Extra dimensions
In Problem #4 of PS2, there was a remark that a photon follows a trajectory given by
x^2+y^2+z^2-c^2t^2=0.
I was thinking about this and thought of something interesting. Suppose the parametric equations for a photon’s trajectory are
x(t)=v_x*t
y(t)=v_y*t
z(t)=v_z*t.
Putting these in to the equation above gives
(v_x^2+v_y^2+v_z^2)t^2-c^2t^2=0
or
v_x^2+v_y^2+v_z^2=c^2.
So, the first equation above is for a photon starting at the origin and traveling in the direction
(v_x/c,v_y/c,v_z/c).
Now, suppose there is a fourth spatial dimension called v. At every point in space, there would be a fourth direction we could go, the v direction. Why we are unaware of it I don’t know. What I do know is that we would know about it (if relativity still held) because we would have
v_v^2+v_x^2+v_y^2+v_z^2=c^2 so that v_x^2+v_y^2+v_z^2<c^2
for the analysis above, implying there would be photons moving at less than the speed of light. Well, that does not seem to happen, but there is not reason to think that the v direction would be an infinite line, but rather a little loop of some radius a. So, a each point in space, you could remain at the same (x,y,z) point and travel around a little loop, going at distance 2 pi a. Really, nothing wrong with that, if a is small enough. How small is small enough? Start back at the first equation
v^2+x^2_y^2+z^2-c^2t^2=0
The three parametric equations would be the same, but since the v direction is a little loop, it parametric equation would have to be
v(t)=mod(v_v*t,2*pi*a)
where mod(a,b) is the modulus and is equal to the fractional part of a/b. Thus, the largest v could be is 2*pi*a. Then, putting in the parametric equations gives
mod(v_v*t,2*pi*a)^2+(v_x^2+v_y^2+v_z^2)t^2-c^2t^2=0 or
c^2=v_x^2+v_y^2+v_z^2+mod(v_v*t,2*pi*a)/t^2
The last term still makes the velocities along the three spatial dimensions less than c, but, since the largest the last term can be is (2*pi*a/t)^2, its effect gets small with time, so after some time t_o, it doesn’t make any difference. suppose a=r_e=2*10^-15 m, the Compton radius of the electron. Then the last term will not matter when 2*pi*a/t_o<<c or t_o>>2*pi*a/c=4*10^-23 s. This time is comparable to the very fastest nuclear decays.
Extra dimensions sounds like a weird idea, but it is a very hot topic in theoretical physics these days.
