildly obvious with WMAP.

 

[Chalnoth]

By the way, you can generate any function with a Fourier Series expansion. With only a few parameters, generally, you can get a pretty good fit. Your ability to put in parameters to an equation and get an approximate fit uses a similar kind of math using spherical Bessel functions or some related idea.

So? Lambda-CDM is most definitely not a Fourier series expansion. A lot of rather detailed physics goes into calculating the precise magnitude of the oscillations of different sizes on the sky.

 

[JDoolin]

So? Lambda-CDM is most definitely not a Fourier series expansion. A lot of rather detailed physics goes into calculating the precise magnitude of the oscillations of different sizes on the sky.

But if you go into the nitty-gritty details, essentially you'll find you're fitting an infinite series expansion of some kind. And you can a get good approximation of the phenomenon with a surprisingly small number of terms.

The key word is model. You're able to model the CMBR, but you aren't predicting anything, and it you're not explaining anything. You just have an equation that fits the data.

 

[JDoolin]

This isn't in any way the case. The big bang theory predicted the thermal spectrum. This was expected back when the CMB was first predicted.

Does the CMB come from the Big Bang, or does it come from hydrogen recombination?

 

[JDoolin]

I don't want to lose the main question of my original post.

So far, no-one has given me an answer to the questions of my original post. I hoped I could find someone who knew how to obtain the following data about the supernovae .

a) right ascension
b) declination
c) luminosity distance
d) redshift

But meanwhile...

Unless you can present a model that has at least as much predictive power as the current cosmology, nobody is going to care.

I disagree. The standard model cosmology does not have that good of a track record. One need only skim through an article about http://en.wikipedia.org/wiki/Dark_energy" to see that it introduces more questions than it answers.

For instance

'The existence of dark energy, in whatever form, is needed to reconcile the measured geometry of space with the total amount of matter in the universe.'

"A major outstanding problem is that most quantum field theories predict a huge cosmological constant from the energy of the quantum vacuum, more than 100 orders of magnitude too large. This would need to be canceled almost, but not exactly, by an equally large term of the opposite sign"

I can't claim to understand these problems exactly. However, they simply don't arise in the Milne model.

For instance, in the Milne model, inflation arises very naturally from acceleration. You take a sphere expanding at the speed of light, and perform a Lorentz Transformation around any event after the Big Bang.

All you need is collisions or explosions to get the necessary Delta V.

If you accelerate toward the center, it will expand the universe. If Milne was correct, and all particles are attracted to the center, then this phenomenon would go on forever, accelerating toward a receding center would make inflation continue forever.

 

[JDoolin]

This is the metric for Minkowski space-time. The Milne metric is a simple coordinate change. Simply take the above metric, and replace:

$$r \to t \sinh r$$
$$t \to t \cosh r$$

...and you will have the Milne metric.

And why would you make that replacement? Time is not flowing slower as you go out from the center. Distance is not getting smaller as you go out from the center.

It is the proper age of particles that are traveling away from you that are going to be aging slower, and the actual distance between particles that are going to be closer together.

The metric does not change when the particles are moving away from one another. The Minkowski metric and the Milne metric are the same. You just have to go back to "Relativity, Gravitation, and World Structure" to find out.

 

[Chalnoth]

But if you go into the nitty-gritty details, essentially you'll find you're fitting an infinite series expansion of some kind. And you can a get good approximation of the phenomenon with a surprisingly small number of terms.

No, this is completely false. The parameters in Lambda-CDM are:

1. Normal matter density.
2. Dark matter density (assumed to be zero-temperature).
3. Cosmological constant value.
4. Scalar spectral index (an inflation parameter).
5. Optical distance to the surface of last scattering.
6. Hubble constant.

Some other parameters are also input from other observations, such as the CMB temperature and the Helium fraction. With these parameters, this is the simplest CMB model, and it fits the data very precisely. Of crucial importance is that with any combination of these parameters, a rather specific sort of power spectrum of the CMB is predicted. The parameters themselves change various ways in which this power spectrum can appear, but the overall pattern is set by these parameters alone.

The second point is that once you measure these parameters, the prediction is that other experiments will measure the same values for the same parameters, to within the experimental error. And they do.

It is true that this simplest CMB model is expected to be a bit wrong. The scalar spectral index, for example, isn't expected to be exactly constant, with the exact behavior depending upon the precise model of inflation (with some inflation models already ruled out by WMAP). Similarly, the dark matter isn't expected to be zero-temperature for many models, but is expected to have some finite, measurable temperature that can, in principle, be measured. The optical distance to the surface of last scattering is also a simplification, as this depends upon precisely how the universe reionized when the stars started turning on.

But the point is this: the simplest model is valid to within experimental errors, and these other issues will require extremely precise measurements of the CMB, as well as other measurements, in order to nail them down further.

The key word is model. You're able to model the CMBR, but you aren't predicting anything, and it you're not explaining anything. You just have an equation that fits the data.

a) The model was put together long before CMB observations were underway. We may not have been able to predict the precise value of every parameter, but the model most definitely predicted the overall shape of the power spectrum of the CMB.
b) Models are explanations.

 

[Chalnoth]

And why would you make that replacement? Time is not flowing slower as you go out from the center. Distance is not getting smaller as you go out from the center.

Don't ask me. These are just the coordinates for the Milne cosmology, which we know is wrong, so I don't see any reason to look into it further.

 

[JDoolin]

Don't ask me. These are just the coordinates for the Milne cosmology, which we know is wrong, so I don't see any reason to look into it further.

Except for the fact that they are not the scale for the Milne cosmology. Someone lied. Milne used the same metric as Minkowski.

 

[Chalnoth]

I don't want to lose the main question of my original post.

So far, no-one has given me an answer to the questions of my original post. I hoped I could find someone who knew how to obtain the following data about the supernovae .

a) right ascension
b) declination
c) luminosity distance
d) redshift

Right ascension and declination are coordinates on the sky. Declination is degrees from the equator as seen from Earth, with negative values used for stars in the southern hemisphere. Right ascension is a value given in hours, typically using the J2000 standard epoch. That is, the hour gives the hour of the day on Jan. 1, 2000 where the star would have been directly overhead.

These values are typically listed for each supernova when a supernova experiment's results are reported. For instance, here is the SNLS first-year release:
http://xxx.lanl.gov/abs/astro-ph/0510447

The table including the supernovae is at the end. The redshift of each supernova is measured by careful measurement of its spectrum. The magnitudes of each supernova are also listed, and the relationship to the luminosity distance is described in the paper.

I disagree. The standard model cosmology does not have that good of a track record. One need only skim through an article about http://en.wikipedia.org/wiki/Dark_energy" to see that it introduces more questions than it answers.

Introducing more questions than it answers is the norm in science, though. This is what learning new things is all about! The fact remains that the parameters in Lambda-CDM are measured to be the same to within experimental errors no matter which set of cosmological observations we use.

For instance, in the Milne model, inflation arises very naturally from acceleration. You take a sphere expanding at the speed of light, and perform a Lorentz Transformation around any event after the Big Bang.

All you need is collisions or explosions to get the necessary Delta V.

No, this won't work, because you won't get the right expansion rate, and you definitely won't come remotely close to a nearly-homogeneous universe.

Look, you can't just say, "Hey, look, in this model the words I use make it sound sort of like it works like this other model! Therefore they're the same!" This isn't the way science works. You have to dig into the details of the model and see what it actually says, not just look at superficial behavior.

 

[JDoolin]

The parameters in Lambda-CDM are:

1. Normal matter density.
2. Dark matter density (assumed to be zero-temperature).
3. Cosmological constant value.
4. Scalar spectral index (an inflation parameter).
5. Optical distance to the surface of last scattering.
6. Hubble constant.

I'm apologize. I have never seen that analysis. I do recall seeing an article which essentially did a curve fitting of the data to some Bessel function, and they were surprised at how big the 17th coefficient or something other was.

In the case that you have a model with these six parameters, then I have to admit that is more interesting.

 

[Chalnoth]

I'm apologize. I have never seen that analysis. I do recall seeing an article which essentially did a curve fitting of the data to some Bessel function, and they were surprised at how big the 17th coefficient or something other was.

Yes, this has been done. This sort of analysis doesn't interest most scientists.

In the case that you have a model with these six parameters, then I have to admit that is more interesting.

I'm glad. You can see how these parameters affect the power spectrum here:
http://space.mit.edu/home/tegmark/cmb/movies.html

Note that these animations were made before WMAP, and the data on the plot is a combination of pre-WMAP data. He also adds the effects of a couple of extra parameters that are not in the simplest-case analysis, so you can see what they do.

 

[twofish-quant]

However, why is the frequency of the light redshifted? Why is it that this light can still be seen? In the standard model, the the events happened long ago, (the plane of last scattering is no longer happening anywhere in the universe) but the light is just now reaching us.

But in the Milne model the redshift is entirely due to the recession velocity of the plane of last-scattering. And the fact that this plane can still be seen is due to time-dilation. The events are still happening; the plane of last-scattering is still there, moving away from us at nearly the speed of light.

Six of one. Half dozen of the other. In GR those explanations are quite equivalent. You can set your coordinate system so that time moves at a constant rate and space is expanding, or you can set your system so that space is fixed and time is slowing down. The really are the same explanation.

On your other detail, you are asking "Can the Milne model predict the lumpiness." Of course not! We can see it, and perhaps hope to find an explanation for it, but the idea that you could or should predict the lumpiness of the CMBR from a metric is mere conceit.

Lamba-CDM predicts the lumpiness of the universe. Once you have a figure for how quickly the universe expands and you make some assumptions about what the universe is made of, then you can rather easily calculate how pressure waves go through the universe, and get lumpiness coefficients.

You can do the same calculation with the Milne model. People have done that calculation and you get nowhere near the observed universe.

If you have a map, you can develop a real theory: With WMAP and COBE, for instance, they could see that the light was a thermal spectrum, and then they were able to come up with a phenomenon that caused it.

Other way around. The CMB was discovered in 1965. The calculations for how to calculate the lumpiness factor of the universe were done in the 1980's. At that point, you have a group of scientists go to Congress and then lobby for money to send up a spacecraft designed to measure the lumpiness of the universe.

 

[twofish-quant]

And why would you make that replacement?

Makes the math simpler.

Time is not flowing slower as you go out from the center. Distance is not getting smaller as you go out from the center.

General relativity says that you can chose whatever clocks and rulers you want and you'll get the same answer. If you want a ruler that shrinks as you move it. That's fine. If you want a clock that speeds up or slows down, that's also fine.

The easily analogy is that you can draw a diagram with square graph paper, or you can draw the diagram with polar coordinates. It's all the same.

The metric does not change when the particles are moving away from one another.

The metric does not change, but it's perfectly fine to do a coordinate transform. The whole point of relativity is that you give me a metric. I can do certain coordinate transforms that I want, and the metric stays the same.

 

[twofish-quant]

Introducing more questions than it answers is the norm in science, though. This is what learning new things is all about! The fact remains that the parameters in Lambda-CDM are measured to be the same to within experimental errors no matter which set of cosmological observations we use.

Coming up with useful questions is more important than coming up with answers. Once you come up with the questions, you then build instruments to answer those questions.

Look, you can't just say, "Hey, look, in this model the words I use make it sound sort of like it works like this other model! Therefore they're the same!" This isn't the way science works. You have to dig into the details of the model and see what it actually says, not just look at superficial behavior.

In particular, cosmic inflation is defined as some dark energy field dumping energy that increases the expansion rate of the universe. It's really inconsistent with the Milne cosmology by definition. In particular, in most inflationary models, the expansion of the universe was exponential and approaches the de Sitter model.

 

[TrickyDicky]

When I apply this transformation to the minkowski metric:
$$r \to t \sinh r$$

$$t \to t \cosh r$$

I don't get this

$$ds^2 = dt^2-t^2(dr^2+\sinh^2{r} d\Omega^2)\$$​

But this

$$ds^2 = dt^2-t^2(\sinh^2{r} d\Omega^2)\$$​

I lose the dr^2 term, what am I doing wrong?

 

[Chalnoth]

When I apply this transformation to the minkowski metric:
$$r \to t \sinh r$$

$$t \to t \cosh r$$

I don't get this

$$ds^2 = dt^2-t^2(dr^2+\sinh^2{r} d\Omega^2)\$$​

But this

$$ds^2 = dt^2-t^2(\sinh^2{r} d\Omega^2)\$$​

I lose the dr^2 term, what am I doing wrong?

Perhaps a sign error? The derivative of the hyperbolic sine and cosine are both positive. Maybe you accidentally made the derivative of cosh negative? Alternatively, maybe it comes from not carrying the product rule through for both $dr$ and $dt$?

 

[TrickyDicky]

From the Minkowski metric with this coordinate transform

$$r \to t \sinh r$$

$$t \to t \cosh r$$
Here's what I get since: $$\cosh^2{r}-\sinh^2{r}=1$$

$$ds^2 = dt^2\cosh^2{r}-dt^2\sinh^2{r}-t^2\sinh^2{r} d\Omega^2 =dt^2(\cosh^2{r}-\sinh^2{r})-t^2\sinh^2{r} d\Omega^2=dt^2-t^2(\sinh^2{r} d\Omega^2)$$​

 

[Chalnoth]

From the Minkowski metric with this coordinate transform

$$r \to t \sinh r$$

$$t \to t \cosh r$$
Here's what I get since: $$\cosh^2{r}-\sinh^2{r}=1$$

$$ds^2 = dt^2\cosh^2{r}-dt^2\sinh^2{r}-t^2\sinh^2{r} d\Omega^2 =dt^2(\cosh^2{r}-\sinh^2{r})-t^2\sinh^2{r} d\Omega^2=dt^2-t^2(\sinh^2{r} d\Omega^2)$$​

Oh, okay, looks like you're not carrying the product rule through.

$$r \to t \sinh r$$

Leads to:
$$dr \to \sinh r dt + t \cosh r dr$$

I'm sure you can figure out the rest.

 

[TrickyDicky]

Oh, okay, looks like you're not carrying the product rule through.

$$r \to t \sinh r$$

Leads to:
$$dr \to \sinh r dt + t \cosh r dr$$

I'm sure you can figure out the rest.

Oops, I must have forgotten the little calculus I know.

Thanks

 

[Chalnoth]

Oops, I must have forgotten the little calculus I know.

Thanks

No worries :) I made a few mistakes in doing this myself before I finally got it right. When you posted, I had to go back to make sure I didn't make a mistake myself...

 

[JDoolin]

Makes the math simpler.



General relativity says that you can chose whatever clocks and rulers you want and you'll get the same answer. If you want a ruler that shrinks as you move it. That's fine. If you want a clock that speeds up or slows down, that's also fine.

The easily analogy is that you can draw a diagram with square graph paper, or you can draw the diagram with polar coordinates. It's all the same.


The metric does not change, but it's perfectly fine to do a coordinate transform. The whole point of relativity is that you give me a metric. I can do certain coordinate transforms that I want, and the metric stays the same.

It is okay to change from Cartesian coordinates to Spherical coordinates, because you are talking about fundamentally different variables. $(x,y,z) \to (r,\theta,\phi)$ is a legitimate transformation, because the first and second sets of coordinates are describing the same information in the different ways:

$$\begin {matrix} z=r cos \theta \\ x=r sin \theta cos \phi\\ y=r sin \theta sin \phi \end {matrix}$$​

This is something true about the mathematical relationships between the radius, the polar angle, and the azimuthal angle.

In Special Relativity, also, you can perform the coordinate transformation:

$$\begin{bmatrix} c t' \\ x' \end{bmatrix} = \begin{bmatrix} \cosh\phi &-\sinh\phi \\ -\sinh\phi & \cosh\phi \\ \end{bmatrix} \begin{bmatrix} c t \\ x \end{bmatrix}$$​

This is also okay, because on the left hand side, we have t' and x'. These are the coordinates of events in another inertial reference frame. t' and x' are fundamentally different from t and x, so it is okay that they have different forms. (You cannot, by the way, say "you want a ruler that shrinks as you move it." or "you want a clock that speeds up or slows down." There are explicit relationships that are already determined.)

Mapping from $(x,y,z) \to (r,\theta,\phi)$ or $(x,t) \to (x',t')$ is mathematically and physically valid. However, when you say:

$$\begin {matrix} t\to t \cosh r\\ r \to t \sinh r \end {matrix}$$​

You are replacing distance with distance, and time with time. Certainly, you preserve all of the information by doing so, but you do not preserve the shape.

For instance: all of the following represent coordinate transformations of the earth.
a) http://en.wikipedia.org/wiki/Peters_map
b) http://en.wikipedia.org/wiki/Albers_projection
c) http://en.wikipedia.org/wiki/Mercator_projection
d) http://en.wikipedia.org/wiki/Globe

Three of these transformations significantly affect the shape of the earth, while the fourth only affects the size and position. The globe represents the true shape, and the others represent convenient distortions of the shape depending on the purpose.. However, they still do not actually map $(x,y,z) \to (x,y,z)$.

If you would like to claim:

$$\begin {matrix} t' \to t \cosh r\\ r' \to t \sinh r \end {matrix}$$​

...then I can ask you why you think this is a convenient distortion of the shape, and what was your purpose in making that distortion. But to claim

$$\begin {matrix} t\to t \cosh r\\ r \to t \sinh r \end {matrix}$$​

... is to claim that t is mathematically different than t, and r is mathematically different from r. This is not legitimate.

Milne, by the way, was also quite disturbed at Eddington's "scale factors" and spent quite some effort in pointing out how ridiculous it was. If Milne knew that a model named after him had been saddled with such a thing, I think he would roll over in his grave.

 

[Chalnoth]

Your resistance to this coordinate change is truly amusing. It is possible in General Relativity to make any coordinate transformation at all, provided the coordinate transformation is one-to-one in some region. That is, the only limitation opposed on coordinate transformations is that they don't throw away information.

As long as your coordinate transformation satisfies this, any calculation you might ever do regarding a physical observable will give the same answer. The use of different systems of coordinates is merely a mathematical convenience of no physical meaning whatsoever. The real world, after all, doesn't have numbers written on it.

 

[JDoolin]

Your resistance to this coordinate change is truly amusing. It is possible in General Relativity to make any coordinate transformation at all, provided the coordinate transformation is one-to-one in some region. That is, the only limitation opposed on coordinate transformations is that they don't throw away information.

As long as your coordinate transformation satisfies this, any calculation you might ever do regarding a physical observable will give the same answer. The use of different systems of coordinates is merely a mathematical convenience of no physical meaning whatsoever. The real world, after all, doesn't have numbers written on it.

I pointed out a genuine flaw in your reasoning, and you dismissed it by pointing and laughing.

I have no resistance to saying

$$\begin {matrix} t' \to t \cosh r\\ r' \to t \sinh r \end {matrix}$$​

...if you can explain what you mean by t, t', r, r'.

But, what you are claiming, is:

$$\begin {matrix} t \to t \cosh r\\ r \to t \sinh r \end {matrix}$$​

This is nonsense, and it certainly wasn't what Milne ever meant.

 

[Chalnoth]

I pointed out a genuine flaw in your reasoning, and you dismissed it by pointing and laughing.

I have no resistance to saying

$$\begin {matrix} t' \to t \cosh r\\ r' \to t \sinh r \end {matrix}$$​

...if you can explain what you mean by t, t', r, r'.

But, what you are claiming, is:

$$\begin {matrix} t \to t \cosh r\\ r \to t \sinh r \end {matrix}$$​

This is nonsense, and it certainly wasn't what Milne ever meant.

You're really making a big deal out of a small lack of rigor? You may note that I actually did use the more correct notation in my first post about this: https://www.physicsforums.com/showpost.php?p=2876300&postcount=32

 

[JDoolin]

You're really making a big deal out of a small lack of rigor? You may note that I actually did use the more correct notation in my first post about this: https://www.physicsforums.com/showpost.php?p=2876300&postcount=32

Let us see if we can find the physical definitions for t', r', t, and r, and then we can have a sensible discussion.

 

[Chalnoth]

Simply give me the physical definitions for t', r', t, and r, and then we can have a sensible discussion.

There is no physical definition for any particular coordinates in General Relativity. These are not physical entities, just labels we place on the system.

However, in this case the primed coordinates are the Minkowski coordinates, with the unprimed coordinates being the Milne coordinates. The Milne coordinates can be thought of as the inside of a light cone in Minkowski space-time. There is a plot here that shows the shape:
http://world.std.com/~mmcirvin/milne.html#time

In any case, coordinates are not physical things, and are merely chosen for their convenience for a particular problem. The Milne coordinates are exactly equivalent to the Minkowski coordinates.