# B Hubble's constant and Minkowski spacetime

1. Jun 13, 2017

### Chris Miller

If Hubble's constant = 160,000 m/sec/million light years and c = 299,792,458 m/sec, then shouldn't any two points in the universe farther than about 1,873,000,000 light years apart be expanding away from each other faster than c?

Since light from sources much farther than this has reached us, does this suggest that H has been increasing since the Big Bang?

Why does the Minkowski spacetime interval not need to take into account H?

E.g., Since light can never reach from one point to another that is expanding away faster than c, there can be no lightcone on which they both lie. Does this preclude all simultaneity, mean that each will only ever exist in
the other's past or future?

2. Jun 13, 2017

### Orodruin

Staff Emeritus
Yes. But this does not mean that anything is actually moving (locally) faster than the c. This is just a separation speed due to the metric expansion.

No. The Hubble constant is decreasing.

Minkowski spacetime is incompatible with an expanding universe. It describes a static universe.

In general, talking about different "points" is completely coordinate dependent.

3. Jun 13, 2017

### Chris Miller

I understand that, but still doesn't it affect how long light will take to travel from one to the other?

So when the universe was the size of a photon... ah, I see. It would've had to have been much higher or it'd still be the size of a golf ball. So how is it light from galaxies > 2 billion light years away managed to reach us?

Thanks. Was wondering. So, not the actual universe then.

I was afraid my vernacular would be wrong. What would be the correct terms for an expanding coordinates system?

4. Jun 13, 2017

### Orodruin

Staff Emeritus
There is no such thing as "the size of a photon". Might as well dispell that notion straight away.

Light from about 45 billion light years away (proper distance) is reaching us. Of course, when this light was emitted, the source was much much closer.

Locally, it is a very good approximation. Globally, it is not.

5. Jun 13, 2017

### Bandersnatch

6. Jun 14, 2017

### timmdeeg

But not Einstein's static universe, right?

To my understanding: The empty universe is expanding in FRW-coordinates which however is equivalent by coordinate transformation to Minkowski spacetime (Milne model). Doesn't this show that one is completely free to interpret increasing distances as expansion of space and as purely motion as well?

7. Jun 14, 2017

### Orodruin

Staff Emeritus
The Milne model describes an empty universe. Our universe is not empty.

Yes, you can interpret Minkowski space as "expanding", but you are then not using the regular conventions of simultaneity, but a very different coordinate system where your time coordinate is the proper time of the interval to some assigned origin. Your coordinate system also does not cover all of Minkowski space. The simultaneity surfaces would be hyperboloids. In 1+1 dimensions, a hyperbola of "simultaneity" would be given by $t^2 - x^2 = \tau^2$, where $\tau$ is the "cosmological time" in regular Minkowski coordinates.

Note that FLRW coordinates are not normal coordinates. If you impose a normal coordinate system locally, you will indeed find that (locally) Hubble's law is satisfied and objects a proper distance $d$ away indeed travel at a velocity $Hd$.

8. Jun 14, 2017

### Chris Miller

Thanks for the link. Interesting article, what I was able to follow. I've seen the inflating balloon analogy. But saying the points don't move relative to the surface of the balloon doesn't really fly for me since there is no universal frame of reference. All motion is relative. And H does cause relative velocities > c. Also, without knowing exactly how H has changed since the BB or even if it's homogeneous throughout the universe, it would seem futile to try to calculate the "current" separation of remote objects or actual size of the universe. I quoted "current" since I'm still having trouble with simultaneity in regions that are "now" far apart enough that light from one can never reach the other. In other words, no matter how powerful the telescope, one can never see the other's clock.

9. Jun 14, 2017

### Chris Miller

Right, thanks. Meant proton.

This is enlightening. Without having a graph of H since that time, no way to know exactly how close?

10. Jun 14, 2017

### Orodruin

Staff Emeritus
Well, you could do exactly the same maths that would go into producing such a plot ...

11. Jun 14, 2017

### Bandersnatch

There's no universal frame of reference, and yet the police can give you a speeding ticket. We always use one convenient FoR or another, and in cosmology that frame is the comoving frame - it's the closest to what 'stationary with respect to the universe' could mean.

Here's a graph of H(t) (truncated for clarity):

Here you can see emission distances calculated, together with recession velocities at emission:

As you can see, the farthest observable light was emitted from then-42 million ly away, by objects initially receding at over 60 times the speed of light.

Here's the calculator from which the outputs were taken:
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7-2017-02-08/LightCone_Ho7.html

That's what the last 20-odd years of observations were for. To establish the expansion history of the universe.

Last edited: Jun 14, 2017