Exploring the Curvature of Light Cones

[Edgar L Owen]

All,

Light cones are always shown with straight sides in textbooks but it seems to me they actually must curve inward and would eventually converge at the point of the big bang if we could see back that far. Thus going back any significant distance in time light cones should curve inward since the universe was considerably smaller and they have to stay within the universe at all times.

For example if the universe has expanded ~1100 times since the CMB (here meaning the surface of last scattering) the curve of light cones back to the CMB should be quite considerable.

Doesn't this mean that
1. we actually see an ~1100x magnfied view of the CMB surface since the curvature of light cones should act as a magnifying lens?
2. The CMB surface we observe was actually 1100 times closer than the distance its light traversed because the radiation we see left it when it was much closer to us than it is now? It's light just had to traverse the large curvature of the light cones from us to it so it took a lot more time to reach us than if it traveled in a straight line?
3. Thus the CMB surface we observe was actually closer than some other distant astronomical objects that were further away when their light was emitted?

I haven't seen this discussed anywhere and wonder whether it's correct or not?

Edgar

 

[Drakkith]

Light cones are always shown with straight sides in textbooks but it seems to me they actually must curve inward and would eventually converge at the point of the big bang if we could see back that far.

If by "light cone" you mean the ones you find in diagrams like these:

Then they work perfectly fine with straight edges, as the slope of the line represents the maximum distance you can travel per unit of time. Generally these are shown as emerging from a single point, which represents a single event in space-time. Also note that this diagram is only applicable in nearly flat spacetime (or so I think). See more below.

Thus going back any significant distance in time light cones should curve inward since the universe was considerably smaller and they have to stay within the universe at all times.

The universe has no known size. Light cones are only limited in how large they can grow by how much time has passed since the event. Since the big bang took place everywhere within the universe (and not at a single location like is commonly thought), light cones corresponding to an event at t = 0 would simply start at any arbitrary location within the universe at that time.

Of course, I haven't taken expansion into account. Things almost certainly get much more complicated when discussing curved spacetime and universal expansion. I assume, possibly incorrectly, that the edge of a light cone will curve outwards over long periods of time because of this, but I could certainly be mistaken.

 

[Chalnoth]

All,

Light cones are always shown with straight sides in textbooks but it seems to me they actually must curve inward and would eventually converge at the point of the big bang if we could see back that far. Thus going back any significant distance in time light cones should curve inward since the universe was considerably smaller and they have to stay within the universe at all times.

In general, yes, it is the case that light won't necessarily travel in a straight line in a curved space-time. A simple example is near a black hole: at a specific radius, light can orbit the black hole. Obviously an orbiting photon won't be traveling in a straight line.

Something similar can be said about light cones in an expanding universe. You can (usually) make the light cones straight if you want to, but for the most part they'll be curved (depends on the coordinates you use).

Doesn't this mean that
1. we actually see an ~1100x magnfied view of the CMB surface since the curvature of light cones should act as a magnifying lens?

In order for the view to be magnified, the light path would have to curve in such a way that the light changes its direction. This is manifestly not the case: the light from the CMB travels directly towards us (unless it travels near a massive object like a galaxy, which will bend the light and cause magnification). The curvature due to the expansion only modifies how much distance it has to traverse to reach us.

2. The CMB surface we observe was actually 1100 times closer than the distance its light traversed because the radiation we see left it when it was much closer to us than it is now? It's light just had to traverse the large curvature of the light cones from us to it so it took a lot more time to reach us than if it traveled in a straight line?

Yes. The light was emitted from a surface that was, at the time, about 42 million light years away. Today that same matter is roughly 46 billion light years away.

3. Thus the CMB surface we observe was actually closer than some other distant astronomical objects that were further away when their light was emitted?

The CMB light is still coming from behind the distant astronomical object, so it still makes sense to think of it as further away.

 

[Bandersnatch]

Have a look at these lightcone graphs from the oft-quoted Davis & Lineweaver paper

As you can see, the nice straight lines of light paths do curve back in an expanding universe. The top graph has the most layman-intuitive coordinates, where distance and time are what we normally think of them to be.

If you don't know what comoving coordinates mean, just ignore the latter two. It is worth keeping in mind, however, that all three represent the same thing.
The graphs below (made by yukterez, who's a member on PF) are analogous to the top two, but animated:
http://yukterez.net/lcdm/lcdm-flrw-animation.gif

If it's not clear, the lightcones show all the light that reaches the observer at their apex. So, you can trace the path of light an say when and where the emission events of all the light that we now observe were. E.g., it can be seen that of all the light we receive now, the emission events that were the farthest away in terms of proper distance happened about 8 Gy ago (where the 'teardrop' shape of the lightcone curves around), and that these events were approx. 5 Gly away. All the earlier (below that point) and later (above) emission events were closer.

In order for the view to be magnified, the light path would have to curve in such a way that the light changes its direction. This is manifestly not the case: the light from the CMB travels directly towards us (unless it travels near a massive object like a galaxy, which will bend the light and cause magnification). The curvature due to the expansion only modifies how much distance it has to traverse to reach us.

But there is a magnification effect at high z, isn't there?
https://ned.ipac.caltech.edu/level5/March02/Sahni/Sahni4_5.html
In particular this graph shows it well:

That is, as the OP guessed, you do get increase in angular diameter of faraway objects, (edit:although that is an effect specific to universes where deceleration switches to acceleration at some point in history (i.e., universes with matter and dark energy)).

 

[Chalnoth]

But there is a magnification effect at high z, isn't there?
https://ned.ipac.caltech.edu/level5/March02/Sahni/Sahni4_5.html
In particular this graph shows it well:
View attachment 109610
That is, as the OP guessed, you do get increase in angular diameter of faraway objects, although that is an effect specific to universes where deceleration switches to acceleration at some point in history (i.e., universes with matter and dark energy).

That really depends upon what you mean by magnification.

First, I don't think that this effect has anything to do with the acceleration of the expansion. I'm pretty sure it will happen in any universe where the rate of expansion decreases sufficiently. That is, if we had no dark energy, I'm pretty sure we'd see this effect. Only if the universe had always been dominated by dark energy might it not be significant.

That said, yes, the angular size of galaxies of the same intrinsic size decreases for a while, then starts increasing again at very large distances. I guess I don't see this as magnification per se because it's still not a matter of light being deflected as with a lens: it's due to the fact that the universe at the time was smaller, but we're considering an object of the same intrinsic size. The apparent size is (very roughly) the size of the object divided by the size of the universe at that time. So same object + smaller universe = bigger apparent size (more formally it's an angle derived from the ratio of the size of the object to the distance between us and the object at the time the light was emitted: because the denominator of that fraction decreases at early times, the apparent size increases).

 

[Bandersnatch]

First, I don't think that this effect has anything to do with the acceleration of the expansion

Yes, you're right. I'm not sure where that came from. Even the graph I posted includes a line for matter-only universe, so a purely decelerating one. I'll correct the post promptly.