Respectfully to the author, I think the answer in the article is among the most confusing ways to answer this question. The simplest answer is that the speed of light, c, is the speed with which...
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Respectfully to the author, I think the answer in the article is among the most confusing ways to answer this question.
The simplest answer is that the speed of light, c, is the speed with which electromagnetic waves propagate in vacuum. It is the inverse of the geometric mean the electric constant and the magnetic constant.
c = √(e₀u₀)⁻¹
Why is the speed of light so fast? Because the magnetic constant is a small number (1.2×10⁻⁶) and the electric constant is a way, way smaller number (8.9×10⁻¹²). So, what's the square root of the product of those numbers? 3.3×10⁻⁹. Let's invert it to get c. About 3×10⁸. Pretty close to 299,792,458 m/s.
OK, but what does it mean? Well, light is an electromagnetic wave, so let's think of fast waves and slow waves. Things that have a lot of inertia move slowly, so a pendulum with a longer chain (greater inertia) will oscillate more slowly than one with a shorter chain (less inertia). Well, the product of the electric and magnetic fields doesn't make for much effective inertia (like a really, really short pendulum), so the resulting electromagnetic wave is probably going to move pretty fast, and it does.
Relativity is neat, but completely unnecessary for answering the question asked.
Edit: I just realized that this makes the famous mass-energy balance actually intelligible. What is the mystical connection in E=mc²?
Well, let's look at the Maxwell formulation. Replacing c with the formulation above, we get:
E=m/e₀u₀.
So, energy is the ratio of mass to the product of the electric constant and the magnetic constant? Nice, but let's keep digging. What if we solve for mass (effective mass, that is):
m = Ee₀u₀.
OK, so the effective mass is proportional to the energy, scaled by the product of the electrical and magnetic constants. It makes sense that the electric and magnetic constants affect effective mass: larger constants would be equivalent to more energy.
Your first explanation for c seems fine to me, but I find the second explanation for the rest mass a little less convincing. (For instance, why should an object's electromagnetic properties be so...
Your first explanation for c seems fine to me, but I find the second explanation for the rest mass a little less convincing. (For instance, why should an object's electromagnetic properties be so important when the bulk of its energy comes from the strong force? In fact, why should it depend on the electromagnetic force at all? Special relativity stills holds in a universe without it.) I don't want to just knock you down though, so let me flesh out your explanation out a bit.
In special relativity, the only inherent property of a particle is its mass m and the only fundamental constant is the speed of light c1. A particle's innate (rest) energy must therefore be be some function of its mass, the speed of light, the particle's position, and arbitrary derivatives of combinations of those quantities (e.g velocity, derivative of momentum, etc.).
However, we are interested in a particle's innate energy, i.e. the portion of its energy that remains constant as we move the particle around. We consequently require its dependence on position, velocity, higher derivatives, and mixed partial derivatives of position be zero. If we further assume that the particle's rest mass doesn't change -- that is, that mass is some inherent property of the particle and not something emergent -- then derivatives of mass must also vanish.
This leaves only two options for a particle's innate energy: 0 or A mc^2 (i.e the only combination of m and c with the correct units, with A some generic multiplicative constant). As it so happens, both options are realized in nature. Photons have rest energy E_0 = 0, while massive particles have rest energy E_0 = mc^2 (the fact that A=1 basically comes down to a combination of convention and happy coincidence). 2
As an aside, let me also address why c has the value c_EM = 1 / sqrt(ε_0 μ_0). As I previously mentioned, the c in relativity has nothing to do with electromagnetism a priori. However, give that (1) photons are massless particles and must have speed v_photon = c and (2) photons propagate at c_EM per Maxwell's equations, we see that c = v_photon = c_EM.
Indeed, massive particles necessarily have non-zero rest energy. To be technical for a minute, the Lorentz group splits into two subgroups, one of which describes massive/time-like particles and the other of which describes massless/light-like particles. Whether an element belongs to one subgroup or the other is equivalent to asking (1) whether a particle is massive or massless which is equivalent to asking (2) whether the particle moves at v=c or v<c which is equivalent to asking (3) whether its rest mass vanishes or not.
Thanks for the detailed explanation. I recognize my hasty dismissal of the point the author was making, as I never read the second part until you suggested it. I had never considered the...
Thanks for the detailed explanation. I recognize my hasty dismissal of the point the author was making, as I never read the second part until you suggested it. I had never considered the possibility that energy disparities between high speed collision energies and molecular bond energies are what create practical higher bounds for velocities well below those imposed by relativity.
I confess, I'm not familiar with another solution for the speed of light aside from the one I discussed. My GR is admittedly weak. Is there another formulation for the value of c that isn't empirical?
As one of my physics professors used to ask: fast compared to what? I'd recommend people skip to the second part. The first part offers some motivation for why E_0 = mc^2 without truly addressing...
As one of my physics professors used to ask: fast compared to what?
I'd recommend people skip to the second part. The first part offers some motivation for why E_0 = mc^2 without truly addressing the question in the title. The second part, in contrast, explains why the speed of light should be much faster than the typical speeds in biological systems.
My takeaway: living things need to be able to turn around easily. That requires operating at speeds where going from +v to -v isn’t a tremendous about of acceleration, force, and energy.
My takeaway: living things need to be able to turn around easily. That requires operating at speeds where going from +v to -v isn’t a tremendous about of acceleration, force, and energy.
Respectfully to the author, I think the answer in the article is among the most confusing ways to answer this question.
The simplest answer is that the speed of light, c, is the speed with which electromagnetic waves propagate in vacuum. It is the inverse of the geometric mean the electric constant and the magnetic constant.
c = √(e₀u₀)⁻¹
Why is the speed of light so fast? Because the magnetic constant is a small number (1.2×10⁻⁶) and the electric constant is a way, way smaller number (8.9×10⁻¹²). So, what's the square root of the product of those numbers? 3.3×10⁻⁹. Let's invert it to get c. About 3×10⁸. Pretty close to 299,792,458 m/s.
OK, but what does it mean? Well, light is an electromagnetic wave, so let's think of fast waves and slow waves. Things that have a lot of inertia move slowly, so a pendulum with a longer chain (greater inertia) will oscillate more slowly than one with a shorter chain (less inertia). Well, the product of the electric and magnetic fields doesn't make for much effective inertia (like a really, really short pendulum), so the resulting electromagnetic wave is probably going to move pretty fast, and it does.
Relativity is neat, but completely unnecessary for answering the question asked.
Edit: I just realized that this makes the famous mass-energy balance actually intelligible. What is the mystical connection in E=mc²?
Well, let's look at the Maxwell formulation. Replacing c with the formulation above, we get:
E=m/e₀u₀.
So, energy is the ratio of mass to the product of the electric constant and the magnetic constant? Nice, but let's keep digging. What if we solve for mass (effective mass, that is):
m = Ee₀u₀.
OK, so the effective mass is proportional to the energy, scaled by the product of the electrical and magnetic constants. It makes sense that the electric and magnetic constants affect effective mass: larger constants would be equivalent to more energy.
Your first explanation for
cseems fine to me, but I find the second explanation for the rest mass a little less convincing. (For instance, why should an object's electromagnetic properties be so important when the bulk of its energy comes from the strong force? In fact, why should it depend on the electromagnetic force at all? Special relativity stills holds in a universe without it.) I don't want to just knock you down though, so let me flesh out your explanation out a bit.In special relativity, the only inherent property of a particle is its mass
mand the only fundamental constant is the speed of lightc1. A particle's innate (rest) energy must therefore be be some function of its mass, the speed of light, the particle's position, and arbitrary derivatives of combinations of those quantities (e.g velocity, derivative of momentum, etc.).However, we are interested in a particle's innate energy, i.e. the portion of its energy that remains constant as we move the particle around. We consequently require its dependence on position, velocity, higher derivatives, and mixed partial derivatives of position be zero. If we further assume that the particle's rest mass doesn't change -- that is, that mass is some inherent property of the particle and not something emergent -- then derivatives of mass must also vanish.
This leaves only two options for a particle's innate energy:
0orA mc^2(i.e the only combination ofmandcwith the correct units, withAsome generic multiplicative constant). As it so happens, both options are realized in nature. Photons have rest energyE_0 = 0, while massive particles have rest energyE_0 = mc^2(the fact thatA=1basically comes down to a combination of convention and happy coincidence). 2As an aside, let me also address why
chas the valuec_EM = 1 / sqrt(ε_0 μ_0). As I previously mentioned, thecin relativity has nothing to do with electromagnetism a priori. However, give that (1) photons are massless particles and must have speedv_photon = cand (2) photons propagate atc_EMper Maxwell's equations, we see thatc = v_photon = c_EM.Indeed, massive particles necessarily have non-zero rest energy. To be technical for a minute, the Lorentz group splits into two subgroups, one of which describes massive/time-like particles and the other of which describes massless/light-like particles. Whether an element belongs to one subgroup or the other is equivalent to asking (1) whether a particle is massive or massless which is equivalent to asking (2) whether the particle moves at
v=corv<cwhich is equivalent to asking (3) whether its rest mass vanishes or not.Thanks for the detailed explanation. I recognize my hasty dismissal of the point the author was making, as I never read the second part until you suggested it. I had never considered the possibility that energy disparities between high speed collision energies and molecular bond energies are what create practical higher bounds for velocities well below those imposed by relativity.
I confess, I'm not familiar with another solution for the speed of light aside from the one I discussed. My GR is admittedly weak. Is there another formulation for the value of c that isn't empirical?
Here is Part 2.
As one of my physics professors used to ask: fast compared to what?
I'd recommend people skip to the second part. The first part offers some motivation for why
E_0 = mc^2without truly addressing the question in the title. The second part, in contrast, explains why the speed of light should be much faster than the typical speeds in biological systems.My takeaway: living things need to be able to turn around easily. That requires operating at speeds where going from +v to -v isn’t a tremendous about of acceleration, force, and energy.