## Saturday, April 7, 2012

### Moving faster than the speed of light

I've been thinking about physics a lot lately, and I'm starting to jot things down, so I don't keep going over the same ground again but also to help me iron out the logical inconsistencies that can creep in when you do a problem in Caput.  A lot of this is really me just thinking about physics that I've read, and doing thought experiments so I can understand it.

This post is thoughts about what would happen if you moved faster than the speed of light.  Start with these premises:
• The only way we know about a particle (anything really) is the effect / force that particle exerts on other particles.
• You push a block with your hand:  the electric / magnetic forces from  the electrons in the atoms in the proteins / molecules in the cells in your hand interact with the electrons in the atoms in the molecules (cellulose) in the block of wood
• The above example is for electricity and magnetism, but (I've read) applies equally well to other, more exotic forces - e.g. the strong nuclear force between quarks in an atom's nucleus
• The forces between particles can be represented as fields.  Fields are vectors that exist through space that indicate the force (magnitude, direction) that a "test" particle would experience if it were at that location
• Imagine 2 charged particles.  From Coulomb's law, we can calculate the force between them.  Or, for each particle we could determine the field it generates throughout space.  Then, the force on each particle is determined by the field generated by the other particle.
• Movement of particles causes changes in the fields
• As the location of 2 particles gets closer together, the force they exert on each other increases.  Similarly, the field strength increases.
• The changes in the fields propagates at the speed of light
The above might sound crazy, but they are well established physics, with tons of experimental evidence.  Given the above it is almost nonsensical to talk about a particle moving faster than the speed of light.  Which is somewhat expected - the above description of reality is based on the tenet that nothing travels faster than light.  But the exercise of investigating what would happen if something moved faster than light helps me understand the relationships.  So, 2 scenarios to imagine:

#### Particle approaches at faster than light

The particle will arrive at a location before the effect of the particle being at the location does. This is just logically inconsistent.

#### Particle moves away faster than light

This situation is harder to rule out.  As the particle recedes, it is not arriving before its effect.  The problem with this one occurs for two situations I can think of:

1.  Imagine another particle, chasing this one.  The "effective" location, based on the fields, is only moving at the speed of light.  In this case, the particle has effectively "disappeared".  The chasing particle sees only the location represented by the field

2. Imagine instead of a single particle, an atom moving faster than the speed of light.  Background:  for a stationary atom emitting radiation, the frequency is intrinsic to the motion of the oscillation of the electron(s) within the atom.  The radiation, regardless of the relative velocity between the emitting atom and the observer, propagates at the speed of light.  The wavelength is determined by the frequency and the speed of light.

Now, for the atom moving faster than the speed of light:  Take the period of oscillation, imagine the first cycle has occurred.  Now, in this period of time, the atom has traveled a distance greater than the wavelength of the radiation, and a new cycle occurs.  So the separation in peaks / troughs between the first and second cycle is greater than the wavelength (as it would be defined for regular sub-luminal speeds).  Furthermore, for the third cycle, the discrepancy is even greater.  So, even though the atom is travelling at constant velocity, the radiation is continuously increasingly red-shifted (chirped down!).   Effectively, as time goes on, the emission of radiation is red-shifted until it would disappear completely.  Now, this description is discrete, but it could be made continuous.

Why would the above be impossible or inconsistent?  Well, the particle, in this case, has effectively disappeared from the universe, since internally it is emitting radiation, but this vanishes / does not appear anywhere else.

The reverse of this is also possible to imagine, in which an atom emitting radiation approaches at faster than the speed of light, and the radiation is continuously increasingly blue shifted.  In this case, leaving aside the issue from above of the particle arriving before its effect, the radiation observed would be increasingly blue shifted over time (chirped up!).  Where is the increased power / energy coming from?  Again, the internal state of the atom is disconnected from the rest of the universe.

1. If you want to use "caput" analogous to the use of vitrum, as in "in vitro" - then you should use the ablative - "in capite."

2. (This is Joel. Megan pointed me to the site.)

For the first part of the post, that's an interesting point but I might have a simple counterexample. Just imagine that instead of a vacuum, we're doing the problem in water. The speed of light in water is about 0.75*c. EM waves should propagate at that speed as well. But of course we know that an electron can travel through water at a speed which exceeds 0.75*c (the reason for Cherenkov radiation). So there is clearly no logical contradiction there; I'm not sure if I see where one would come in once we move back into a vacuum.

(Not that superluminal particles are possible; they of course aren't. Also, I'll mention that my inspiration for thinking about the effect in water comes from a paper that Shel Glashow put out before the OPERA superluminal neutrino results were conclusively put down that pointed out that a superluminal particle would emit Cherenkov-like radiation leading to an energy spectrum inconsistent with the OPERA results. I'll confess that I didn't read the paper, only the APS synopsis:
http://physics.aps.org/synopsis-for/10.1103/PhysRevLett.107.181803 )

So, when you write "The particle will arrive at a location before the effect of the particle being at the location does" is that right? Won't the EM field arising from when the particle is at point 'A' arrive at point 'A' at the exact same time as the particle (sure the particle is moving faster than the field, but the distance form point 'A' to point 'A' is 0). It is only true to say that the particle will arrive before the effects of where it previously was, but that might not be a problem.

Anyway, those were just the thoughts that popped into my head and I certainly might have missed something. Haven't thought too much yet about the red/blueshifting ideas you mention at the end.

3. Hi Joel! Thanks for the thoughts. Good point about Cherenkov radiation. Reading the wikipedia page (http://en.wikipedia.org/wiki/Cherenkov_radiation), I'm reminded that when light propagates through a material it is by causing the electrons / nuclei in that material to respond to the electric field of the light. So, for example, when light strikes a surface, you get an initial response from the electrons at the surface. That initial response then triggers further electric fields that propagate into the material, and the process repeats. The slower than c speed is due to the "responsiveness" of the material's electrons to the applied electric fields.

In that respect, if we zoom in and view the subatomic area around the charged Cherenkov particle, the electric fields in its immediate vicinity are still travelling at c. It seems that they are "damped" out at further distances - at any further distance, the electric field is dominated by the intervening matter. It is only when that matter (electrons / nuclei) move in response to the cherenkov particle that the field from the cherenkov particle propagates.

That's my initial thoughts, remembering what I've read about dielectric responsiveness of materials, I'll need to go back and see what I can pull out of my texts about Cherenkov.