Intergalactic Space & the Minkowski Diagram

[DiracPool]

Couple of questions as to how to interpret the Minkowski diagram when it comes to intergalactic space.

1) To begin, I'll make some pre-assumptions to the question, which you may correct if I'm wrong:

1a) Within any given galaxy, the distances between it's constituent stars is roughly equal over time due to the gravitational well holding the galaxy together. Thus, for the most part, we can have some sense of an absolute or stable sense of space and distance between objects within a given galaxy (or galaxy cluster).

1b) Between any two given galaxies, the distance between the galaxies and, hence, the distances between the constituent stars in the galaxies, increases over time, and increases at a rate proportional to the distance between the galaxies.

2) My supposition based on these pre-assumptions is that the Minkowski diagram/metric only holds true for the first condition (1a). Why? Say I'm Bob sitting in galaxy A, and I send Alice off through the galaxy by applying some force to her. Well, I'm confident that, in this instance I'm bound by the Minkowski metric and I can't send her off at a velocity that is going to exceed the speed of light. In fact, when I look at the Minkowski diagram, I see that the hyperbolic lines of constant time asymptote at the null, or light cone, lines and there's NO WAY I can send her off faster than the speed of light. I'm trying pretty hard to send her off as fast as I can, and I've sent her away from me at 99.999c, but I just can't seem to get her going any faster. That is, until I reach the end of the galaxy. Once she shoots into intergalactic space, she's now got the "tailwind" of the scale factor to push her over c. So that means that once my beloved Alice is some way into intergalacitc space, I will see her as moving away from me at faster than the speed of light? Where am I wrong here?

3) Relativistic velocity addition--I'm still sitting here in galaxy A, and my girl Alice is now hanging out in galaxy Y pretty close to the cosmic horizon. In fact, the galaxy she's sitting in is traveling away from me at .7c. Just for fun, I tell her to throw a baseball across her galaxy in the same direction she is traveling away from me, at .5c. So what happens now? Can we really believe that relativistic velocity addition here is going to "add up" when we have this scale factor thing to contend with? What is my observation of the speed of the baseball going to be?

4) Considering the above argument, my real question is it seems to me that the Minkowski diagram only applies to "intra" galactic processes. Do we have an equivalent diagram for "inter" galactic processes? For example, the current theory is that there are, in fact, galaxies receding from us at faster than c beyond the cosmic horizon. Considering such, the traditional Minkowski diagram breaks down because, obviously, a (massive) galaxy receding from us at 1.25c is not going to fit into a traditional Minkowski diagram framework.

5) What is the equivalent diagram we deal with here for intergalactic processes? My first guess is that it would a FRW-related thing, but the FRW is not a metric-related equation that is designed to warp the shape of the light cone in the same way that the Schwarzschild solution to GR field equations provide a modification of the flat space metric. Is it? That is, the Schwarzschild solution modifies the flat space metric by changing the unity constant placed before the variables "t" and "x" in the equation (tau)^2 = (1) t^2 - (1) x^2 with (tau)^2 = (function) t^2 - (function) x^2.

I like the Minkowski diagram. I think it tells us a lot about space and time locally, and we like to get all fancy about it because it adds to the SR component of the GPS system and the atomic clock thing on the airplane experiment. But does it work "universally" for interactions between objects in intergalactic space? Or does it break down there in some inexplicable way? Or does it break down in an explicable way where we can use some sort of "modified" Minkowski diagram in order to model these interactions?

 

[PeterDonis]

My supposition based on these pre-assumptions is that the Minkowski diagram/metric only holds true for the first condition (1a).

It doesn't even hold for that case, because of the galaxy's gravity well. Spacetime is curved, not flat; the Minkowski metric describes flat spacetime.

For curved spacetimes, there is no possible "diagram" that has all of the nice properties of the Minkowski diagram for flat spacetime. You have to pick and choose. The various possible "diagrams" are called "coordinate charts", and different charts are suitable for different purposes.

 

[sweet springs]

Even inside our galaxy we observe the cases that Minkowsky space or SR is inapplicable, for example black holes. Minkowsky space is applicable only in small area as we regard the ground flat though the Earth is a globe.

 

[Ibix]

Think of a street atlas of your town. It uses a simple Euclidean system which is strictly only valid at a point, but can be extended to larger regions as long as the errors are small enough. You can get a street atlas covering the UK without too much trouble, but I don't think you can do the same for the US. Ignoring the curvature of the Earth will come back to bite you eventually.

Similarly, a Minkowski diagram is strictly applicable only at an event, but you can extend it to larger regions of space time as long as the errors are small enough. The FRW metric assumes a uniform distribution of matter in the universe, so we're talking on the scale where galactic clusters look like sand grains. Within a sand grain you can use a Minkowski diagram in exactly the same way you can use the street atlas (so a trip to Andromeda is fine, as long as you stay away from black holes). But if you try to stretch too far then the errors become apparent and you need to do something more sophisticated, as Peter says. And, as with maps of the whole Earth, there is no one natural more sophisticated way to do it.

 

[PeterDonis]

Within a sand grain you can use a Minkowski diagram in exactly the same way you can use the street atlas

No, you can't, because within the "sand grain" are individual stars and other gravitating objects, and the "sand grain" itself has a significant gravity well due to the combined effect of all those gravitating bodies, and spacetime is not flat because of that.

 

[Ibix]

As long as you don't get too close to any gravitating body (i.e. stay where the weak field approximation is good enough) a Minkowski diagram should do, shouldn't it? I'd have to do the turnaround on rockets, obviously, no tricks with orbits around things.

 

[DiracPool]

Within a sand grain you can use a Minkowski diagram in exactly the same way you can use the street atlas (so a trip to Andromeda is fine, as long as you stay away from black holes). But if you try to stretch too far then the errors become apparent and you need to do something more sophisticated

Well, that's the question I'm asking. I have no idea, btw, what your follow-up sentence means...

And, as with maps of the whole Earth, there is no one natural more sophisticated way to do it.

What does the GR modified Minkowski diagram look like and what are it's parameters? Specifically, I'm guessing that the "coordinate charts" PeterDonis is referring to are some sort of special implementation of the Minkowski diagram that somehow warp the shape of the light cone in some measured way, say close to a black hole. But how does this morphing manifest itself and are there restrictions as to, say, how warped the light cone can become?

More generally, though, is there a well-accepted "coordinate chart" that describes the larger observable universe that may average out and account for gravitational anomalies on a more global scale? I mean, are we talking "Kruskal coordinates" here, or a Penrose diagram?

More specifically, I'm interested in item #3 of my initial post; let's say I'm is sitting in galaxy A and I'm watching Alice in Galaxy Z which is receding from me at .7c, solely due to the expansion of the scale factor. Now Alice throws a baseball in the direction of her expansion away from me at .5c. How fast do I observe this ball to be receding from me? It doesn't seem that relativistic velocity addition would apply here, would it? It would seem to me that I would cease to see the ball as it approached .3c in Alice's galaxy Z.

Finally, at this point, and I know I'm asking for too much, what is it about the property of "normal" local space that limits the propagation of light, or any object for that matter (no pun intended) to the speed of c, whereby we can view moving object in the intergalactic realm as moving faster than the speed of c?

 

[sweet springs]

More specifically, I'm interested in item #3 of my initial post; let's say I'm is sitting in galaxy A and I'm watching Alice in Galaxy Z which is receding from me at .7c, solely due to the expansion of the scale factor. Now Alice throws a baseball in the direction of her expansion away from me at .5c. How fast do I observe this ball to be receding from me? It doesn't seem that relativistic velocity addition would apply here, would it? It would seem to me that I would cease to see the ball as it approached .3c in Alice's galaxy Z.

In case Alice is nearby us, speed of the ball is $$\frac{v_1+v_2}{1+v_1v_2/c^2}=\frac{1.2}{1.35}c=0.88..c$$. I assume this stands even she is in Galaxy Z where the speed of the galaxy is $$v_1<c$$. In case $$v_1>c$$ we cannot apply the formula of velocity addition.

 

[PeterDonis]

As long as you don't get too close to any gravitating body (i.e. stay where the weak field approximation is good enough) a Minkowski diagram should do, shouldn't it?

Sure, for a small local area around your spaceship, but not if you want a diagram on a scale covering galaxies.

 

[PeterDonis]

I'm guessing that the "coordinate charts" PeterDonis is referring to are some sort of special implementation of the Minkowski diagram

No. It's the other way around; what you are calling the "Minkowski diagram" is a special case (a very special case) of a coordinate chart. See Chapter 2 of Carroll's lecture notes on GR for a good general discussion of coordinate charts:

http://arxiv.org/abs/gr-qc/9712019

is there a well-accepted "coordinate chart" that describes the larger observable universe

Yes, the FRW coordinate chart--actually there are several different, closely related charts that can be used on FRW spacetime, see here.

let's say I'm is sitting in galaxy A and I'm watching Alice in Galaxy Z which is receding from me at .7c, solely due to the expansion of the scale factor. Now Alice throws a baseball in the direction of her expansion away from me at .5c. How fast do I observe this ball to be receding from me? It doesn't seem that relativistic velocity addition would apply here, would it?

No, certainly not, because the "velocities" here are not relative velocities in the SR sense; they are coordinate velocities in the FRW coordinate chart, and they can't be directly measured.

When you say that Alice is receding from you at .7c, what that actually means is that the light you are currently seeing from Alice has a redshift $z$ of 0.7 (in other words, the observed wavelength of any particular spectral line in the light you are observing from Alice is $1 + z = 1.7$ times the reference wavelength of the same lines measured in the lab). The redshift gets converted to a velocity by the formula $v = cz$. (In fact, this formula is only an approximation valid at small redshifts, where 0.7 is still reasonably "small" for this purpose. At larger redshifts, the conversion formula to velocity is model-dependent; see here.) However, even this conversion is not really saying anything physically meaningful; it's done mainly for historical reasons, because everyone is used to quoting speeds instead of redshifts. What the redshift $z$ really means, physically, is that the universe expanded by a factor $1 + z$ during the time it took the light Alice emitted to travel from her to you.

When you say Alice throws a baseball in the direction away from you at .5c, however, that is a relative velocity between the baseball and her in a local inertial frame centered on her at the event where she throws the ball (which we are assuming is the same as the event where the light you are seeing from Alice was emitted). But that local inertial frame does not cover you as well; you are too far away from Alice (because her light has a measurable redshift when you see it); so you and Alice do not have a well-defined relative velocity in that frame (certainly not a relative velocity of .7c). So adding the .5c and the .7c, whether you use relativistic velocity addition or any other formula, is meaningless, because they are two different things--it would be like adding apples and truck tires.

The question of "how fast do I observe the baseball receding from me?" is not well-defined either, because you don't observe how fast it's receding. The best you can do would be to observe its redshift; but here there is a further problem, because unlike Alice, the baseball is not "comoving"--it is not following a worldline on which the universe appears homogeneous and isotropic. All of the conversion formulas from redshift to velocity (or even from redshift to how much the universe expanded during the light's travel time, as above) assume "comoving" observers on both ends. You and Alice are both "comoving", so you can convert her redshift to a speed (or better, an expansion factor, as above); but you can't even do that with the baseball, because it's not "comoving".

I assume this stands even she is in Galaxy Z

No, it most certainly does not. See above.

 

[sweet springs]

If speed of Alice in Galaxy X (If I can say it as "speed", v_1) and speed of the ball to Alice(v_2) cannot be added as is shown in #8, I am frustrated to know the right answer.
Is it a wrong question?

Dusts falling into center is another case of comoving coordinate. When Alice in Dust X throw a ball, we cannot say about speed of the ball either? The two co-moving coordinates have different nature?

 

[PeterDonis]

Is it a wrong question?

Yes. Light from the ball will have a well-defined redshift when we observe it, but there is no well-defined way to translate that redshift into a "speed".

 

[sweet springs]

Though velocity addition is not allowed, is redshift addition, i.e. redshift of ball thrown by redshifted Alice, allowed and the formula of which is similar to #8 ?

 

[PeterDonis]

is redshift addition, i.e. redshift of ball thrown by redshifted Alice, allowed and the formula of which is similar to #8 ?

No. Redshifts don't add. As I noted, light emitted by the ball will have a well-defined redshift when it reaches us, but it won't be derived by just adding Alice's redshift relative to us and the ball's redshift relative to Alice. In fact, the actual derivation is somewhat complicated.

 

[sweet springs]

In addition to usual velocity or peculiar velocity, recession velocity and redshift velocity are new to me. Thanks.