[hamdani yusuf]

Which doesn't imply a variation in c.

If we insist that speed of light in vacuum is constant while also accepting that space is stretching, it implies that we also need to stretch the time by the same amount. So far, I haven't found any source for the latter.

[alancalverd]

We don't insist, we define c  to be a constant. Then we can measure everything else.

[Kryptid]

If we insist that speed of light in vacuum is constant while also accepting that space is stretching, it implies that we also need to stretch the time by the same amount. So far, I haven't found any source for the latter.

That doesn't follow. It takes light one second to travel 299,792,458 meters. Of course, over the course of one second, a portion of space of that distance will expand to be ever-so-slightly longer than 299,792,458 meters. As such, it will take ever-so-slightly longer than one second for light to travel that distance because it can only travel 299,792,458 meters in one second.

[Halc]

If we insist that speed of light in vacuum is constant while also accepting that space is stretching

Nobody insists any such thing. Per relativity theory, light moves at c only relative to a flat (Minkowskian) inertial frame, and an expanding metric is not an inertial one, and probably isn't Minkowskian either.
Since spacetime is locally flat anywhere, light moves locally at c.

Similar stories for other kinds of non-inertial metrics like a rotating frame or an accelerating one, either of which can also be mapped onto flat Minkowskian spacetime.

As such, it will take ever-so-slightly longer than one second for light to travel that distance because it can only travel 299,792,458 meters in one second.

A more pointed example illustrating the non-constant speed of non-local light in an expanding metric:

Light emitted from the creation of the hydrogen in your body was emitted near 'here' at the recombination event 13.8 billion years ago and is currently ~45 GLR away, meaning it has averaged a velocity of over 3c. Meanwhile, the CMB light that we detect here today was emitted 13.8 billion years ago from material that was at a proper distance that was much closer then than where Andromeda is now. which is an average velocity of around 0.0001c

An example closer to home which doesn't involve expanding space: If you put a reflector on the surface of Mars and Mercury, light shone to Mercury will make the round trip at a speed a bit less that c, and the round trip to Mars at a speed a bit above c.Both are as measured by an Earth observer, and both are examples of light speed being dependent on gravitational potential.

[Eternal Student]

Hi.

It takes light one second to travel 299,792,458 meters. Of course, over the course of one second, a portion of space of that distance will expand to be ever-so-slightly longer than 299,792,458 meters. As such, it will take ever-so-slightly longer than one second for light to travel that distance because it can only travel 299,792,458 meters in one second.

      I'm not sure if you meant to do it but this is actually a perfect way of suggesting that light will change its speed. 

    Let's keep it simple and have light travel at 3 x 10⁸ m/s precisely.    Let's assume a constant rate of expansion for space set at 10% per second.
    Let's have some light emitted out to space by a man with a flashlight on a planet.   We'll assume the planet is co-moving with the expansion of space.   We also ought to make it clear whose time we're using,  we'll use co-moving time, so that will be the time recorded by a stopwatch that stays with the man on the planet.
    Now, you ( @Kryptid ) claim light will travel 3 x 10⁸ m  after 1 second, that's the distance you can measure between the planet and the light after 1 second.    The problem is, the space which it has covered will now also start to expand.   So after 2 seconds,  the light has travelled on a bit further.   However the total distance between the planet and the light will be a bit more than  6  x 10^{8 m/s}   (2 lots of c).   Specifically, it will now be this far away from the planet:       
           3 x 10⁸             +          (1.1  x  3 x10⁸)                      =           6.3 x 10⁸ m
[Distance moved t=1 to t=2]   +   [110% of the distance travelled t=0 and t=1]
   
    Between t=1 and t=2, which is just 1 second for the man on the planet,  the light became  3.3 x 10⁸ m  further away from the planet,   so it's average speed over that second was   110% of c.

    A similar calculation shows that the average speed of the light would be 3.63 x 10⁸ m/s between t=2 and t=3.   In general the speed of light seems to be increasing by 10% every second.

    Anyway, it can be some fun to think about it for a while.    It's fixable but usually leaves us forced to talk about the movement of light through that bit of space which is local to it.   Specifically we cannot consider the rate of change of physical distance from any point that is even just slightly remote from the particle  - which can take a page to fully explain and develop the idea.  [ Sean Carroll in "Spacetime and geometry" takes most of a chapter].
    When we say light has the speed 3 x 10⁸ m/s we don't really mean that it will be 3x10⁸ metres away from where it was emitted after 1 second.   We wouldn't be able to say anything about the physical distance between the light and the place of emission unless we also knew something about the curvature and especially the evolution of the scale factor in this region of space and over this time.
    So, that can be used as an alternative way to mitigate the earlier comment from @hamdani yusuf

If we insist that speed of light in vacuum is constant while also accepting that space is stretching, it implies that we also need to stretch the time by the same amount. So far, I haven't found any source for the latter.

   No, we don't.   Mostly because the speed of light isn't as restrictive as you might imagine.   It's a description of the movement of the light through space that is absolutely local to it (which is a different bit of space at every instant of time).   It certainly isn't enough on its own to tell us about the physical distance between the light and any place that is remote from that light.   If space is expanding rapidly with time, this distance (from the light to the place of emission) can grow much faster than 3 x 10⁸ m/s, that isn't a problem and that does happen.  We do not need to scale time in any way because we are under no obligation to keep that rate of change at 3 x 10⁸ m/s,  the speed of light never meant that, it only meant that light has a local speed of 3 x 10⁸ m/s.

Best Wishes.

[Overlap with @Halc who posted just before I did.   At a glance, we have said similar things, just in different ways and it won't hurt to have two different styles of explanation].

[alancalverd]

That doesn't follow. It takes light one second to travel 299,792,458 meters. Of course, over the course of one second, a portion of space of that distance will expand to be ever-so-slightly longer than 299,792,458 meters. As such, it will take ever-so-slightly longer than one second for light to travel that distance because it can only travel 299,792,458 meters in one second.

Er, no. A second is defined by the frequency of an atomic (currently cesium) clock, and a meter is defined as the distance that light travels in one somethingth of a second, so if c were to vary, our measure of distance would change by the same amount and we wouldn't know about it.

Except that if c were to increase, the number associated with the length of the stick in your hand would decrease.

So the abandonment of material standards means that we no longer have the ability to determine whether c is constant!

[Eternal Student]

Hi.

   First of all, I think 3 people in succession have now said something slightly opposing @Kryptid .   So, just to be clear, we don't dislike you @Kryptid .    Quite the contrary, people probably wouldn't have bothered making comments if they didn't think you and your ideas were sensible and could be amenable to discussion.

    So, let's challenge the comment made by @alancalverd slightly instead.   It's not his fault, it's mainly just the "SI council" of people who make changes to the measurement system are often about 10 years behind the state of the art in science.  (I don't know if they are called the 'SI council' but that's what I will refer to them as).   I suppose they have their reasons, for example why change a system that works well enough until there's a good reason for it, especially if it could be expensive for all instititutions to re-callibrate all their equipment etc.   They may also feel that certain things are simply beyond their jurisdiction and control.   For example they probably concentrate on providing a system that works here on Earh and over a few years that we can call "now" rather than suggesting all precise callibrations and definitions refer only to a region of outer space that is well behaved.

    Anyway, this is how it is:   The SI council changed the definition of a metre much as described by @alancalverd .   That happened in 1983 and was extremely useful and sensible because it puts things on a footing that agrees nicely with the theory of special relativity.   Sadly, it did NOT pay a lot of attention to the later theory of General Relativity and has left a lot of "mud in the water".
    So that's a fairly bold claim and I'd better have some evidence for this instead of just walking away.

1.  It's not just me who thinks the current SI definition of the metre fails to recognise the impact or role of General Relativity.   The CIPM (International committee for weights and measures) are aware of these limitations.

The CIPM issued a clarification in 2002:

    Its (..the speed of light..) definition, therefore, applies only within a spatial extent sufficiently small that the effects of the non-uniformity of the gravitational field can be ignored (note that, at the surface of the Earth, this effect in the vertical direction is about 1 part in 1016 per metre). In this case, the effects to be taken into account are those of special relativity only.

   (The CIPM)   ....Considers the metre to be a unit of proper length and thus recommends this definition be restricted to "lengths ℓ which are sufficiently short for the effects predicted by general relativity to be negligible with respect to the uncertainties of realisation"....

Various extracts from  https://en.wikipedia.org/wiki/Metre#Speed_of_light_definition

2.     Credible physicists have also written about this:
    This is an article by various contributors and held on the University of California website:
https://math.ucr.edu/home/baez/physics/Relativity/SpeedOfLight/speed_of_light.html
     
   We'll only need a few sentences from it:
.....in the presence of more complicated frames and/or gravity, relativity generally relinquishes the whole concept of a distant object having a well-defined speed.  As a result, it's often said in relativity that light always has speed c, because only when light is right next to an observer can he measure its speed? which will then be c.  When light is far away, its speed becomes ill-defined.  But it's not a great idea to say that in this situation "light everywhere has speed c", because that phrase can give the impression that we can always make measurements of distant speeds, with those measurements yielding a value of c.  But no, we generally can't make those measurements.  And the stronger gravity is, the more ill-defined a continuum of observers becomes, and so the more ill-defined it becomes to have any good definition of speed.

     I'll need to elaborate on this a bit:   If we are considering a solution to the EFE with a scale factor that changes significantly every second,  then even 1 light second, is still too far away to be able to treat this as the local vicinity of the light now.   We can't just measure the physical distance between the light and the point of emission and divide by 1 second.    Where we seek precision over the definition of the speed of light, or the metre, then even 1/300 000 000 th of a second is too long and has allowed the light to move too far away.   Specifically, the SI definition of the metre works here on Earth and right now only because the scale factor won't have changed very much over  1/300 000 000  of a second in this region of space.


3.     Let's just see how the physical distance from the point of emission and the particle of light evolves from first principles.
        We'll have a simple  FRW metric for a universe with 0 spatial curvature (which is what we think we do have in our universe). 
        dS²   =   a(t)² ( dr² + r² dΩ² )  - c²dt²

[Reference:  https://en.wikipedia.org/wiki/Friedmann%E2%80%93Lema%C3%AEtre%E2%80%93Robertson%E2%80%93Walker_metric#General_metric ]

   a(t) = scale factor;       r = reduced circumference co-ordinate   ---> it's a co-moving co-ordinate based on a radius as used in spherical polar co-ordinates upto some constants or choice of units;     dΩ² is based on the polar and azimuthal angles for the co-moving co-ordinate system,  dΩ² = dθ² + Sin² θ dφ².
    We'll centre the co-moving spherical co-ordinate system at the place of emission of the light at t=0 and will only be considering light that progresses along the radial direction, so dθ = dφ = 0 and the metric simplifies to:
     dS²   =   a(t)² dr²    - c²dt²

  Light is a null ray,   dS² = 0,    so  a(t)² dr²   =  c²dt².   taking the positive square root is appropriate for an outward travelling ray which is what we will be considering  (the light was emitted at r=0 and t=0 and travels outward).   So we have, 

       dr   =    c dt / a(t)         

   That's the relationship between a differential change in time, dt, and a differential change in radial co-ordinate, dr.   This holds at all times and at any place along the path taken by the light.   Note that this t and r are the co-moving times and r co-ordinate.

    So from time t=0 to t = an arbitary t,  the  total change in the co-moving r co-ordinate for the particle of light is given by a simple integration.....

(... LaTeX mathematics support is still not working....so it's not so simple....  I'll just insert an image....)


Delta r.jpg (8.76 kB . 438x126 - viewed 602 times)

The initial radial co-ordinate of the light  r(at t=0) = 0,   the radial co-ordinate of the light at time = t  is r(t) = R_H    (I've got to call it that, that's the image I could find - they were calculating a cosmological horizon so a letter H made sense for them).

   Recall that this, r(t) is a co-moving co-ordinate,   we multiply by  a(t)  = the scale factor at time t,  to obtain the physical distance  R_H(t)    between the  particle of light and the place of emission at time=t.


Delta R.png (14.25 kB . 528x139 - viewed 568 times)

In the situation where a(t) is strictly increasing with time,  i.e.  we say the universe is expanding,   then the integrand above is strictly greater than  dT / a(t)  while  T<t.       ( I can't get the tilda above the t and have used a capital T,   just evaluate the integral knowing that  a(T) < a(t)  while  T varies between 0 and t  in the integral ).   
    We have (omitting limits on the integral because LaTeX isn't avalaible),     a(t) . (1/a(t)) . c . ∫ dT    =   c ∫dT    =    ct .
   So we have  R_H  =  the physical distance from the point of emission to the particle of light  >  ct.   Hence the speed of the light  (if we define it as physical distance between point of emission and the particle of light after 1 second)    >  c.
    We get equality if the scale factor a(T) in the aforementioned integrals remained effectively constant throughout the whole time between t=0 and t.   If the universe expanded during this time interval, then we don't.    Hence, the SI definition of the metre doesn't actually work when you consider General Relativity and an expanding universe.  I mean it is what it is - that's how they defined it and what we have to use - but it just doesn't quite define what we would really want it to define.   It gives a satisfactory definition of a metre and an appropriate speed of light follows from this ONLY in a region of spacetime (which means over a small time and over a small region of space) where the scale factor can be assumed to hold constant.
    It turns out that having an expanding universe is only one way in which GR challenges the SI definition of a metre.   Other spatial curvatures (which are often described as gravitational fields) can also present problems.

    The safest way to talk about "the speed of light" is to refer only to what someone could measure as a differential rate of change while they are right at the same location as the light.   The SI definition of the metre is just some  "mud in the water".    We shouldn't measure the distance from an initial position to a final position, not even over a tiny time interval like 1/300 000 000  th of a second   because GR suggests that various things about the metric or gravity can still mess up that distance.
    Maybe the SI council will adapt the way a metre or the speed of light is defined in the future but provided the Universe doesn't suddenly return to expansion rates like the epoch of inflation or become very close to an extreme source like a gravitation like a black hole, then what we have as the definition of the metre will, I suppose, just have to do.
    Anyway, that's why it might seem like there are at least two slightly different sorts of replies that may seem slightly contradictory:
     The General Relativists   (e.g.  @Halc )   suggesting the speed of light is only a local thing.   Also me who will stand by the statement that in 1/300 000 000 of a second light may not have travelled 1m  of physical distance from where it was, we need to know about gravity or the evolution of the metric before we could determine that physical distance (e.g. assume the scale factor ≈ constant).   
     The Practical-ists   (e.g. @alancalverd )  suggesting that the metre is the metre as set by the SI council and that's the deal.   The speed of light follows.   So, regardless of where you are, how fast the universe is expanding, or the existence of unsual gravitational fields in the region, light will have travelled 1 m  after 1/300 000 000   of a second.
   
Best Wishes.

[Halc]

and a meter is defined as the distance that light travels in one somethingth of a second

I think I can compress ES's post to a short line.
A meter is what light travels in one somethingth of a second relative to an inertial frame.

The original official definition might have omitted that presumption since one can get light to move pretty much any distance you want in a second if the right kind of non-inertial frame is chosen.

because only when light is right next to an observer can he measure its speed? which will then be c.

Technically, even then the speed cannot be measured, and is only presumed. A meter is best measured by light reflected from one end to the other and back, in two somethingths of a second. Since an inertial frame is required for this, obviously you can't have both ends of the meter stick be comoving at once.

Your quote refers to non-inertial frames as "more complicated frames", and any expanding frame would be one of those.

[paul cotter]

Although I have followed the various metre arguments I am now getting confused. I think i'll return to the avoirdupois system- maybe that Parisian standards lab would sell me that hunk of platinum/iridium at a knock down price as they have no further use for it?

[Eternal Student]

Hi.

I think I can compress ES's post to a short line.

   I do like it and it is a bit shorter than my post.   I did have to read the line twice, make inertial frame a bit more obvious and I wouldn't have had to.

Best Wishes.   

[hamdani yusuf]

We don't insist, we define c  to be a constant. Then we can measure everything else.

What is the difference?

[hamdani yusuf]

Light emitted from the creation of the hydrogen in your body was emitted near 'here' at the recombination event 13.8 billion years ago and is currently ~45 GLR away, meaning it has averaged a velocity of over 3c. Meanwhile, the CMB light that we detect here today was emitted 13.8 billion years ago from material that was at a proper distance that was much closer then than where Andromeda is now. which is an average velocity of around 0.0001c

What causes the asymmetrical difference? It speeds up in one case, while slows down in the other case?

[alancalverd]

We don't insist, we define c  to be a constant. Then we can measure everything else.

What is the difference?

You need to ask the UK House of Lords.

Our corrupt, incompetent government wants to spend taxpayers' money shipping invaders to Rwanda, but this would be illegal as several legitimate Rwandan refugees have been granted asylum here and you cannot deport anyone to an known unsafe country.

So instead of insisting that Rwanda is safe,  a procedure that could be tested in court or overturned by Parliament by the evidence, His Majesty's so-called Government has attempted to bypass civilised procedure by defining Rwanda as a safe place.

Anticipated developments on this theme include "black = white" and "2 + 2 = 5", by definition. You could base optics and mathematics on these axioms but not everyone is a politician, economist, or totally stupid. Problem here is that the only intelligent life in Parliament resides in the Lords, not the Commons.

However when it comes to physics, we have to start somewhere, and if we define c as constant we can derive the equations that seem to predict pretty much everything that happens in the universe, to an acceptable degree of precision.

[Eternal Student]

Hi.

What causes the asymmetrical difference? It speeds up in one case, while slows down in the other case?

    The light was trying to travel towards the place you measure distance from, or away from that place.  For an expanding space, this matters.

    The terms "speed" or "velocity" were used informally.   Ratios of change in physical distance with time were being described.   In a flat Euclidean or Minkowskian space this is exactly what we can call a speed or velocity.

Best Wishes.

[hamdani yusuf]

However when it comes to physics, we have to start somewhere, and if we define c as constant we can derive the equations that seem to predict pretty much everything that happens in the universe, to an acceptable degree of precision.

If the word "define" is replaced by "insist", would the statement become false?

[hamdani yusuf]

The light was trying to travel towards the place you measure distance from, or away from that place.  For an expanding space, this matters.

In an expanding space, everything that's stationary in the space-time continuum always moves away from one another over time.
IMO, the observer is always on the receiving end.

[alancalverd]

If the word "define" is replaced by "insist", would the statement become false?

It becomes different. This was actually the reason for the foundation of the National Physical Laboratory and international standards ever since.

I can define a meter as the length of a particular stick (indeed we did just that for many years) and if everyone signs up to the same standard, we can make things that fit together. A problem arose during the First World War when artillery shells made in Coventry didn't fit the cartridges made in London. Each factory insisted that their machines were calibrated to trading standards but it turned out that the Coventry Municipal Inch, good enough for dressmaking and housebuilding, was significantly different from the Woolwich Arsenal Inch, so the Imperial Statute Inch was defined as the one held in a London laboratory.

Subtle, but it saved lives.

[hamdani yusuf]

The required characteristics of a standard is consistency from time to time. Its value in the past should be the same as its value in the future.

[alancalverd]

Depends on its purpose. If you want to replace yesterday's screw with one made today, yes, it is important that the thread standard hasn't changed, but if you want to privatise the water supply, you have to alter the standard so that every investor can make a profit and pass on the cost of achieving adequate sterility to the  customer.

[hamdani yusuf]

Depends on its purpose. If you want to replace yesterday's screw with one made today, yes, it is important that the thread standard hasn't changed, but if you want to privatise the water supply, you have to alter the standard so that every investor can make a profit and pass on the cost of achieving adequate sterility to the  customer.

Even making good standards is just an instrumental goal, serving to help achieving the common terminal goals among the users of the standards.