Why massive particles may travel faster than light.
There is a lot of hue and cry these days at an experiment
that has detected a neutrino travelling faster than light. Many physicists are
skeptical and are asking for extraordinary evidence as they feel this is an
extraordinary result. Many are calling for more accurate measurements of time,
distance and masses involved. I believe this may not be an extraordinary
circumstance as in my humble interpretation of special relativity it only
prohibits massive bodies from traversing the light speed barrier
and do not say anything about particles born travelling at speeds greater than
the speed of light. My interpretation is
as follows:
Special relativity holds that when massive bodies travel at
speed
, their length in the direction of motion contarcts
by a factor of (sq root of (1 - v squared / c squared)) . Similarly
their mass increases by the same factor, and finally their clock slows down by
the same factor.
The above three effects can be readily
understood if we consider a 2 – dimensional world (for the sake of this
discussion) and add time as a third dimension to this world. We need to make
just one conjecture to explain the above three effects in this 2+1 space time
continuum. The conjecture is that as a body travels in space with a velocity v , it tilts up in the time dimension. The
angle of inclination depends upon the velocity. I will illustrate this
conjecture with a diagram:
Figure 1 A cross section of 2+1 space time
Let us suppose that a rocket R1R2 is
travelling at velocity v toward right. According to the conjecture only its
‘shadow’ R1R3 is visible to the inhabitants of the 2D
space. Obviously length L of the rocket
visible to the inhabitants is smaller than its rest length Lo. Now
special relativity tells us that L = Lo(sq root of (1 - v squared / c squared)).
From Eq.2 when v = 0, K = 0 and when v =
c, K = Lo. That is when the rocket is at rest, there is no tilt and
hence no length contraction. As the speed of the rocket increase the tilt
increases, therefore the length visible to 2D people shrinks. Now, when the
rocket is tilted, another effect comes into being. As the Force applied to the
rocket to accelerate it in the right direction (in space) is actually being
applied at an angle to the space dimension it produces less acceleration in x
direction as compared to the expected value (From F = ma law).
From Eq.2 force
available for acceleration in space dimension is reduced by a factor (sq root of (1 - v squared / c squared)), the 2D people interpret it by saying
the mass of the rocket has increased by the same factor. Now as v approaches c,
the inclination angle gets steeper and steeper and the force component
available in space becomes smaller and smaller. Eventually at v = c the rocket
is completely vertical and travelling only in the time dimension, and to space
people it seems that no matter how much force they apply to the rocket it wont
budge an inch, and they therefore conclude that its mass has become infinite
and its length has become zero.
Finally we can explain the time dilation
by considering the following fact. If a boat is traveling at a speed v, which
is the same as the speed of the water in the river, but in the opposite
direction, to an observer standing on the bank of the river the boat looks
still.
Figure 2 standing boat
Now when the rocket is vertical it is
moving in the time dimension with the speed at which time seems to be moving to
the inhabitant of 2D space but in the opposite direction, consequently giving
the impression to the observers that the rocket time is stationary.
However, if a body travels at a speed
faster than the speed of “time river”, its length may become negative and its time will start flowing in
the reverse direction, and in presence of these two effects its mass will
become meaningless. (in other words, instead of offering resistance to the force, the rocket will start attracting the force, whatever that might mean) (Assuming that the
angle will keep on increasing).
On the other hand, the angle of 90 degree
may be some kind of a limit, after which the movement of the body would remain
vertical, but only increase in speed. In this case, a massive body born
travelling at a speed greater than the speed of light will remain undetectable
to 2D space people. However, if, in rare cases, they are somehow able to detect
the birth and demise of such particles, they may indirectly conclude that it
must have traveled at speeds faster than that of light.