Conditions for spacetime to have flat spatial slices

[PeterDonis]

What about a larger clock? You take the motions of the planets around the solar system, and mark them against the background of distant stars. Or you take the motions of stars in the galaxy and mark them against the background of distant galaxies. I would think that as the clock becomes larger, as you take into account more and more of universe, it's "proper time" gets closer to the coordinate time.

The proper time of individual "small" clocks are measured in reference to this larger celestial clock.

I'm sorry to keep harping on this, but I really think this is a very misleading way of looking at things. First of all, as I noted in my last post, the "time" I measure by observing the motion of stars and galaxies against a background of more distant objects will be affected by my particular position in a gravitational field. But more important, "proper time" is a property of specific worldlines--more precisely, it's an invariant quantity associated with a specific segment of a specific worldline between two specific events. That means that in order to talk about proper time, you need to first specify the worldline and the events. There is no "proper time of the universe" or "proper time of the Milky Way galaxy" per se; you have to pick out a specific worldline (say, the worldline of the center of mass of the Milky Way galaxy) and specify events on it (say, the event of two successive "meridian passages" of the Solar System, as observed at the center of mass of the Milky Way) before you can talk about proper time. (The same goes for "proper distance", which is the spacelike analogue of proper time, of course; you have to first specify a particular spacelike curve, and two events on it, before you can talk about proper distance.)

When you talk about "coordinate time", I suspect what you are really thinking about (though you may not realize it at first) is the proper time as experienced by a particular observer traveling on a particular worldline that has some special property you're interested in. I think it would make for a much clearer discussion, and probably help you to conceptualize what's going on, if you would explicitly state, in every case, who those particular observers are and what particular worldlines they're traveling on, instead of using terms like "coordinate time" as though they automatically have a well-defined physical meaning.

 

[JDoolin]

I'm sorry to keep harping on this, but I really think this is a very misleading way of looking at things. First of all, as I noted in my last post, the "time" I measure by observing the motion of stars and galaxies against a background of more distant objects will be affected by my particular position in a gravitational field. But more important, "proper time" is a property of specific worldlines--more precisely, it's an invariant quantity associated with a specific segment of a specific worldline between two specific events. That means that in order to talk about proper time, you need to first specify the worldline and the events. There is no "proper time of the universe" or "proper time of the Milky Way galaxy" per se; you have to pick out a specific worldline (say, the worldline of the center of mass of the Milky Way galaxy) and specify events on it (say, the event of two successive "meridian passages" of the Solar System, as observed at the center of mass of the Milky Way) before you can talk about proper time. (The same goes for "proper distance", which is the spacelike analogue of proper time, of course; you have to first specify a particular spacelike curve, and two events on it, before you can talk about proper distance.)

When you talk about "coordinate time", I suspect what you are really thinking about (though you may not realize it at first) is the proper time as experienced by a particular observer traveling on a particular worldline that has some special property you're interested in. I think it would make for a much clearer discussion, and probably help you to conceptualize what's going on, if you would explicitly state, in every case, who those particular observers are and what particular worldlines they're traveling on, instead of using terms like "coordinate time" as though they automatically have a well-defined physical meaning.

You're saying that I'm talking about the proper time as experienced by a particular observer traveling on a particular worldline, and you are absolutely right.

You can visualize as the moon goes round the earth, the Earth goes around the sun, the sun goes around the milky way, and the milky-way undulates with other galaxies in a local cluster. By doing so you are invoking a reference frame of a particular observer traveling on a particular worldline.

What I'm saying is that any hypothetical observer who can back up and see all of these things at the same time is imagining a Minkowski reference frame of the size and scope of whatever events he visualizes.

But to attempt for a much clearer discussion:

We imagine that colonies are erected on several planets and moons in the solar system, and we decide to share an interplanetary time standard.

The solar body with the straightest worldline is the sun, so we imagine a hypothetical observer at the center of the sun, facing the star Polaris. And each time Earth completes a complete circle around the sun, from the perspective of this observer, we call it a solar year.

Colonies on Earth, Ganymede, and Mars all share the same time standard, but the intersolar time committee has its work cut out. None of them can actually see the solar system from the perspective of their hypothetical observer at the center of the sun, so they have to do a few tricky calculations. But they are able to do it, and when they come on the news in the morning, they are able to report the exact time as would be measured by that hypothetical observer.

I think what we have here is a spacetime that is "locally Minkowski" where the locality is quite large, but we can make it larger by taking more care in defining our solar observer:

One problem that occurs over the next few hundred years is this: over time, Polaris moves across the sky relative to the background galaxies. This means that the coordinate system as defined is very slowly rotating, to maintain a facing toward Polaris. So instead of using Polaris to define our constant reference direction, we use a background of distant galaxies.

By switching to a more distant reference point, we keep our East, North, and Up more constant, so we've increased the size of our "Local Minkowski-ness" Errors that would have crept up in dozens of years, now will take hundreds of years.

There are also slight errors that creep up over time due to the fact that the sun's worldline is not quite straight, the sun is actually under constant acceleration toward the center of the galaxy.

By switching to a reference frame based on the worldline of the center of the galaxy, we improve things; errors that would have crept up in hundreds of years, will now take thousands or millions of years, perhaps.

Ideally, we would use a worldline of an observer which never accelerates at all, and construct a coordinate system from that observer's reference frame. I think this is theoretically possible construction.

Not only is it a theoretically possible construction, but it's the construction of spacetime that comes most naturally to mind whenever any person first realizes that the it is actually the Earth that goes around the sun; not vice versa.


What we automatically imagine, is the view "http://video.google.com/videoplay?docid=8842256077873416888#" " and as larger and larger regions come into our view, the space becomes more and more Minkowski. Though perhaps there are regions right around gravitational bodies where the space is stretched, from the distance, you see the birds-eye view, where hills and valleys can be ignored, and placed into a large Euclidian space with a "global time" (meaning--everyone within the system can reference a single hypothetical gravity-immune observer's positon and time mapping, for every event in the system.)

 

[PeterDonis]

What we automatically imagine, is the view "http://video.google.com/videoplay?docid=8842256077873416888#" " and as larger and larger regions come into our view, the space becomes more and more Minkowski. Though perhaps there are regions right around gravitational bodies where the space is stretched, from the distance, you see the birds-eye view, where hills and valleys can be ignored, and placed into a large Euclidian space with a "global time" (meaning--everyone within the system can reference a single hypothetical gravity-immune observer's positon and time mapping, for every event in the system.)

Thanks, this does make it a lot clearer. Something like this is done today with what is called the "Earth-centered inertial" (ECI) frame, which is constructed as the local inertial frame of someone moving along the geodesic worldline of the Earth's center of mass, but with (I believe this is correct) the time coordinate scaled so that it matches the proper time rate at the Earth's surface (on the "geoid", which is the equipotential surface "at sea level", more or less). If you took this frame and rescaled the time so that it matched the proper time rate of an observer at Earth's position in the Sun's gravity well, but far away from the Earth (for example, orbiting the Sun in the Earth's orbit but on the opposite side of the Earth), you would have something closer to what you are talking about, at least with reference to the Earth. Then, as you say, you could continue to "correct" the time coordinate by rescaling to match the proper time rate far outside the Sun's gravity well, the galaxy's gravity well, the gravity well of the Local Group of galaxies, etc.

There's a catch to this, though: the Universe is expanding. Suppose we make all these corrections and arrive at a local inertial frame which tracks the proper time of an observer at Earth's spatial location "now" who is moving exactly with the cosmological "Hubble flow", i.e., this observer's only "motion" is due to the expansion of the Universe. The way to distinguish such an observer, experimentally, is that only this observer (and the family of observers like him, at different spatial locations, e.g. at some point in the Andromeda galaxy) sees the Universe as isotropic--it looks the same in all directions. The most sensitive measure of this that we know of currently is the CMBR, which does *not* look isotropic to us because the Earth is not "at rest" with respect to the expansion of the Universe as a whole; but we can certainly construct what the local inertial frame of an observer that was just passing Earth now but that *did* see the CMBR as isotropic would look like.

So we have what I'll call the "Earth-centered cosmological" (ECC) reference frame, which is the local inertial frame of this observer just passing Earth now that sees the CMBR as isotropic, so his only "motion" is due to the expansion of the Universe. Now consider a similar observer a million parsecs from Earth (distance "now" as measured in the ECC frame). The current estimate of the Hubble constant is about 70 km/sec per million parsecs, so this second observer, in the ECC frame, would appear to be moving away from Earth at 70 km/sec. Pick the event on the second observer's worldline that has the time coordinate "now" according to the ECC frame, and call that event E; the origin of the ECC frame will be the event of Earth "now", which we'll call event O.

Now consider the following: the time since the Big Bang at event O, according to the ECC frame, is 13.7 billion years. How much proper time has the second observer experienced since the Big Bang at event E? The problem is that the "obvious" answer in the ECC frame is *wrong*. The obvious answer is that, since the second observer is moving in the ECC frame, time dilation will make his proper time since the Big Bang at event E *less* than the Earth's proper time since the Big Bang at event O. However, the actual general relativistic cosmological models that best match the data say that the second observer's proper time since the Big Bang at event E will be the *same* as the Earth's proper time since the Big Bang at event O.

The resolution of this "paradox" is, of course, that the expansion of the Universe means spacetime, as a whole, is not flat, even though each spatial slice (hypersurface of constant "cosmological time") in the spacetime of the Universe as a whole is flat (i.e., Euclidean), at least according to our current best-fit model. Because spacetime is not flat, the local inertial frames at different events on the "ECC" worldline (e.g., the observer "at rest" at the Earth's position in the Universe as a whole now, vs. the observer "at rest" at the Earth's position in the Universe as a whole a billion years ago) do not "line up" with each other; the frame of the observer in the past will appear to be moving, relative to the frame of the observer now, just as the frame of an observer at a different spatial position now will appear to be moving. So there is no such thing as a "global Minkowski" frame for the Universe that works everywhere and at all times; the best you can do is to set up one that works "locally" around a particular event of interest (such as the Earth now), and since the constant-time slice of the Universe is Euclidean, you *can* in principle use this frame to cover the entire Universe (or at least everything we can see), as long as you remember that you can't draw correct deductions over extended time ranges using this frame (e.g.., how much proper time since the Big Bang for an observer a million parsecs away). (And if it turns out that we have to change our best-fit model with future data, such that the constant-time slices of the Universe turn out not to be Euclidean, then a frame like the "ECC" frame would be limited in the spatial range it could cover, as well as the time range.)

 

[JDoolin]

What we automatically imagine, is the view "http://video.google.com/videoplay?docid=8842256077873416888#" " and as larger and larger regions come into our view, the space becomes more and more Minkowski. Though perhaps there are regions right around gravitational bodies where the space is stretched, from the distance, you see the birds-eye view, where hills and valleys can be ignored, and placed into a large Euclidian space with a "global time" (meaning--everyone within the system can reference a single hypothetical gravity-immune observer's positon and time mapping, for every event in the system.)

I wanted to cover this analogy a little further before moving on. We imagine three observers, a man standing on the top of a hill, overlooking a wide landscape, a plane flying at about 4 or 5 miles, and an observer on the moon.

In the man's point-of-view, he looks at the hills and valleys around him and sees curvature he is going to have to deal with, because he is walking. On the moon, the observer sees a round earth.

But in between, you have an observer who looks down and basically sees a flat surface until it disappears into the clouds in the distance.

So there is a scope where small-scale curvature is evident, and a scope where large scale curvature is evident, and in between, a scope where, though both can be detected, neither curvature is really obvious.

You can watch thehttp://video.google.com/videoplay?docid=8842256077873416888#" , and estimate that the small-scale curvature effects would probably disappear if you were looking at a patch of ground about 10^3.5 meters across (about 3 km). The large scale curvature begins to become obvious between 10^6.5 meters (about 3000 km). At scales between 10^3.5 and 10^6.5 meters, the Earth basically looks flat.

Is this analogy applicable to General Relativity? We have a curvature at the small scale (on the scale of stars and planets) that curve space in their region. But if we pull back, we can take a photo from the distance, at the scale of a solar system, or a galaxy, or perhaps even a galaxy cluster, and treat it as flat space.

Then we move yet further out, to where we can observe 1 million parsecs, (3.26 million light years) and this curvature amounts to something on the order of 70 km/second

http://upload.wikimedia.org/wikipedia/commons/2/2f/Local_Group.JPG
The image shows a region about 1 megaparsec in radius; 2 megaparsecs across​

First of all, to notice motion of 70 km/second on this scale would take thousands of years. (Using doppler effect it's not so difficult, but I'm talking about side-to-side motion.) At 70 km/second it takes ~4300 years to go just one light-year. But we're looking at a map on the scale of millions of light years.

Furthermore, 70 km/second per megaparsec is just 0.023% of the speed of light.

I don't know exactly how one calculates the relation between hubble's constants and curvature in General Relativity. However, if I can relate this velocity to Special Relativity, you could go out an awful lot of megaparsecs before this curvature becomes noticeable. Special Relativistic effects such as time-dilation become just barely noticeable at around 10% of the speed of light. That would be around 400 million megaparsecs or 1.4 billion light years.

So where the span over which the Earth appears flat is from about 3 kilometers* to 3000 kilometers, the span over which the universe looks flat is between say 1 Astronomical unit* to 1.4 billion light-years. So the difference between the local curvature and the global curvature based on Earth geological features is a factor around a thousand, while the difference between the local curvature and the global curvature based on general relativity features is a factor of around a trillion.

*There are some enormous features of the earth, such as the grand canyon and the large mountain ranges which will not look flat on the scale of 3 kilometers. There are also enormous features of the universe, such as black holes where you'd have to back away further than 1 AU to have them look flat.

 

[JDoolin]

Now consider the following: the time since the Big Bang at event O, according to the ECC frame, is 13.7 billion years. How much proper time has the second observer experienced since the Big Bang at event E? The problem is that the "obvious" answer in the ECC frame is *wrong*. The obvious answer is that, since the second observer is moving in the ECC frame, time dilation will make his proper time since the Big Bang at event E *less* than the Earth's proper time since the Big Bang at event O. However, the actual general relativistic cosmological models that best match the data say that the second observer's proper time since the Big Bang at event E will be the *same* as the Earth's proper time since the Big Bang at event O.

I can't vouch for the data, but let's see whether we agree about what the "obvious" answer is. I'll give you my "obvious" answer, then you can see if you agree.

Let's take a galaxy, for instance, at 400 megaparsecs (approximately 1.3 billion light years away). According to Hubble's Law, this galay should be moving away from us at 70 X 400 = 28,000 km/second=.093c. So the time-dilation factor at 1.3 billion light years away would be:

\frac{1}{\sqrt{1-(v/c)^2}}=1.004​

So where our galaxy aged 13.7 billion years, the distant galaxy would age 13.65 billion years. Now we also need to account for the travel-time of the light, so we can subtract about 1.3 billion years from this figure. The actual age of the galaxy we see should be about 12.35 billion years old.

Take notice that at 1.3 billion light years, the effect of 70 km/s per megaparsec is quite small. The effect of the delay due to the speed of light is huge by comparison.

I've shown that the "obvious" answer (at 1.3 billion light years distance) is that the second observer ages 13.65 billion years, while the Earth ages 13.7 billion years. 13.65 billion and 13.7 billion are very close. This means that any observer within 1.3 billion light years should have approximately the *same* amount of proper time since the Big Bang event.

So now I need some clarification on what you've said above. "the second observer's proper time since the Big Bang at event E will be the *same* as the Earth's proper time since the Big Bang at event O" When you say this, are you already taking into account the speed of light delay? For instance, the image of a galaxy 1.3 billion light years away should not look 13.7 billion years old, but should look 12.4 billion years old, because the light originated from an event 1.3 billion years ago.

 

[PeterDonis]

I've shown that the "obvious" answer (at 1.3 billion light years distance) is that the second observer ages 13.65 billion years, while the Earth ages 13.7 billion years. 13.65 billion and 13.7 billion are very close. This means that any observer within 1.3 billion light years should have approximately the *same* amount of proper time since the Big Bang event.

According to the "naive" calculation in the ECC frame, yes. But the difference will get larger as you go out to larger distances in the ECC frame. In the "cosmological" frame, the frame in which the FRW metric is written, *all* observers at the same cosmological time t have experienced *exactly* the same proper time since the Big Bang. See next comment.

So now I need some clarification on what you've said above. "the second observer's proper time since the Big Bang at event E will be the *same* as the Earth's proper time since the Big Bang at event O" When you say this, are you already taking into account the speed of light delay? For instance, the image of a galaxy 1.3 billion light years away should not look 13.7 billion years old, but should look 12.4 billion years old, because the light originated from an event 1.3 billion years ago.

Remember that I said event E and event O are, by definition (or perhaps by construction) *simultaneous* in the ECC frame. That means event E is the event on the second observer's worldline that would be assigned the same time coordinate in the ECC frame as event O; in other words, event E is the event at which the Earth observer's "surface of simultaneity" at event O intersects the second observer's worldline. So the "speed of light delay" is already taken into account (at least, I think that's how you're using the term).

However, the way the ECC frame is constructed, its "surface of simultaneity" at event O exactly matches the (Euclidean) surface of constant cosmological time t = 13.7 billion years in the FRW frame, the (curved) spacetime metric in which cosmology is normally done. So according to the FRW frame, the second observer's proper time since the big bang at event E will be *exactly* 13.7 billion years, *not* 13.65 billion years.

 

[JDoolin]

So we have what I'll call the "Earth-centered cosmological" (ECC) reference frame, which is the local inertial frame of this observer just passing Earth now that sees the CMBR as isotropic, so his only "motion" is due to the expansion of the Universe. Now consider a similar observer a million parsecs from Earth (distance "now" as measured in the ECC frame). The current estimate of the Hubble constant is about 70 km/sec per million parsecs, so this second observer, in the ECC frame, would appear to be moving away from Earth at 70 km/sec. Pick the event on the second observer's worldline that has the time coordinate "now" according to the ECC frame, and call that event E; the origin of the ECC frame will be the event of Earth "now", which we'll call event O.

Now consider the following: the time since the Big Bang at event O, according to the ECC frame, is 13.7 billion years. How much proper time has the second observer experienced since the Big Bang at event E? The problem is that the "obvious" answer in the ECC frame is *wrong*. The obvious answer is that, since the second observer is moving in the ECC frame, time dilation will make his proper time since the Big Bang at event E *less* than the Earth's proper time since the Big Bang at event O. However, the actual general relativistic cosmological models that best match the data say that the second observer's proper time since the Big Bang at event E will be the *same* as the Earth's proper time since the Big Bang at event O.

You've double-defined event O. Saying that both the big bang is "O" and Earth now is "O." I want to define four events. You've defined "E" as the "the event on the second observer's worldline that has the time coordinate "now" according to the ECC frame" That one is good. The other events I'm defining are:

O: The Big Bang.
N: The event of here and now. Earth Circa 2010, 13.7 billion years after O.
P: The event that we are currently seeing, looking at the galaxy that is moving along a path from O to E.

Now, in my previous post, I only considered what to expect out to about 1.3 billion light years. And I only gave lip-service to the speed-of-light delay. In this post, I want to make "The obvious answer" a little more complete. Specifically, I think the "obvious answer" out to 1.3 billion light years is pretty close to what we see. It is out beyond 6 billion light years where the "obvious answer" is clearly wrong.

Here is the problem with the "obvious answer". If you are observing galaxies out beyond 7 billion light years (which we do), (Event P at 7 billion light years--Event E at 14 billion light years, it means that the velocity of those stars is greater than the speed of light. If this "obvious answer" were correct, we should not see any stars beyond 7 billion light years distance.

However, we see here:

http://www.astro.ucla.edu/~wright/sne_cosmology.html

We have supernovae all the way out to 12 giga-parsecs (39 billion light years.) That means the "obvious" answer cannot be right. It should be noticed though, that this is under the assumption that the hubble-constant is a global parameter. A careful analysis of the graph indicates that hubble's constant is 70 km/s per megaparsec in a region nearby, out to about 6 billion light years, but seems to be much smaller as you go out beyond that region.

 

[Mentz114]

The big bang didn't happen 'somewhere'. It was 'everywhere'. So you're free to choose any point as the 'center', and choosing the point you're at is most convenient. ( sorry about all the 'quotes', but it's a subtle distinction).

 

[JDoolin]

Remember that I said event E and event O are, by definition (or perhaps by construction) *simultaneous* in the ECC frame. That means event E is the event on the second observer's worldline that would be assigned the same time coordinate in the ECC frame as event O; in other words, event E is the event at which the Earth observer's "surface of simultaneity" at event O intersects the second observer's worldline. So the "speed of light delay" is already taken into account (at least, I think that's how you're using the term).

I posted, and then saw you had posted. I'm still not quite sure how the event E takes into account the speed-of-light delay. Hopefully the diagram I drew, which includes point P will show what I mean by taking into account the speed of light delay.

The event E which intersects the Earth observer's "surface of simultaneity" will not be seen on Earth for a long, long time. However, the event P which intersects the observer's "past light cone" is what is seen on Earth NOW.

 

[JDoolin]

The big bang didn't happen 'somewhere'. It was 'everywhere'. So you're free to choose any point as the 'center', and choosing point you're at is most convenient. ( sorry about all the 'quotes', but it's a subtle distinction).

PeterDonis brought up the idea of an ECC, Earth Centered Cosmological frame, in which the anisotropy of the CMBR disappeared. It appears that if we adjusted our velocity by about 600 km/second, in the appropriate direction, then we would be able to make this anisotropy disappear.

I would agre that once we did this, then a "point" in our space-ship is an actual center of the universe. Since we're free to set off that space-ship at any time we want, and any place we want, we can choose any event to be the center of the universe. But if we choose a point, we have to choose a specific velocity as well. A "point" in space, is a worldline in space-time.

Now, the interpretation of General Relativistic Cosmology is that all of these cosmologically centered worldlines are parallel (or at least they don't cross). The divergence of those cosmologically centered worldlines is not due to real motion of the particles. It is due, instead to an increasing scale factor. In the past, that scale factor goes right down to zero. However, the unscaled distance remains more-or-less the same throughout time.

This is how I have come to understand the idea, though I realize that perhaps my understanding is not complete.

But how can I reconcile this with my idea that the big bang happened "somewhere." Honestly, I don't think I can. If you trace back a set of straight lines to where they intersect, you get a "somewhere" But if you trace back a set of lines that curve, and do not cross, then you get an "everywhere". My diagram in post 37 assumes that the two lines do cross somewhere in the past. Is there any chance that we could even use this as an approximation? That we could maybe just "pretend" that two galaxies traveling at velocity away from each other were once, far in the past at the same place, at the same time? I guess we wouldn't have to call it the big bang; just the meeting event. And maybe they never met, but just assume that the majority of the time they were moving away from each other, they acted as though they met.

 

[Mentz114]

I think you are on the right track. If the big-bang started at a singularity, then all the constituents that make up the universe were once in same place ( the only place) at t=0.

I don't have time to reply to this at length, but this document says it much better, and has loads of good diagrams.

http://ls.poly.edu/~jbain/philrel/philrellectures/12.RelativisticCosmology.pdf

 

[JDoolin]

I think you are on the right track. If the big-bang started at a singularity, then all the constituents that make up the universe were once in same place ( the only place) at t=0.

I don't have time to reply to this at length, but this document says it much better, and has loads of good diagrams.

http://ls.poly.edu/~jbain/philrel/philrellectures/12.RelativisticCosmology.pdf

The attached diagrams came from the referenced article. They both are graphs of space vs. time. The Robertson Walker diagram has elements that go to infinite speed. The "conformal" version has all elements stationary. Are these two diagrams supposed to be different diagrams of the same thing, or are they different geometries in the same coordinates?

Is the time parameter in the Robertson Walker diagram supposed to be proper time of particles, or does it represent coordinate time for a hypothetical observer?

 

[Mentz114]

The attached diagrams came from the referenced article. They both are graphs of space vs. time. The Robertson Walker diagram has elements that go to infinite speed. The "conformal" version has all elements stationary. Are these two diagrams supposed to be different diagrams of the same thing, or are they different geometries in the same coordinates?

Same geometry, different coords.

Is the time parameter in the Robertson Walker diagram supposed to be proper time of particles, or does it represent coordinate time for a hypothetical observer?

Those diagrams illustrate the horizon that exists in an expanding FRW universe. The first diagram uses scaled coordinates, so everything starts in the same place and remains the same distance apart. The straight red lines represent horizons. As time progresses, each worldline loses touch with the others as they move outside its future light cones.

In the second diagram, comoving coordinates are used, so we can pick any worldline as our reference frame. In these coords all the other matter appears to be receding. The horizon lines now become the red curves. I think the point is that the physics is the same whichever of these coords we use.

I haven't got time to do it right now but I'll get the actual transformations and post them.

Is the time parameter in the Robertson Walker diagram supposed to be proper time of particles, or does it represent coordinate time for a hypothetical observer?

In both diagrams, the vertical axis is coordinate time.

 

[PeterDonis]

Is the time parameter in the Robertson Walker diagram supposed to be proper time of particles, or does it represent coordinate time for a hypothetical observer?

I'm not sure what "coordinate time for a hypothetical observer" means. The Robertson-Walker coordinates are constructed in such a way that coordinate time directly represents the proper time of observers who are "at rest" in those coordinates--in other words, observers whose values of the spatial coordinates remain constant (and who see the universe as isotropic for all time). As you noted, the actual proper distance between any two such observers increases with time because of the increase of the scale factor; this is a more precise way of saying that the universe is "expanding".

The "conformal" coordinates in the other diagram can be constructed from the Robertson-Walker coordinates in two steps, which I'll describe here in highly "heuristic" fashion: (1) take the initial singularity, which is a point (a single event), and "stretch it out" into a line; (2) take the Robertson-Walker time coordinate and "stretch" it by an amount that increases as the singularity is approached, so that after the stretching is done, all light rays travel on 45 degree lines. This makes it easy to see the causal structure of the spacetime.

 

[PeterDonis]

You've double-defined event O. Saying that both the big bang is "O" and Earth now is "O."

I didn't mean to double-define O; I meant O to always refer to the Earth now. Sorry if I wasn't clear.

Here is the problem with the "obvious answer". If you are observing galaxies out beyond 7 billion light years (which we do), (Event P at 7 billion light years--Event E at 14 billion light years, it means that the velocity of those stars is greater than the speed of light. If this "obvious answer" were correct, we should not see any stars beyond 7 billion light years distance.

However, we see here:

http://www.astro.ucla.edu/~wright/sne_cosmology.html

We have supernovae all the way out to 12 giga-parsecs (39 billion light years.) That means the "obvious" answer cannot be right. It should be noticed though, that this is under the assumption that the hubble-constant is a global parameter. A careful analysis of the graph indicates that hubble's constant is 70 km/s per megaparsec in a region nearby, out to about 6 billion light years, but seems to be much smaller as you go out beyond that region.

This is what I was referring to as the curvature of the FRW spacetime: as you go back in time, the local inertial frames on a given worldline (say, that of the cosmological observer who is at Earth's spatial location now) don't "line up" with each other. The changing slope of the graph is evidence of that.

 

[PeterDonis]

Hopefully the diagram I drew, which includes point P will show what I mean by taking into account the speed of light delay.

Yes, it's clear, and it matches what I understood you to be saying. Event E is spacelike separated from the event you've labeled N, but that doesn't prevent us from calculating the proper time between the big bang (which you've labeled O) and event E, along the "straight" worldline connecting them. As we've seen, that calculation gives an answer which can't be right.

 

[JDoolin]

In the conformal map (see post 42, above), the light rays are always parallel to the light cones. In the Robertson Walker coordinates, the light rays don't follow the light cones. Is there a mistake in one of the diagrams?

The light rays come to a full stop in the Robertson Walker diagram, stopping and turning around and coming the opposite direction. Is that expected?

(Now I see, Mentz post #43, the lines are not light rays but horizon lines? I'm not quite clear on what that means.)

 

[PeterDonis]

In the conformal map, the light rays are always parallel to the light cones. In the Robertson Walker coordinates, the light rays don't follow the light cones. Is there a mistake in one of the diagrams?

The light rays come to a full stop in the Robertson Walker diagram, stopping and turning around and coming the opposite direction. Is that expected?

Yes, it's expected, and no, there's no mistake in the diagrams. The light rays actually do follow the light cones in the Robertson Walker coordinates, but it's hard to see because the bottom of the diagram is crunched together (that's one of the reasons conformal coordinates are useful). The two opposite "sides" of the past light cone actually originate from the *same* event, the initial singularity (since everything originates there), but they "come out" in different directions that aren't quite opposite--there's a little bit of "tilt" in each ray towards the other one. As the expansion of the universe decelerates, and the light cones tilt inward (until they're vertical at "now"), the two light rays are bent back together to meet at "here and now".

Part 3 of Ned Wright's cosmology tutorial at http://www.astro.ucla.edu/~wright/cosmo_03.htm gives more details about these diagrams.

 

[JDoolin]

Yes, it's clear, and it matches what I understood you to be saying. Event E is spacelike separated from the event you've labeled N, but that doesn't prevent us from calculating the proper time between the big bang (which you've labeled O) and event E, along the "straight" worldline connecting them. As we've seen, that calculation gives an answer which can't be right.

So does it seem to you that if we go out to about one or two billion light years that we have what appears to fit a Minkowski approximation, and then only once you get out past 6 billion light years does it seem that the space is noticeably stretching? Because if you agree with me that this is essentially the case, then I can begin to discuss the invisible elephant in the room, which is acceleration.

The "obvious" calculation we're doing makes the assumption of a uniform, peaceful expansion at the beginning of the universe. This is highly unlikely, because one should expect nuclear explosions, or matter-anti-matter explosions, of great force (or possibly even greater unknown particle-interactions) in the instants immediately after the big bang.

I don't know how to treat these interactions in General Relativity, but I do have some idea how to treat them in Minkowski spacetime. You've seen that the calculation without acceleration cannot be right. Would you humor me long enough to see that the calculation with acceleration actually could be right?

 

[PeterDonis]

So does it seem to you that if we go out to about one or two billion light years that we have what appears to fit a Minkowski approximation, and then only once you get out past 6 billion light years does it seem that the space is noticeably stretching?

I don't know that I disagree, but I don't know that I would put it this way either. The "stretching" doesn't happen in space; it happens in time, as the scale factor changes. Obviously you can kind of convert time to space because looking farther away means looking at things the way they were a longer time ago; but I'm not sure I would characterize what we see as we look out that far as "stretching". Maybe my further comments below will help to clarify what I'm getting at.

Because if you agree with me that this is essentially the case, then I can begin to discuss the invisible elephant in the room, which is acceleration.

We have to be careful with terminology here as well, because the cosmological observers we've been discussing (the ones at rest in the Robertson-Walker coordinates) are not accelerating; that is, they don't feel any acceleration. And again, I'm not sure that the fact that the local inertial frames of cosmological observers at different times don't "line up" is most usefully viewed as an "acceleration", although that term is often used (as in, the expansion of the universe is now accelerating, but was decelerating earlier in its history).

The "obvious" calculation we're doing makes the assumption of a uniform, peaceful expansion at the beginning of the universe. This is highly unlikely, because one should expect nuclear explosions, or matter-anti-matter explosions, of great force (or possibly even greater unknown particle-interactions) in the instants immediately after the big bang.

I don't know how to treat these interactions in General Relativity, but I do have some idea how to treat them in Minkowski spacetime. You've seen that the calculation without acceleration cannot be right. Would you humor me long enough to see that the calculation with acceleration actually could be right?

The calculations we've been doing are kinematic; they don't get into the detailed dynamics of what's going on with the matter-energy in the universe. The Robertson-Walker models abstract all that out by treating the matter-energy in the universe as a perfect fluid, which can take one of three simple forms characterized by different equations of state relating the pressure, p, to the energy density, rho:

(1) "Matter dominated": a fluid with zero pressure, p = 0. This is a good approximation at the cosmological level for non-relativistic matter (meaning, the average speed of the individual "particles" in the fluid, which are basically galaxies or galaxy clusters, is much less than the speed of light).

(2) "Radiation dominated": a fluid with equation of state p = 1/3 rho. This is the equation of state for a "fluid" made of pure radiation (for example, the CMBR).

(3) "Vacuum dominated": a fluid with equation of state p = - rho. This is the equation of state for a "fluid" which is due to a cosmological constant (other terms used are "vacuum energy" or "dark energy").

The current best fit model for the evolution of the universe is: first the "inflation" phase, in which the equation of state was vacuum dominated with an extremely large effective energy density rho, meaning that the universe "expanded" exponentially; then, after the phase transition that ended inflation, a radiation dominated phase, which lasted roughly until the time of "recombination" (electrons and nuclei combining into atoms, which made the universe basically transparent to photons) when the universe was about 100,000 years old (all these times are very approximate, I'm going from memory here); then a matter dominated phase, which lasted until a few billion years ago (I believe); and finally, another vacuum dominated phase but with a very, very small effective energy density, causing the expansion of the universe to start "accelerating" again (it had been decelerating during the radiation and matter dominated phases).

The reason I go into all this is to illustrate that nowhere in any of this did I have to specify what, exactly, was going on *within* the cosmological fluid, in terms of nuclear reactions, explosions, whatever. The only thing that matters for the overall dynamics of the universe is the equation of state, and the effective equation of state of the cosmological fluid, on an overall level, can remain the same while the fluid undergoes violent internal changes. (Part of how this can work is that the identity of the individual "particles" that compose the fluid changes over time: in the early universe, they were elementary particles like quarks, electrons, and photons; then they were atoms of hydrogen, helium, and a few other elements; and for the past few billion years, at least, they've been galaxies and galaxy clusters. But on a gross cosmological level, all of these different "fluids" can be described by the same simple equations of state I gave above.)

So as far as the overall dynamics of the universe is concerned, we can get away with *not* modeling all the details you mention. (We do have to model them, at least to some extent, in order to predict finer details like the ratios of abundances of different elements that we should expect to see in intergalactic space.) Another way of saying this is that, in fact, the Robertson-Walker models do *not* make the assumption of a "peaceful" expansion of the early universe; they only make the assumption that, whatever might be going on at a detailed level, it can be adequately modeled at an overall level by a fluid with one of the above equations of state. That assumption appears to be working pretty well so far.

I'm not sure if you want to get into the actual derivation of the Robertson-Walker metric; the Wikipedia page, http://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric, has a decent (if brief) discussion. Also, technically, these models do not go all the way back to the initial singularity (the word "singularity" means the equations break down and the model can't make predictions). Currently the actual models, as I understand it, more or less assume that the inflation phase started with a state the size of the Planck length or thereabouts (not a zero-size initial singularity), which began expanding exponentially; how that state came to be is not known, though there are various proposals with no real way of testing any of them experimentally at this time.

 

[JDoolin]

Yes, it's expected, and no, there's no mistake in the diagrams. The light rays actually do follow the light cones in the Robertson Walker coordinates, but it's hard to see because the bottom of the diagram is crunched together (that's one of the reasons conformal coordinates are useful). The two opposite "sides" of the past light cone actually originate from the *same* event, the initial singularity (since everything originates there), but they "come out" in different directions that aren't quite opposite--there's a little bit of "tilt" in each ray towards the other one. As the expansion of the universe decelerates, and the light cones tilt inward (until they're vertical at "now"), the two light rays are bent back together to meet at "here and now".

Part 3 of Ned Wright's cosmology tutorial at http://www.astro.ucla.edu/~wright/cosmo_03.htm gives more details about these diagrams.

Mea Culpa. I did not notice that on the lower triangles both the left side and the right side are going in the same direction.

Thank you for the link to Ned Wright's page, but that leads me to further questions. Earlier, I was saying that I did not know how to handle acceleration in general relativiy. Ned Wright handles a change-of-reference frames in the GR by doing a Galilean Transformation (attached)

http://www.astro.ucla.edu/~wright/cosmo_03.htm]Note[/PLAIN] that this is not a Lorentz transformation, and that these coordinates are not the special relativistic coordinates for which a Lorentz transformation applies. The Galilean transformation which could be done by skewing cards in this way required that the edge of the deck remain straight, and in any case the Lorentz transformation can not be done on cards in this way because there is no absolute time. But in cosmological models we do have cosmic time, which is the proper time since the Big Bang measured by comoving observers, and it can be used to set up a deck of cards. The presence of gravity in this model leads to a curved spacetime that can not be plotted on a flat space-time diagram without distortion. If every coordinate system is a distorted representation of the Universe, we may as well use a convenient coordinate system and just keep track of the distortion by following the lightcones.

I'm not sure what he's saying, but he seems to be saying that since Lorentz Transformation does not apply to these coordinates, that Galilean Transformation actually DOES apply. Also, this suggests that the vertical component of the graph is NOT the coordinate time of the central observer, but rather it is the "cosmological time" which is the "proper time since the Big Bang measured by comoving observers."

So effectively, you're doing a velocity transformation, but it is unusable as acceleration, because the two reference frames are "comoving."

So my question is, if you want to handle a real acceleration; how is this handled? I presume it is not done by either galilean transformation or by lorentz transformation.

We have to be careful with terminology here as well, because the cosmological observers we've been discussing (the ones at rest in the Robertson-Walker coordinates) are not accelerating; that is, they don't feel any acceleration. And again, I'm not sure that the fact that the local inertial frames of cosmological observers at different times don't "line up" is most usefully viewed as an "acceleration", although that term is often used (as in, the expansion of the universe is now accelerating, but was decelerating earlier in its history).

I know RW coordinates are not accelerating. The question is whether they can handle acceleration. And I'm not talking about gravitational acceleration. I'm talking about acceleration due to the pressure caused by primordial particles decaying, creating pressure, reaching critical mass, exploding, etc. One can handle these accelerations in Minkowski space-time rather easily. But in R-W coordinates, it appears that Lorentz Transformation will not do the job. So will Galilean Transformation do it?

 

[JDoolin]

I don't know that I disagree, but I don't know that I would put it this way either. The "stretching" doesn't happen in space; it happens in time, as the scale factor changes. Obviously you can kind of convert time to space because looking farther away means looking at things the way they were a longer time ago; but I'm not sure I would characterize what we see as we look out that far as "stretching". Maybe my further comments below will help to clarify what I'm getting at.



We have to be careful with terminology here as well, because the cosmological observers we've been discussing (the ones at rest in the Robertson-Walker coordinates) are not accelerating; that is, they don't feel any acceleration. And again, I'm not sure that the fact that the local inertial frames of cosmological observers at different times don't "line up" is most usefully viewed as an "acceleration", although that term is often used (as in, the expansion of the universe is now accelerating, but was decelerating earlier in its history).

Actually, I'm not quite sure what you mean by observer's lining up.

The calculations we've been doing are kinematic; they don't get into the detailed dynamics of what's going on with the matter-energy in the universe. The Robertson-Walker models abstract all that out by treating the matter-energy in the universe as a perfect fluid, which can take one of three simple forms characterized by different equations of state relating the pressure, p, to the energy density, rho:

(1) "Matter dominated": a fluid with zero pressure, p = 0. This is a good approximation at the cosmological level for non-relativistic matter (meaning, the average speed of the individual "particles" in the fluid, which are basically galaxies or galaxy clusters, is much less than the speed of light).

(2) "Radiation dominated": a fluid with equation of state p = 1/3 rho. This is the equation of state for a "fluid" made of pure radiation (for example, the CMBR).

(3) "Vacuum dominated": a fluid with equation of state p = - rho. This is the equation of state for a "fluid" which is due to a cosmological constant (other terms used are "vacuum energy" or "dark energy").

I don't understand the idea of a pressure less than zero. Also, what is rho?

The current best fit model for the evolution of the universe is: first the "inflation" phase, in which the equation of state was vacuum dominated with an extremely large effective energy density rho, meaning that the universe "expanded" exponentially; then, after the phase transition that ended inflation, a radiation dominated phase, which lasted roughly until the time of "recombination" (electrons and nuclei combining into atoms, which made the universe basically transparent to photons) when the universe was about 100,000 years old (all these times are very approximate, I'm going from memory here); then a matter dominated phase, which lasted until a few billion years ago (I believe); and finally, another vacuum dominated phase but with a very, very small effective energy density, causing the expansion of the universe to start "accelerating" again (it had been decelerating during the radiation and matter dominated phases).

To me, it seems like there should be a radioactive decay dominated era. You have a matter-dominated era, where the particles are not, on average, old enough to have decayed. Then particle start decaying, and the pressure builds, and you have cascading reactions. This would have, on average, a pressure far beyond the pressure in the radiation dominated era. Also, the net effect of these interactions would be to mash the matter in the universe together, providing the seeds for stars and galaxies.

In the attachment, I took the liberty of modifying the RW diagram to show worldlines of particles flying out of a "secondary bang" imagining some region where a large number of particles reached critical density at the same time, producing an enormous reaction.

If we don't have a good explanation already for our 600 km/s peculiar velocity with the CMBR, this could explain it. Also, this would explain why nearby, we have a hubble's constant of 70 km/s per megaparsec, and further out we have a much smaller Hubble constant.

 

[PeterDonis]

I know RW coordinates are not accelerating. The question is whether they can handle acceleration. And I'm not talking about gravitational acceleration. I'm talking about acceleration due to the pressure caused by primordial particles decaying, creating pressure, reaching critical mass, exploding, etc. One can handle these accelerations in Minkowski space-time rather easily. But in R-W coordinates, it appears that neither Lorentz Transformation will not do the job. So will Galilean Transformation do it?

Wright calls the transformation he uses a "Galilean transformation" because, loosely speaking, it doesn't "change the time direction" the way a Lorentz transformation does; it leaves the surfaces of constant cosmological time invariant, and just transforms the space coordinates within each surface. In General Relativity, this kind of transformation is perfectly legitimate, as long as you transform *everything* accordingly. It just so happens that this particular transformation, in addition to leaving the surfaces of constant cosmological time invariant, also preserves the isotropy of space--the fact that everything looks the same in all directions. So it actually leaves the entire Robertson-Walker metric invariant; the metric looks exactly the same in the transformed coordinates as it does in the original coordinates.

This means that, for example, the cosmological observers--the ones "at rest" in the Robertson-Walker coordinates--are still not accelerating in the transformed coordinates; all that's changed is which particular set of constant space coordinates each observer is at. So I think the simple answer to your question is that Wright's "Galilean transformation" is not intended to "handle accelerations".

However, as I said in my previous post, I don't think a model of the universe as a whole is meant to handle accelerations in the sense you mean, because it simply doesn't model things to that level of detail. See my next post responding to your next post for more on this, since it fits in better in response to what you said in that post.

 

[PeterDonis]

Actually, I'm not quite sure what you mean by observer's lining up.

I mean that the local inertial frames at, say, the event you labeled N (the Earth now), and another event P, a point on the worldline of the cosmological observer that passes through N, but ten billion years to the past of N (according to that observer's proper time) are different inertial frames, the same way that the inertial frames of observers moving at different velocities are different inertial frames. If we extended both local inertial frames far enough in time that they overlapped, their coordinate lines would not line up, just as the coordinate lines of observers moving at different velocities do not line up.

I don't understand the idea of a pressure less than zero. Also, what is rho?

Ned Wright has a page that explains briefly how the negative pressure of the vacuum works here: http://www.astro.ucla.edu/~wright/cosmo_constant.html.

Rho is the energy density of the cosmological fluid, as seen by an observer at rest in the cosmological coordinates. Some people write \rho to mean the mass density, so the energy density would then be \rho c^{2}; I like to work in units where c = 1, so energy density and mass density are the same thing.

To me, it seems like there should be a radioactive decay dominated era. You have a matter-dominated era, where the particles are not, on average, old enough to have decayed. Then particle start decaying, and the pressure builds, and you have cascading reactions. This would have, on average, a pressure far beyond the pressure in the radiation dominated era. Also, the net effect of these interactions would be to mash the matter in the universe together, providing the seeds for stars and galaxies.

What you say about pressure here isn't right. The pressure in the radiation dominated equation of state is actually a limiting case of the pressure due to highly relativistic particles, which is what would be produced by reactions such as you describe. Basically, as the particles in the fluid become more relativistic, the pressure builds from zero (the non-relativistic limiting case) to 1/3 rho (the relativistic limiting case).

I believe you're correct, though, that internal reactions in the cosmological fluid would have the effect of producing "clumps" of higher density, which would then contract gravitationally to form stars and galaxies. I don't know offhand how large this effect is believed to be compared with the effect of "primordial density fluctuations", which are basically quantum fluctuations in the initial state at the beginning of the inflation phase, magnified by many orders of magnitude during the inflation.

In the attachment, I took the liberty of modifying the RW diagram to show worldlines of particles flying out of a "secondary bang" imagining some region where a large number of particles reached critical density at the same time, producing an enormous reaction.

Again, I believe this is basically correct, but I don't know how large an effect it would be compared to others. The only caveat I can see is that, because the universe is expanding, there may be limits on how high the density and pressure can get in such a localized "bang" compared to what they were in the actual initial "bang".

If we don't have a good explanation already for our 600 km/s peculiar velocity with the CMBR, this could explain it. Also, this would explain why nearby, we have a hubble's constant of 70 km/s per megaparsec, and further out we have a much smaller Hubble constant.

I agree your suggestion could in principle explain the peculiar velocity of a particular system of matter now (e.g., the solar system). The variation in the Hubble constant, however, is due to the change in the expansion rate of the universe, and is already accounted for in the basic Robertson-Walker models. (Different models predict different specific "curvatures" in the diagram, but they all predict *some* curvature.)

 

[Mentz114]

JDoolin, you should be aware that the FRW model is an large scale approximation whose symmetries are isotropy and homogeneity. On a smaller scale than cosmic we know that clusters of galaxies have peculiar ( ie not Hubble flow) motions and galaxies within clusters also have this kind of motion, superimposed on the flow. PeterDonis makes this point also.

GR allows us to write models of the whole universe and also to work out how this universe would look to different observers, which is where the coordinate transformations come in. A great benefit is that the essential physics and symmetries of the model will not be lost when we do coordinate transforms. You've seen two examples in FRW spacetime. The one that corresponds best to our experience is the comoving frame which sees all other matter as receding with cosmological red-shift.
It's also possible to define a frame where clock rates change with time so everything looks static.

I think my point is - some of your questions, good though might be, are outside the range of what GR can tell us.

On the question of Gallilean vs Lorentz transformations - this depends on what geometry you choose for your 3D hyperslices, when making a transformations. As you can see in the lecture notes, there are Euclidean ( Galliean relativity) or hyperbolic ( SR) as possibilities. No doubt the coords Ned Wright was using had Euclidean spatial hyperslices (Painleve coords, probably).

 

[PeterDonis]

No doubt the coords Ned Wright was using had Euclidean spatial hyperslices (Painleve coords, probably).

No, Painleve coordinates apply to the Schwarzschild spacetime. Wright was using the Robertson-Walker coordinates for the critical density case, where the spatial slices are Euclidean. Wright doesn't specifically write down the metric in the coordinates he's using, but I believe it would look like this:

d\tau^{2} = dt^{2} - a(t) \left[ dr^{2} + r^{2} \left( d\theta^{2} + sin^{2} \theta d\phi^{2} \right) \right]

Every "cosmological observer" has constant spatial coordinates r, theta, phi in this coordinate system; and it should be obvious from the metric above that the time coordinate t directly reflects the proper time experienced by these observers. All the time variation (the "expansion" of the universe), and hence all the curvature of spacetime, is reflected in the dynamics of the scale factor a(t).

The spatial coordinate in Wright's "Galilean transformation" diagram would be r in the metric above; his transformation is basically just a spatial translation to put a different cosmological observer at the origin r = 0.

 

[Mentz114]

No, Painleve coordinates apply to the Schwarzschild spacetime.

Yes, there is a Painleve chart for FRW dust. Do you want me to post it ?

Every "cosmological observer" has constant spatial coordinates r, theta, phi in this coordinate system; and it should be obvious from the metric above that the time coordinate t directly reflects the proper time experienced by these observers.

Phew - how can say that when you've just seen two different coordinate transformations of FRW representing observer frames with completely different proper times ?

 

[PeterDonis]

Yes, there is a Painleve chart for FRW dust. Do you want me to post it ?

Yes, please.

Phew - how can say that when you've just seen two different coordinate transformations of FRW represnting observer frames with completely different proper times ?

Just read it off the metric I wrote: for an observer at rest in the spatial coordinates, dr = dtheta = dphi = 0. So the metric reads d\tau^{2} = dt^{2}. Any transformation that leaves the metric in the same form (which, as I understand it, Wright's "Galilean transformation" does) will preserve that property. But of course, a transformation into some other coordinates where the metric looks different (such as Wright's "special relativistic" coordinates) may not.

 

[Mentz114]

Just read it off the metric I wrote: for an observer at rest in the spatial coordinates, dr = dtheta = dphi = 0. So the metric reads d\tau^{2} = dt^{2}. Any transformation that leaves the metric in the same form (which, as I understand it, Wright's "Galilean transformation" does) will preserve that property. But of course, a transformation into some other coordinates where the metric looks different (such as Wright's "special relativistic" coordinates) may not.

You're right about the raw coordinates, no transformation is needed because it the spatial slice is already conformally flat. I didn't read your post properly. Anyhow, I did put a 'probably' after my Painleve speculation

I'll look up the Painleve chart as soon as my coffee is brewed and consumed.

 

[Mentz114]

FRW in Painleve chart

I had a problem editing my previous post so I made a new one.

<br /> \begin{align*}<br /> dt_p&amp;=dt\\<br /> dx_p&amp;=dx-\frac{2\,x}{3\,t}dt\\<br /> dy_p&amp;=dy-\frac{2\,y}{3\,t}dt\\<br /> dz_p&amp;=dz-\frac{2\,z}{3\,t}dt<br /> \end{align*}<br />

which gives a metric,

<br /> \left[ \begin{array}{cccc}<br /> \frac{4\,{z}^{2}+4\,{y}^{2}+4\,{x}^{2}-9\,{t}^{2}}{9\,{t}^{2}} &amp; -\frac{2\,x}{3\,t} &amp; -\frac{2\,y}{3\,t} &amp; -\frac{2\,z}{3\,t}\\\<br /> -\frac{2\,x}{3\,t} &amp; 1 &amp; 0 &amp; 0\\\<br /> -\frac{2\,y}{3\,t} &amp; 0 &amp; 1 &amp; 0\\\<br /> -\frac{2\,z}{3\,t} &amp; 0 &amp; 0 &amp; 1<br /> \end{array} \right]<br />
The spatial part is static now.

The Einstein tensor transformed by this frame field has only one non-zero component,
G_{00}=\frac{4}{3\,{t}^{2}}
which up to a factor is the SET of static non-interacting dust whose density varies with time. The Riemann scalar is 80/27t^4 which shows there is an unremoveable curvature singularity at t=0.