How does the speed of light affect clocks?

[Sonderval]

If the speed of light would change in the universe without any other natural constant changing, would all clocks be affected in the same way by this?
This is implied by Einstein in this paper on page 368
http://myweb.rz.uni-augsburg.de/~eckern/adp/history/einstein-papers/1912_38_355-369.pdf where he says:
"Die Ganggeschwindigkeit ω der Gravitationsuhr ist also c proportional, wie dies für Uhren jeder Art der Fall sein soll." (The speed ω of the gravitational clock is thus proportional to c as it should be for clocks of all kinds.)

If this is true, how exactly would we detect a change in the speed of light (if it were variable for some reason)?

 

[Ibix]

If the speed of light would change in the universe without any other natural constant changing

It can't. At least one other has to change. As I understand it, unless that constant is one of the dimensionless constants (most likely the fine structure constant) then you can't see any change in physics - it's just a complicated way of re-defining your unit system.

 

[Sonderval]

Thanks, Ibix,
but if I change c and none of the other constants (G, ħ...), does that not imply a change in the fine structure constant?

 

[PeterDonis]

if I change c and none of the other constants (G, ħ...), does that not imply a change in the fine structure constant?

No. I can adopt units in which $c = 1$ (by defining my units of length and time appropriately) and still keep the same values for $G$ and $\hbar$ (by defining my units of mass, energy, momentum, etc. appropriately). This value of $c$ is certainly different from, for example, the SI unit value; but the fine structure constant in SI units is the same as it is in $c = 1$ units.

 

[Ibix]

Thanks, Ibix,
but if I change c and none of the other constants (G, ħ...), does that not imply a change in the fine structure constant?

I was counting the fine structure constant among "the other constants". Essentially, if you change a dimensionless constant then physics changes. If you don't, only your unit system changes. So changing c isn't interesting - as Peter says, you can always set c=1. It's changing the fine structure constant (which may or may not change c - depends on your unit choice) that is interesting.

 

[Sonderval]

It seems I was phrasing things very badly. So if I understand things correctly, the correct way of stating things would be to look at all dimensionless constants and instead of talking about "changing c" I should talk about changing these constants. So instead of saying, "let's increase c by 10%", I would look at all dimensionless constants (like fine structure) and change all the constants that depend on c in the appropriate way. So I would instead change the fine structure constant (reducing it by about 10%) and the other constants in the appropriate way.

In this case, how should I interpret the Einstein quote? Or is that a totally different thing because he is basically talking about the slowing down of clocks in a gravitational field and the "speed of light" is a map speed, not a local speed in a Lorentz frame?

 

[Dale]

So instead of saying, "let's increase c by 10%", I would look at all dimensionless constants (like fine structure) and change all the constants that depend on c in the appropriate way. So I would instead change the fine structure constant (reducing it by about 10%) and the other constants in the appropriate way.

I went through and calculated this once. If the fine structure constant changes then different types of clocks are affected differently, leading to measurably different physics. See:

https://www.physicsforums.com/showpost.php?p=2011753&postcount=55

https://www.physicsforums.com/showpost.php?p=2015734&postcount=68

However, all clocks would still time dilate appropriately.

 

[Sonderval]

@Dale Absolutely great, that was exactly what I was looking for.
PS: Would be great if you published/posted that calculation with the different constants somewhere.

 

[Dale]

@Dale Absolutely great, that was exactly what I was looking for.
PS: Would be great if you published/posted that calculation with the different constants somewhere.

Yes, I wish I had written that up. I will dig through my computer and see if I still have the file.

 

[kith]

If the fine structure constant changes then different types of clocks are affected differently, leading to measurably different physics.

This seems non-trivial to me. I don't know how this actually works but if I naively try to write down the coupling constants for the other interactions in an analoguous fashion as the fine structure constant, I get
$$\alpha = \frac{k g²}{\hbar c}$$
If this is the case and we dial up c but keep all other (dimensionful) constants the same, the strength of all interactions is dialed down the same, leaving their relative strengths unchanged.

I don't know whether QFT works as I naively suspected above and even if it does, gravity may be special with respect to this. In any case, if the physics does change when c is dialed up, there seems something non-trivial going on to me.

 

[Dale]

OK, so here are the calculations referenced above at:

https://www.physicsforums.com/showpost.php?p=2011753&postcount=55

https://www.physicsforums.com/showpost.php?p=2015734&postcount=68

I apologize in advance, they were not that hard to find on my computer, but they are a mess. It was done in Mathematica without a lot of commenting

First, I am investigating only two dimensionless constants, the fine structure constant and the gravitational coupling constant: $\alpha= \frac{e^2}{2hc\epsilon_0}$ and $\alpha_G=\frac{2\pi G m_e^2}{hc}$. I considered the impact of changing c, G, h, and $\epsilon_0$, but I left the electron and proton mass and charge unchanged.

The first comparison was the comparison of the SI meter, based on the speed of light, to a meter bar. The SI meter is equal to c times the SI second, and the SI second, based on an atomic clock, is proportional to $h/E_{hyperfine}$ where the energy of the hyperfine transition is proportional to $\alpha^4 c^2 m_e^2/m_p$. So overall the length of the SI meter is proportional to $\frac{h m_p}{c m_e^2 \alpha^4}$. Similarly, the length of a meter bar is proportional to the Bohr radius, so the length of a meter bar is proportional to $\frac{h}{m_e c \alpha}$. Simplifying, the ratio of the length of a meter bar to the length of the SI meter is proportional to $\alpha^3 m_e/m_p$ and since I was leaving the masses the same (and $m_e/m_p$ is dimensionless anyway) this works out to a comparison that does not depend on any of the dimensionful constants.

The second comparison was the SI second to a pendulum. The SI second is again proportional to the hyperfine transition energy. A pendulum clock is proportional to $\sqrt{m_{bar}^3/GM}$ where $m_{bar}$ is the length of a meter bar. I had to use Mathematica to simplify it but the ratio is proportional to $\frac{c^2 h^2 m_p\sqrt{G M \epsilon_0^5}}{e^5\sqrt{m_e}}$. I then had to make a table to show how $\alpha_G$ and $\alpha$ change when that ratio changes. It is not obvious, but it turns out that the ratio only changes when the dimensionless constants change.