[EvaH (OP)]

John asks:

How can the speed of light be constant when it supposedly slows down passing through a dense liquid?

What do you think?

[Bored chemist]

The speed of light in a vacuum is constant.

[Halc]

And in more general terms, the local speed of light (in a vacuum) is constant.

For example, consider a really tall building of some height that's known quite accurately.  One guy sends a light pulse from the bottom to the top and back (using a mirror up there) and gets time X which is slightly less than twice the height of the building divided by c.  Another guy does the same from above using a mirror at the bottom and gets time Y which is slightly more than twice the height of the building divided by c. This is a non-local test, sending light into regions of different gravitational potentials where their respective clocks run at different rates, so of course X cannot equal Y and thus the computed non-local speed of light (time to make identical trips over identical distances) cannot be measured identically by each person.

[Kryptid]

This is a non-local test, sending light into regions of different gravitational potentials where their respective clocks run at different rates, so of course X cannot equal Y and thus the computed non-local speed of light (time to make identical trips over identical distances) cannot be measured identically by each person.

Interesting. Does this mean that the speed of light is affected by gravity? Or, alternatively, could it be interpreted that the building is measured to be a different height by the two different observers?

[RobC]

Light travels about 186,000 miles/sec. in a vacuum. Why this speed, and not some other speed?

[Janus]

This is a non-local test, sending light into regions of different gravitational potentials where their respective clocks run at different rates, so of course X cannot equal Y and thus the computed non-local speed of light (time to make identical trips over identical distances) cannot be measured identically by each person.

Interesting. Does this mean that the speed of light is affected by gravity? Or, alternatively, could it be interpreted that the building is measured to be a different height by the two different observers?

It means the the "coordinate speed" for light varies with position in the Gravitational field. The local or proper speed of light is always c.
Thus on the top floor of the building you will measure light locally as traveling at c, but the coordinate speed of light at the ground level would be less than c. 
On the ground floor you would measure the local speed of light as being c, but the coordinate speed for light at the top floor would be greater than c.
The height difference between top and bottom floors would be measured as being the same for both observers.

Look at it this way:  Each of you have an an ideal clock and a local light clock.  Your ideal clock and light clock are in sync.
The top floor observer will measure light as taking 2L/c for his light clock to complete one tick, and his ideal clock is set to match this tick rate.  The ground observer also has to agree that the top floor ideal clock and light clock remain in sync.  But due to gravitational time dilation, that the ideal clock ticks fast compared to his own, This means that the light clock must tick fast also.  Since L is the same for both light clocks, this can only happen is the light bouncing back and forth in the top floor light clock does so faster than c.
Conversely, the upper floor observer would conclude that the light bouncing back and forth in the ground floor light clock is doing so at less than c.

[Bored chemist]

Light travels about 186,000 miles/sec. in a vacuum. Why this speed, and not some other speed?

Because it depends on the electromagnetic properties of a vacuum.
https://en.wikipedia.org/wiki/Electromagnetic_wave_equation

[RobC]

The link gave the following - "Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light".

It is a criticism of Wiki that it tends to explain concepts to people who already understand them. Typically not defining the constituents of the formulae.

The derivation of the speed of light was not clear to me.

Is there a simpler explanation?

[Halc]

Interesting. Does this mean that the speed of light is affected by gravity?

It is frame dependent, and gravity is part of your frame.  Light slowing down in a gravity well (effectively stopping in coordinate speed while approaching a black hole event horizon) is part of what makes light refract around a massive body.

Or, alternatively, could it be interpreted that the building is measured to be a different height by the two different observers?

They share the same tape measure. I suppose if you really tried you could frame it in such a way. Admittedly they don't measure the distance into a strong gravity well by dangling a tape measure into it, but rather by running the tape measure in a circle at constant altitude and then dividing that by 2π.  Anyway, I don't see how our two different observers are going to disagree about the building height so long as they use the same method to measure it.  They have to measure different elapsed times since the lower clock is more dilated, so they're going to measure different speeds of light.

Another example is very distant light due to space expansion.  Near the Hubble radius, light shone our way is going to move at c local to the local frame there, but that source is receding from Earth at c (Hubble's law), so the distance between that light when sent and an hour later is no different.  Light coming our way is effectively stopped even though it's locally moving at c.  Light shining the other way recedes from us at 2c or more, just like the stars.  All this is dependent on the coordinate system used to measure time and distance, but an inertial coordinate system is not an option since there cannot be an event horizon in such a frame, and there is very much a real event horizon out there beyond which light can never get here no matter how long you wait.

[alancalverd]

The link gave the following - "Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light".

It is a criticism of Wiki that it tends to explain concepts to people who already understand them. Typically not defining the constituents of the formulae.

The derivation of the speed of light was not clear to me.

Is there a simpler explanation?

We know that a moving charge generates a magnetic field, and a changing magnetic field induces charge (Oersted, Faraday). Maxwell combined and generalised these experimental findings and deduced that an electromagnetic wave can be generated by an accelerating charge, and is selfpropagating in vacuo.

This turns out to be true by experiment (Hertz, Marconi et al...) and the constants that determine the speed of propagation are obviously universal and independent of direction since a vacuum is inherently isotropic and all vacua have identical properties.   

[evan_au]

Is there a simpler explanation?

Probably the simplest equation to calculate the speed of light in a vacuum:
c = 1/√(ε₀μ₀)
where:
- c: velocity of light in a vacuum
- ε₀: permittivity of a vacuum
- μ₀: permeability of a vacuum
Look at just the last 2 lines of this section: https://en.wikipedia.org/wiki/Electromagnetic_wave_equation#The_origin_of_the_electromagnetic_wave_equation

This is just as simple as the equation for the speed of a vibration on a string (eg a guitar string):
v = √(T/ρ)
where:
- v: velocity of a wave in the string
- T: tension of the string
- ρ: density of the string
Look at just the last 3 lines of this section: https://en.wikipedia.org/wiki/String_vibration#Derivation


For more information:
https://en.wikipedia.org/wiki/Vacuum_permittivity
https://en.wikipedia.org/wiki/Vacuum_permeability

[Bored chemist]

Is there a simpler explanation?

Probably the simplest equation to calculate the speed of light in a vacuum:
c = 1/√(ε₀μ₀)
where:
- c: velocity of light in a vacuum
- ε₀: permittivity of a vacuum
- μ₀: permeability of a vacuum
Look at just the last 2 lines of this section: https://en.wikipedia.org/wiki/Electromagnetic_wave_equation#The_origin_of_the_electromagnetic_wave_equation

This is just as simple as the equation for the speed of a vibration on a string (eg a guitar string):
v = √(T/ρ)
where:
- v: velocity of a wave in the string
- T: tension of the string
- ρ: density of the string
Look at just the last 3 lines of this section: https://en.wikipedia.org/wiki/String_vibration#Derivation


For more information:
https://en.wikipedia.org/wiki/Vacuum_permittivity
https://en.wikipedia.org/wiki/Vacuum_permeability

Nice analogy.

[PmbPhy]

Interesting. Does this mean that the speed of light is affected by gravity? Or, alternatively, could it be interpreted that the building is measured to be a different height by the two different observers?

Yes. See; http://www.speed-light.info/speed_of_light_gravity.htm

[Yahya A.Sharif]

Speed of light in vacuum is constant .Speed in a liquid is slow but also constant.speed of light doesn't slow down , it has constant speeds for each medium

[alancalverd]

Interesting. How do ε and μ relate to speed?

[evan_au]

How can the speed of light be constant when it supposedly slows down passing through a dense liquid?

c = 1/√(ε₀μ₀)

In a medium such as water, the permittivity is no longer ε₀, but some larger value.
- How much larger is given by relative permittivity ε_r (ε_r = 1.77 for visible light in water)
- With a larger value on the denominator, the speed of light is lower in that material
- Its similar with permeability and μ_r (only μ_r for water is almost identical to a vacuum)
See: https://en.wikipedia.org/wiki/Relative_permittivity
https://en.wikipedia.org/wiki/Permeability_(electromagnetism)

Does this mean that the speed of light is affected by gravity?

From the viewpoint of a distant observer watching events deep in a gravitational well:
- The speed of light will appear slower
- The speed of a vibration in a string will appear slower
- Both are affected by time dilation deep in a gravitational well.

Edit: fix broken URL

[Bored chemist]

Interesting. How do ε and μ relate to speed?

It's still the reciprocal of the square root of the product, just like I cited earlier.

Light travels about 186,000 miles/sec. in a vacuum. Why this speed, and not some other speed?

Because it depends on the electromagnetic properties of a vacuum.
https://en.wikipedia.org/wiki/Electromagnetic_wave_equation

And as Evan pointed out here

c = 1/√(ε0μ0)

[alancalverd]

Interesting. How do ε and μ relate to speed?

It's still the reciprocal of the square root of the product, just like I cited earlier.

Not sure you have grasped the right end of the stick there, BC.

Halc said the speed of light in a medium depended on the speed of the medium.

Now we know that c= √1/(μ₀ε₀) and s_m = √1/(μ_mμ₀ε_mε₀), so the question is how do μ_m and ε_m, which are dimensionless quantities, vary with the speed of flow of the medium m, and speed with respect to what?

[evan_au]

it supposedly slows down passing through a dense liquid?

You can see the effects of this in a rainbow.
- Red and blue light slow down by (slightly) different amounts
- So sunlight passing through water droplets in the air are diffracted refracted by (slightly) different angles
- This separates out sunlight into its constituent rainbow of colors

The same effect was recently used to estimate the density of matter in intergalactic space
- Fast Radio Bursts generate a wideband radio signal that lasts about 1 or 2 ms at the source
- As it travels through the extremely good vacuum of intergalactic space, the low frequencies travel slightly slower than the high frequencies
- So the radio signal when it arrives at Earth has the low frequencies delayed by seconds to a minute out of a billion years or so travel time, so the effect is very subtle
- They used a prototype for the Square Kilometer Array project to identify the source for these radio bursts.
- This allowed the researchers to locate half of the universe's baryonic matter as a cold neutral gas in intergalactic space; they estimate a density of about 1 atom in the volume of a typical residential room.
- Because it is neutral, cold, and low density, it had been impossible to measure using previous techniques.
See: https://www.abc.net.au/news/2020-05-28/astronomers-find-universe-missing-matter/12291788

Correction: Thanks to hamdani yusuf

[alancalverd]

Granted that ε and μ are frequency-dependent in a medium, as noted by Newton*, but that still doesn't address Halc's assertion that the propagation speed of light depends on the speed of the medium.

*Another mystery. Newton bought his prism from a stall in Cambridge market (the market still exists and is reputedly the oldest 7-day charter market in Europe). If nobody had noticed refractive dispersal before 1666, why was anyone making and selling glass prisms in a general market?