# Is the speed of expansion of the universe faster than light?

Gold Member
... as they have moved a distance even more than 28 billion light years in less than 13.7 billion years . isn't that wierd huh?
It looks weird if one does not take into account two very important factors:
1. When we say that something is 28 billion light years away now, we mean that we are calculating this distance to the object, where it will be now, but what we see (now) is the light that has traveled for 13 billion years, which then was emitted when the object was only 4 billion light years away from us.

2. Space is expanding and everything moving inside, including light, has to make its way 'upstream' this expansion, hence it will take light a much longer time to travel a distance, than the first obvious conclusion (think of running at 10 km/h after someone walking at 5 km/h, it will take you longer time to reach that person, than if he stood still).

The brown line on the diagram is the worldline of the Earth. The yellow line is the worldline of the most distant known quasar. The red line is the path of a light beam emitted by the quasar about 13 billion years ago and reaching the Earth in the present day. The orange line shows the present-day distance between the quasar and the Earth, about 28 billion light years.

http://en.wikipedia.org/wiki/Metric_expansion_of_space#Understanding_the_expansion_of_Universe

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it means there are some objects in the universe which can't be ever seen or reached by us right?
now then how did they get this far i mean in the time of big bang they were together so after big bang they got farther more then 13.7 billion light years radius (they are also objects and they also follow relativity) because that's the age of universe and nothing can exceed the velocity of light . so what made em get this far?
The theory of inflation says that objects were close enough to interact with each other when the universe was first created, but a phase change in the quantum field released energy into the universe, causing the distance between these objects to expand at faster than the speed of light for a short period of time.

This is why two objects which were 13.7 billion light years away from each other 13.7 billion years ago still had about the same temperature.

oh you mean the friends can never see each other again ? now you get a lesson don't buy house near a friend because time can change things ...... i km can become 13.7 billion light years. i feel bad for them both :(

JDoolin
Gold Member

The brown line on the diagram is the worldline of the Earth. The yellow line is the worldline of the most distant known quasar. The red line is the path of a light beam emitted by the quasar about 13 billion years ago and reaching the Earth in the present day. The orange line shows the present-day distance between the quasar and the Earth, about 28 billion light years.

http://en.wikipedia.org/wiki/Metric_expansion_of_space#Understanding_the_expansion_of_Universe
I notice that the Ben Rudiak Gould's lambda CDM image cuts off at the bottom at 700 million years ABB (After Big Bang). If I'm reading it correctly, the expansion rate of the universe is fastest where the cone is flared out, and slowest if the cone is vertical.

I thought I recalled a gif animation on Ned Wright's page (though it may have been somewhere else) that went all the way back to the first instant. The particles started out co-located, yet not causally connected. Then they moved away from each other at great speed (via the stretching of space), and then slowed down to a velociy where light could travel between them.

Extrapolating from Gould's image, if the base of the cone goes horizontal, something like that may have happened before 700 million years.

Rishavutkarsh said:
oh you mean the friends can never see each other again ? now you get a lesson don't buy house near a friend because time can change things ...... i km can become 13.7 billion light years. i feel bad for them both :(
When the cone flares out enough, can signals cease between two previously neigboring (i.e. causally connected) particles? I wonder whether the theory is similar to the black hole situation (but more symmetrical), where the person falling into the black hole falls in in finite time, while the person watching sees him fall closer and closer into the event horizon, but never falling in.

In the lambda CDM cosmological model, where the stretch of space exceeded the speed of light, perhaps both parties would see the last light of the other eternally redshifting to infinity, while continuing to experience their own time normally.

The red line does not look like a null geodesic if infinitesimal segments are considered.Strictly speaking we should consider light cones at each and every point of the null geodesic[red-line], maintaining the time axis parallel to itself. [Incidentally all infinitesimal segments lying on the light cone do not make 45 degrees with the time axis. We can always wrap smooth curves on a light cone which are of a mixed character----to be precise any smooth curve[with delta_t not equal to zero] wrapped over the light cone is of a mixed character[generally speaking] comprising spacelike and null segments].

Is the following Wikipedia statement[in your link] in conformity with the above considerations?
"In particular, light always travels locally at the speed c; in our diagram, this means that light beams always makes an angle of 45° with the local grid lines."

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Dale
Mentor
The red line does not look like a null geodesic if infinitesimal segments are considered.
It is hard to tell exactly, but it looks pretty close to me.

By introducing the concept of conformal time with the FRW metrics we have light cones whose generators make 45 degrees with the time[conformal ] axis.
We consider a general type of a FRW metric [in simplified form] :
$${ds}^{2}{=}{dt}^{2}{-}[{a}{(}{t}{)}]^{2}{[}{dx}^{2}{+}{dy}^{2}{+}{dz}^{2}{]}$$
Writing,
$${dt}{=}{a}{(}{t}{)}{d}{\eta}$$
we have for the null geodesics,
$${d}{\eta}^{2}{=}{dr}^{2}$$ ----------- (1)
where,
$${dr}^{2}{=}{[}{dx}^{2}{+}{dy}^{2}{+}{dz}^{2}{]}$$
Equation (1) gives us a picture of the light cone as we know in special relativity so far as the coordinate speed of light is concerned..
Alternatively we may do the following:
$${ds}^{2}{=}{dt}^{2}{-}[{a}{(}{t}{)}]^{2}{[}{dx}^{2}{+}{dy}^{2}{+}{dz}^{2}{]}$$
Or,
$${ds}^{2}{=}{dt}^{2}{-}{dL}^{2}$$------ (2)
Where,
$${dL}^{2}{=}{[}{a}{(}{t}{)}{]}^{2}{[}{dx}^{2}{+}{dy}^{2}{+}{dz}^{2}{]}$$
Again we have the Special Relativity picture at each point of time with equation (2).

The Wikipedia model seems to have used unmodified time[and not conformal time]. If the distance represented is the coordinate distance, we should get[rather we are supposed to get] the Special Relativity picture of the null geodesics[so far as infinitesimal sections are concerned]] after transforming to conformal time[from unmodified time]. But from the picture it is not clear [and quite difficult to say]whether the red line will finally cater to the required properties of the null geodesics.[after the transformation.]
[If Wikipedia has used physical distance for the picture the situation would become much more difficult]

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We consider the following equation again:
$${ds}^{2}{=}{dt}^{2}{-}[{a}{(}{t}{)}]^{2}{[}{dx}^{2}{+}{dy}^{2}{+}{dz}^{2}{]}$$
For a null geodesic:
$$\frac{dx}{dt}{=}\frac{1}{{a}{(}{t}{)}}$$

dx/dt should be a variable thing[function of time]----this does not seem to hang with the constant 45 degree depiction in the Wikipedia model:"In particular, light always travels locally at the speed c; in our diagram, this means that light beams always makes an angle of 45° with the local grid lines."

[Only x-coordinate has been considered in the above relation. One may consider both x and y coordinates to get a better picture]

If the picture represents a plot of conformal time against coordinate distance we get the same light cone picture as we have in Special Relativity.The red line contains segments that do not make 45 degrees with the conformal time axis

The same holds true if coordinate time[unmodified] is plotted against physical distance[The Special Relativity picture of the light cone should hold true-the red line does not seem to be following it every where--it is not at 45 degrees to the time axis at all points]

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JDoolin
Gold Member
Here's what the wikipedia article says about the diagram.

"The narrow circular end of the diagram corresponds to a cosmological time of 700 million years after the big bang; the wide end is a cosmological time of 18 billion years, where one can see the beginning of the accelerating expansion which eventually dominates in this model. The purple grid lines mark off cosmological time at intervals of one billion years from the big bang. The cyan grid lines mark off comoving distance at intervals of one billion light years. Note that the circular curling of the surface is an artifact of the embedding with no physical significance; space does not actually curl around on itself."​

Looking path of the light through each individual rectangle, it appears that sometimes the light covers only 1B LY in 1 billion years, and in others (notably the time between 1 billion years and 2 billion years) the light covers several billion light years.

Is that consistent with $c d\tau^2 = c^2 dt^2 - a(t)^2dx^2$?

To follow the path of a photon, set $d\tau = 0$, and derive:

$$\frac{\mathrm{d} x}{\mathrm{d} t} = \frac{c}{a(t)}$$

I'm pretty sure this $$\frac{\mathrm{d} x}{\mathrm{d} t}$$ refers to how fast the beam travels according to the purple grid lines (representing Δt= 1BYears) , and cyan grid lines (representing Δx=1B LY).

Here's what the wikipedia article says about the diagram.

"The narrow circular end of the diagram corresponds to a cosmological time of 700 million years after the big bang; the wide end is a cosmological time of 18 billion years, where one can see the beginning of the accelerating expansion which eventually dominates in this model. The purple grid lines mark off cosmological time at intervals of one billion years from the big bang. The cyan grid lines mark off comoving distance at intervals of one billion light years. Note that the circular curling of the surface is an artifact of the embedding with no physical significance; space does not actually curl around on itself."​

Looking path of the light through each individual rectangle, it appears that sometimes the light covers only 1B LY in 1 billion years, and in others (notably the time between 1 billion years and 2 billion years) the light covers several billion light years.

Is that consistent with $c d\tau^2 = c^2 dt^2 - a(t)^2dx^2$?

To follow the path of a photon, set $d\tau = 0$, and derive:

$$\frac{\mathrm{d} x}{\mathrm{d} t} = \frac{c}{a(t)}$$

I'm pretty sure this $$\frac{\mathrm{d} x}{\mathrm{d} t}$$ refers to how fast the beam travels according to the purple grid lines (representing Δt= 1BYears) , and cyan grid lines (representing Δx=1B LY).
"The cyan grid lines mark off comoving distance at intervals of one billion light years."

One billion light year is a comoving distance--it does not change with time as we proceed upwards along the time axis.But the graduations are widening as we go up along the time axis--so these graduations mark off the physical distances[the increasing physical distances] corresponding to the comoving distance of one billion light years [with the advancement of time]

The quantity [physical distance/time] is increasing for the light ray.

[Physical distance= a[t]*comoving distance[comoving distance=coordinate distance between labels that do not change with time]]

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JDoolin
Gold Member
The quantity [physical distance/time] is increasing for the light ray.

[Physical distance= a[t]*comoving distance[comoving distance=coordinate distance between labels that do not change with time]]
Is it increasing in the diagram? If I understood the idea correctly, we have:

$$c^2 d\tau^2 = c^2 dt^2 - a(t)^2 dx^2$$

which simplifies to:

$$c^2 = \frac{a^2 dx^2}{dt^2} \overset ? = constant$$

which would be the speed of light (squared) when using the physical distance.

But the speed of light (squared) in the comoving distance would be

$$\frac{dx^2}{dt^2} = \frac{c^2}{a(t)^2}$$

which would be slowing down as a(t) increases.

(Edit: Now that I look at the diagram again, it does appear that the physical distance speed of light is increasing, i.e. $c^2 = \frac{a^2 dx^2}{dt^2} \neq constant$. Is that a particular feature of the Lambda-CDM model, or is it just a badly drawn speed-of-light line?)

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bcrowell
Staff Emeritus