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I've recently read that what most people learned to think of as the 'speed of light' is actually the 'speed of causality', and that light just happens to travel at that speed (through free-space.) I'm also aware that light travels measurably slower through transparent materials such as water or glass.

This leads me to ask: is light traveling slower through such materials, because the speed of causality is reduced in such materials? And if so, what does this tell us about how the speed of causality is determined by permittivity and permeability of various materials?

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    $\begingroup$ What about the particles that travel faster than light through a given material? $\endgroup$
    – Triatticus
    Nov 12, 2023 at 23:51
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    $\begingroup$ @Triatticus That sounds like the beginning of a decent answer. $\endgroup$ Nov 13, 2023 at 0:33
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    $\begingroup$ Although I've used the phrase "speed of causality" myself, it's perhaps a little too metaphysical. As I said in physics.stackexchange.com/a/291346/123208 c is fundamentally a scaling factor that relates pure time-like intervals to pure space-like intervals. $\endgroup$
    – PM 2Ring
    Nov 13, 2023 at 4:13

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The speed of causality in water is still $c$. Although light does not travel at $c$ in water, it is possible to have particles travel through water faster than light does. Such particles emit a specific type of radiation called Cherenkov radiation shown here

blue glow of Cherenkov radiation

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    $\begingroup$ I have always been captivated by the beauty of Cherenkov radiation, but despite my best efforts my partner refuses to allow a nuclear reactor in our home. $\endgroup$
    – J. Murray
    Nov 13, 2023 at 1:17
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    $\begingroup$ I think it is important to note that even in matter light really propagates at the speed of causality, $c$, but its effective optical speed, so to speak is less than $c$. This is because between two consecutive scatterings on atoms/molecules/crystal lattice, etc. the speed is $c$, but due to the multiple scatter the effective speed gets slowed down, and this what is observed. $\endgroup$
    – hyportnex
    Nov 13, 2023 at 1:32
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    $\begingroup$ @hyportnex - In more standard terminology, one could say that "photons" propagate at the speed of causality, but "beams of light" (which consist not just of photons, but also of briefly excited matter) don't. It's analogous with the speed of a "free electron" versus speed of "electricity" in a conductor. $\endgroup$ Nov 13, 2023 at 6:40
  • $\begingroup$ @JirkaHanika to the extent I understand this, and I admit readily that is not very deep, photons do not propagate, and I try to avoid that potentially terminology; see physics.stackexchange.com/questions/90646/… $\endgroup$
    – hyportnex
    Nov 13, 2023 at 14:56
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    $\begingroup$ For everyone, I purposely used the word “light” in my answer instead of “photon”. It is easier to avoid all of that just by treating the light classically. $\endgroup$
    – Dale
    Nov 13, 2023 at 22:36
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The "speed of light" inside a medium refers to different things, so they can't all be the speed of causality.

First, there are the group and phase velocities. In a medium where the index of refraction changes based on frequency (a dispersive medium), then these two velocities are different. The group velocity is often touted as the speed that information can travel because it is the speed at which the modulation of a signal travels.

But neither of those is the "speed of causality". The initial front of a new signal through any medium still has to travel at $c$.

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  • $\begingroup$ I'm not 100% sure how to interpret 'initial front'. I want to guess that you mean the interface where the light (signal) first meets the new material (or exits it.) $\endgroup$ Nov 13, 2023 at 4:41
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    $\begingroup$ @CognitiveHazard Yes, where the intensity of the wave first goes from zero to non-zero. There is a reference on this page, at least: en.m.wikipedia.org/wiki/Front_velocity $\endgroup$ Nov 13, 2023 at 8:01
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    $\begingroup$ +1 for mentioning the front velocity. Various experiments have verified that in some materials the group velocity of laser pulses can be larger than $c$. This is not a fundamental problem: no information propagates faster than light. However, front velocity carries information and so it must always be less than $c$. $\endgroup$
    – Quillo
    Nov 13, 2023 at 8:13
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Although your question only concerns water and other such media, it can also be directly associated with Maxwell’s equations showing that c is inversely proportional to the permeability and permittivity of free space. The permittivity of space can be thought of as the resistance of space to the formation of fields, or perhaps as the viscosity of space. Paraphrasing from Arvin Ash; Why are µ0 and ɛ0 these exact values? These are the constants of nature. These are properties of free space that tell us how fast magnetic fields and electric fields can interact with each other. This sets a limit on how fast these fields can propagate through space. In a different substance, or in a different universe, these constants could be different.

Of course we know that the permeability and permittivity of space changes near any gravitational body where time slows. Unlike in water, in this case the speed of causality changes along with c.

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Different wavelengths of visible light travel in water at different speeds. This is why we have rainbows (as well as other important things).

Does it mean that causuality for red events happens at different pace than the causuality of blue events?

Water (as well as most other materials) almost have no refraction for X-rays and even less for gamma rays. Do we have a separate X-ray causuality?

Not really.

The familiar c is the speed of causuality everywhere, it is the light that takes small rests at favorable atoms.

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  • $\begingroup$ "Does it mean that causuality for red events happens at different pace than the causuality of blue events?" Excellent didactic point. Thank you. $\endgroup$ Nov 14, 2023 at 5:18
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The main thing to note is that when electromagnetic waves propagate in some medium other than vacuum, then the medium itself may be driven by the electromagnetic waves into some wave-like oscillation. For materials such as glass and water this is what happens. The consequence is that although we say in everyday language that "light" is propagating through the glass, or "light" is propagating through the water, the situation in full is not quite as simple as that. Really a wave of polarization of the material is propagating along with a wave of disturbance of the electromagnetic field. The overall result is a wave whose speed can be less than the maximum speed of causality. It is ok in everyday language to refer to this wave as "light" but in more careful physical analysis it would be called a "wave of electromagnetic field and material polarization".

The causal phenomena going amongst all the atoms or molecules are still going at the maximum speed, and in such a way as to keep everything in step in this wave of electromagnetic field and material polarization.

You can also have other things moving through the medium at faster than the speed $c/n$ (where the refractive index $n > 1$), but slower than $c$. For example: electrons, protons, neutrons, neutrinos, or anything in principle (if it does not disturb the medium too much). Also if $n \simeq 1.5$ is the refractive index at optical frequencies, you can still have electromagnetic waves of some other frequency (e.g. gamma rays) propagating at $c$ in the medium.

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