# I Expansion Rate of Cosmic Bubbles

1. Dec 10, 2017

### Thuring

I’ve understood that within the theory of Eternal Inflation the Bubble Universes that form from a drop of “vacuum” energy expand at the Speed of Light.

My question is:

Why would expansion be at the speed of light and not at a speed proportional to the level of vacuum energy still present in the bubble?

After all, I thought the extremely high vacuum energy was why the “original” mega-universe expands so super-duper quickly and why our bubble expanded so quickly until almost all the vacuum energy decayed and turned into normal mass and energy. Yet, the region of our universe further than the horizon is leaving us faster than light, relative to us, and will get faster.

2. Dec 10, 2017

### bapowell

The wall of the bubble expands at c; this is true even in static spacetimes. The expansion rate of the space inside the bubble is another matter, depending on the energy density of the bubble interior.

3. Dec 10, 2017

### Thuring

I'm don't understand how the inside and wall could expand at different rates.

4. Dec 10, 2017

### Orodruin

Staff Emeritus
You are confusing movement of the bubble wall (the bubble getting bigger due to the wall moving) with the metric expansion inside the bubble (the bubble getting bigger due to distances inside the bubble intrinsically expanding).

5. Dec 10, 2017

### Thuring

I believe I know what a metric is, essentially relative distances within the bubble (dS). Wouldn't this relative distance increase between any two points be , say, 1/2 of the wall speed? I would think that the metric expansion and wall expansion would be related. Am I comparing 3-D space with 4-D space?

There is an interesting concept here that I am missing.

6. Dec 10, 2017

### Orodruin

Staff Emeritus
No. As bapowell already said, these two are completely unrelated concepts. Nothing needs to "move" for the metric expansion inside (or outside) the bubble to take place. Nor does there need to be metric expansion for a bubble wall to expand.

7. Dec 11, 2017

### Staff: Mentor

That's the metric within the bubble. It's not the same as the metric outside the bubble.

It's best not to think of the bubble wall as having a "speed" at all in any global sense. The bubble wall is a null surface, because it's a causal boundary and any causal boundary has to be a null surface. (At least that's my understanding of this general class of models.) Since it's a null surface, it will move at the speed of light relative to any nearby observer; but that doesn't tell you anything about how the null surface is related to the spacetime geometry inside the bubble.

No, they aren't. Spacetime is curved. You are trying to apply intuitions that only work in flat spacetime.

8. Dec 11, 2017

### Thuring

So the metric expansion is a 4-D expansion; whereas the wall movement is a 3-D spacial (physical) expansion? The metric would be :-c2 dr2 = -c2 dt2 + a(t)2 d∑2 where a(t)2 d∑2 would be the physical wall expansion? For instance, the spacial portion may increase, but be compensated for by the time component.

Why would a null surface travel at the speed of light (if it's not getting too complex)?

(Yeah, intuitions seem to often fall apart with physics.... that's my favorite part.)

9. Dec 11, 2017

### Orodruin

Staff Emeritus
No. I do not understand where you got the notion of 3D or 4D expansion from, but it is not something that is standard nomenclature and you probably should not be using these terms. What would you mean by 4D expansion?

Both the expansion of the bubble itself and the metric expansion are physical. The bubble being a null surface means you can never get out of the bubble (the event horizon of a black hole is also a null surface although for different reasons).

From your posts I would say that you need to acquire a deeper understanding of space-time geometry before you can fully understand what is going on in this scenario.

10. Dec 11, 2017

### Thuring

X, Y, Z, time .... 4-D as in -c2 dr2 = -c2 dt2 + a(t)2 d∑2 Minkowski space-time
X, Y, Z 3-D as in ds = dx2 + dy2 + dz2 Euclidean space

11. Dec 11, 2017

### Orodruin

Staff Emeritus
I am sorry, this makes no sense. You still have not explained what you would mean by 4D expansion versus 3D expansion. Metric expansion has a precise mathematical meaning in GR. You cannot just throw nomenclature around without properly defining things.

12. Dec 11, 2017

### Thuring

I guess this is going to have to be answered at a later time.

I got most of what was said, but I don't know why what was said, was true. That extra step back is probably not simple, so that's why people stopped with a couple sentences.

I need further study.

What topic would this be called? For instance : Cosmology / Spacial Mathematics / ..... ???

Thanks for trying all

13. Dec 11, 2017

### Staff: Mentor

I said something more restricted than that. I said a null surface travels at the speed of light relative to any nearby observer. A more technical way of putting this would be that, in any local inertial frame that contains a segment of the null surface, the surface moves at the speed of light--because that's what "null" means in a local inertial frame (since in a LIF things work just like in SR, and in SR a null surface obviously travels at the speed of light).

However, this concept of "speed" is only meaningful within a single LIF. There is no meaningful notion of "speed" in a curved spacetime that applies globally, which means there's no meaningful notion of "speed" for the bubble wall that applies everywhere inside the bubble. So if, for example, our actual universe is a bubble inside an eternally inflating region, we are not anywhere near the bubble wall (because if we were we would observe it), and therefore there is no meaningful notion of "speed" of the bubble wall relative to us.

The spacetime we are talking about--an eternally inflating spacetime with bubble regions inside it--is not Minkowski spacetime, or even close to it.

This is why I said earlier that you should not try to apply your intuitions in this case; they are intuitions that might work for Minkowski spacetime, but do not work in a curved spacetime such as the one we are discussing.

"Space" depends on how you choose coordinates in spacetime. Most such choices in most spacetimes do not lead to a Euclidean space.

14. Dec 11, 2017

### kimbyd

I don't think it's exactly the speed of light, but it's close enough. I talked with a string theorist a long time ago about the similar situation of quantum vacuum decay. Apparently what happens is the domain wall rapidly accelerates as the domain initially forms, approaching the speed of light in less than a second. I don't know the exact dynamics, but it would be easy to explain if there's no friction-like force proportional to velocity which opposes the acceleration. There probably can't be such an opposing force due to Lorentz invariance.

The rate at which the domain wall accelerates is likely a function of the difference in energy, but as long as the effective mass of the domain wall is low, it can accelerate to close to the speed of light in no time.

15. Dec 11, 2017

### Thuring

After reading the answers many times, I think I'm getting some understanding. At the very least, I know my ideas were wrong.

Too bad you guys had to repeat some things so many times for it to sink into my wooden head.

Thanks again....

16. Dec 12, 2017

### nikkkom

Since moving bubble wall represents a process of conversion of old _vacuum_ to the new one (+some particles), it should be Lorentz-invariant.

"Moving at any speed other than speed of light" is not Lorentz-invariant - the speed will be different depending on chosen reference frame. Which would lead to a nonsensical situation of vacuum decay rate being frame-dependent, while vacuum itself is not frame-dependent (vacuum looks the same no matter how fast you move).

The only Lorentz-invariant velocity is speed of light.

17. Dec 12, 2017

### kimbyd

I strongly suspect that this changes a bit once you consider the possible dynamics that could cause such a vacuum transition to occur. It's probably mostly right, but as long as the configuration that makes up the domain wall between vacuum states has some non-zero effective mass, it won't quite be moving at the speed of light. But it will be so close as to be indistinguishable very soon after the vacuum state with lower energy starts to expand.

18. Dec 12, 2017

### nikkkom

Okay, a thought experiment. If wall is moving slower than speed of light, a spaceship can accelerate to a velocity such that it moves at the same speed as the wall. Therefore, from the spaceship's point of view, it sees a _stationary_ boundary between old and new vacuum. Which should not be possible, because old vacuum wants to decay into the new one - the wall must start moving.

19. Dec 12, 2017

### Orodruin

Staff Emeritus
If I remember correctly, what @nikkkom is saying is true for a wall that is a plane. Once you take into account the bubble dynamics (essentially when energy in the wall is comparable to the energy difference between the true and false vacua inside the bubble), the wall will generally not move at the speed of light. However, any bubble that is supercritical will start expanding so close to the speed of light that the difference is irrelevant for all practical purposes.

20. Dec 12, 2017

### kimbyd

That's not inconsistent as long as the wall is always accelerating. Which means it will asymptotically approach the speed of light.

21. Dec 12, 2017

### Orodruin

Staff Emeritus
Here is a link to Sidney Coleman's seminal paper Fate of the false vacuum: Semiclassical theory in the KEK library. He discusses a setting that is not eternal inflation, but where the false vacuum has zero energy density and essentially gives Minkowski space as the solution. In figure 4 you can see exactly the acceleration of the wall mentioned by @kimbyd.

Side note: This paper contains one of my favourite quotes across all physics papers that I have read. It highlights a somewhat morbid sense of humour in Coleman (my clarifications in square brackets):
Edit: Here is a link to the same paper from Princeton with much nicer formatting.

22. Dec 12, 2017

### kimbyd

I found this paper which attempts to do a detailed treatment of the dynamics of domain walls:
https://arxiv.org/abs/0811.0866

It's quite dense, unfortunately, but it describes a picture where once a new domain starts to form, it transitions from no acceleration to constant acceleration, which means the domain wall asymptotically approaches the speed of light.

23. Dec 12, 2017

### Orodruin

Staff Emeritus

24. Dec 13, 2017

### bapowell

25. Dec 13, 2017

### kimbyd

This is off the top of my head, but it probably stems from the boundary conditions. Consider that you're transitioning from a state with $H^2 = \Lambda$ (with the appropriate constant conversion factor I'm not going to bother with right now) to a state with a lower cosmological constant. This results in an immediate decrease in the energy density associated with the cosmological constant, but the rate of expansion would (initially) remain the same. If this energy density didn't go into any other matter/radiation field, or if there was an equal amount of energy density in matter and radiation produced but not all of it ended up within the expanding bubble, then there would be a discrepancy that would show up as negative curvature.

So if I were to guess, the details of the spatial curvature are all about how the transition to a lower vacuum energy interacts with matter/radiation, and how that matter/radiation is distributed as the boundary wall expands.