# Is there an objective difference between space expansion and reduction in speed of light

I just read an article about the cosmological model of an inflationary universe. As you probably know it basically states that space itself is subject to an expansion over time.

Since there is no moving matter or energy involved, this might even happen at a "speed" faster than light. So far, so good and obscure. What strikes me is the principle that the speed of light as a fundamental constant can only be expressed as a function of space-time. Where do we know that the one is constant but the other can suddenly be variable?

Is there any reason why the point of view of an expanding space is preferred over, say, a reduction in the speed of light or an increase in the "speed" of time? Is there any objective difference or is it just the good old rubber metaphor being stretched (pun intended) too far?

• This is a very interesting question. Feb 2 '16 at 18:12
• I think you mean "expanding" rather than "inflationary". We don't observe inflation today. I think the answer to your question is related to the conservation of energy - we see a red shift. Feb 15 '16 at 12:58
• I am not aware of the difference between inflation and expansion- so you might be quite right with the first part of your answer, but the second part, I am still quite unsure about: red shift is a change in frequency which is either a function of space (in case of constant speed) or a function of speed (in case of constant space), right? Feb 15 '16 at 15:46
• @choeger The difference between expansion and inflation is how fast. The universe inflated at a very fast rate in the first trillionth, trillionth, trillionth of a second. When the inflationary epoch ended, the universe was suddenly a hundred trillion trillion times wider. That's like having a meter stick go from one meter to stretching to the nearest quasar in less than a second. That's inflation! The current expansion rate of the universe is much slower. Only 67 kilometers per second over a distance of 3.26 million light years. Apr 6 '16 at 7:42

In physics the "speed" of anything depends on the coordinate system you choose, since speed is measured as change in coordinate position in some interval of coordinate time. Even in the special theory of relativity, which doesn't take into account gravity and hence involves no spacetime curvature, the notion that the speed of light is always equal to the same constant (labeled $c$ in physics and astronomy) would only true in a special class of coordinate systems known as inertial frames, it is quite possible to define a "non-inertial" coordinate system in special relativity such as Rindler coordinates in which the speed of light does not have the same value $c$. In the general theory of relativity, which models gravity in terms of mass/energy curving spacetime, you can only have "local inertial frames" defined on very small patches of spacetime (specifically, the limit as the size approaches zero)--see this article on the "equivalence principle" for the conceptual details of how local inertial frames can be defined by observers in freefall measuring events in their immediate neighborhood (like an observer looking at events within an elevator in freefall). Such observers will always measure the local speed of light within a vacuum in their local region to be equal to $c$, regardless of the larger-scale properties of the spacetime they're embedded in, like the "expansion of space".

But if you try to define a global coordinate system on a large region of curved spacetime, this coordinate system is always a non-inertial one, so there is no guarantee that the coordinate speed of light in this coordinate system will be equal to $c$, and indeed the coordinate speed of light may vary from one region of spacetime to another depending on what coordinate system you choose (the equations of general relativity work in all smooth coordinate systems, as long as you define the metric correctly relative to your chosen coordinate system). In the basic model of curved spacetime in cosmology (the FLRW model), the simplifying assumption is made that matter is a sort of uniform fluid filling all of space, so that if you pick the right definition of simultaneity (multiple definitions are always possible in relativity due to the relativity of simultaneity), you will find that the density of this fluid is identical at every point in space at any given moment of cosmic time. This obviously isn't completely true to life, but it's expected that on large scales the density of matter is close to uniform at any given cosmic time, so it's seen as a reasonable approximation. The expansion of space basically means that the density of the fluid gets lower as time passes, and that if two objects are at rest relative to the local fluid in their immediate neighborhood, then the proper distance between them will grow with time (proper distance corresponds to what you would measure if you laid a bunch of short rulers end-to-end between the two objects at a particular moment in time, and then added up the distances).

As it happens, this cosmological model has a further nice feature (as discussed in the 'proper distance' link above which is based on this paper, see p. 99 of the 'Full Refereed Journal' link). The most "natural" coordinate system to use in this model is one in which the time coordinate corresponds to the proper time measured by a set of observers who have been at rest relative to the cosmic fluid since the big bang, and the spatial coordinate is such that the coordinate distance between any such observers at a given time corresponds to their proper distance at that time. If you use such a system, it works out that the overall coordinate velocity of any object can be broken down into a sum of two velocities:

1. The "recession velocity" at any given space, which is the velocity that an observer at rest relative to the cosmic fluid would be moving (the rate that their proper distance from the origin of the coordinate system is growing as a function of time, where we can assume the origin corresponds to our own location in space).

2. The "peculiar velocity" of any object which is not at rest relative to the cosmic fluid, which is just the same as the velocity of that object as measured in the local inertial frame of an observer at the same location who is at rest relative to the cosmic fluid. So, the peculiar velocity of a light ray must always be $c$.

So, if we know the recession velocity $v_{rec}$ at some distant location in space, then a light ray emitted directly towards us from that location will have an overall velocity $v_{rec} - c$ in this coordinate system, and a light ray emitted directly away from us will have an overall velocity $v_{rec} + c$. So from the perspective of this coordinate system, it makes sense to say as a shorthand that the light itself always travels at $c$, but space is also expanding away from us and this accounts for why the light is redshifted, and also why light originally emitted at distance $d$ won't necessarily take a time of $d/c$ to reach our own location. But this neat way of describing things is specific to both the cosmological model being assumed and the coordinate system used, things may not work out so neatly in other choices of spacetime or other coordinate systems. The only really general statement you can make about the speed of light is the one I mentioned earlier, that regardless of what global coordinate system you use and what the speed of a light ray works out to be in that system, it's always true that in a local inertial frame defined on a small patch of spacetime, light traveling through that patch always has a speed of $c$ as measured in that local frame.

• Thanks for the great answer. I was astounded to just learn from a question I asked astronomy.stackexchange.com/a/18610/13071 that the recession velocity between us and Andromeda is indeed just about the same as the peculiar velocity (I assumed it would be vastly smaller, a million or billion times smaller). Oct 6 '16 at 16:23

To say the "speed of light" is (mostly) about light is a common misunderstanding of that speed. It's actually the speed of causality, and it has far more implications that merely how fast photons propagate. It is also the conversion factor from mass to energy. (E = mc^2)

When space expands the photons within it don't suddenly go faster to cover the expansion. Think of it this way. Let's say you have a beetle crawling across a rubber sheet and you stretch that rubber sheet to double its original size. Does the beetle suddenly start crawling twice as fast? No, the beetle still plods along at its same speed. Just like light crossing the universe. The universe expands and light has to take more time.

• but you are only guessing that light travelling through stretched space is like a beetle crossing a rubber band. It may be more like a frog hopping across pebbles at a constant hop rate. As the pebbles stretch out the frog crosses the pond more quickly. Dec 7 '18 at 19:01