an anything travel faster than the speed of light? (so called ‘superluminal’ velocities)?
According to the theory of relativity and the principal of causality, the answer to this question is ‘no’. The principal of causality says that a cause must happen before its effect. Yet, in July 2000, a team of physicists led by L. J. Wang made head-lines when they claimed to have made a light pulse travel much faster than the speed of light, so fast that it went backwards in time. This result would appear to violate both relativity and caus-ality, however, all is not as it seems.
To achieve their result the physicists passed a light wave through a specially prepared medium (caesium gas) to produce ‘anomalous dispersion’, In normal dispersion, such as occurs in glass, the blue light component in a light ray is slowed more than the red, In anomalous dispersion, the
red is slowed more than the blue. The effect of slowing the red is to change the way that the com-ponents of the light add together, to make the overall wave pulse appear to shift backward in time. It is thus a wave interference effect rather than a genuine superluminal velocity.
Wang et al. point out that this is the case, and that since it is a wave effect, no object with mass could travel this way. Additionally, because of the nature of the effect, no information could travel faster than light speed this way either. They note that their effect does not violate relativity or caus-ality. Perhaps surprisingly, scientists have been performing this type of experiment for almost twenty years, but the light pulses have been so distorted that the results are inconclusive. The success of Wang et al. has been to design an experiment that avoids the distortions.
C
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A
B String
x y
Before collision
String After collision
A
B
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however, their velocities in the x direction are still equal but opposite and almost unchanged. Due to the symmetry of the collision, the velocities of the spacecraft in the y direction, although very small, are also equal but opposite. As a result the total momentum in both the x and y directions is zero, hence momentum has been conserved.
The situation is different as seen from the point of view of one of the spacecraft, however, and the difference is due to time dilation and its effect on the y velocities.
As seen by a passenger of spacecraft A, prior to the collision space-craft B speeds toward it with a velocity of 0.6c (actually 0.55c — see the note in the margin), strikes it a sideways blow and then departs at a slight angle to its original direction. Due to the presence of the string, the passengers of spacecraft A can identify that they are now moving slowly away from the string — covering, say, 10 metres in one second, giving a velocity of 10 m s–1. Looking across, the passenger sees that the clocks in spacecraft B are running slow, so that it covers 10 metres in
s.
The velocity of spacecraft B is thus = 8 m s−1. Therefore, the y velocities of the spacecraft are not identical and momentum is not con-served in the y direction.
Algebraically:
py before collision = 0
py after collision = mAvA + mBvB
= m(10) + m(−8)
= 2m where m = mA = mB Hence, momentum is not conserved.
Einstein believed very strongly that momentum must be conserved in all inertial frames of reference. In order to solve this dilemma he suggested that the mass of an object must increase, or dilate, at relativistic speeds by a factor that compensates for the effect of time dilation on speed measurement. We can use this idea to derive a formula for mass dilation.
Referring back to the spacecraft problem, assume that total momentum is conserved in the y direction, as seen by the passenger of spacecraft A.
momentum before collision = momentum after collision 0 = mAvA + mBvB (as seen by A)
=
hence = and
so that, as seen by A, mB =
This expression shows how the mass of spacecraft B increases with speed, as seen by the passenger of spacecraft A. By substituting this new expression back into the problem we can see how the momentum works out.
In relativity, velocities do not simply add together. For example, if two cars travel toward each other, each with a velocity of 90 km h−1, then the velocity of one car as seen from the other will be 180 km h−1. However, two spacecraft approaching each other with velocities of 0.9c will have a closing velocity of 0.99c, rather than 1.8c. The formula that applies here is:
combined velocity = .
(The proof of this formula is not within the scope of this course.) Using this expression we can see that the two spacecraft referred to in the text, each with velocity 0.3c, actually approach each other at 0.55c, not 0.6c as
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pBy after collision= mBvB
=
=
= Hence, we can now say:
py before collision = 0 py after collision = pAy + pBy
= mAvA + mBvB
= mA
= 0 Hence, momentum is now conserved.
The masses of the two spacecraft were originally identical, so the expression relating the masses can be generalised to the form
mv =
where
m0= mass measured in the rest frame of reference
= rest mass
mv= mass as seen from the frame of reference in relative motion to the rest frame
v = velocity c = speed of light
This effect is called mass dilation and can be generally stated as follows: The mass of an object within its own rest frame is called its rest mass m0. Measurements of this mass mv, made from any other inertial reference frame in relative motion to the first, are always greater. The degree of mass dilation varies with velocity as shown in figure 5.14.
The effect can be most simply stated as: moving objects gain mass.
Experimental evidence for mass dilation came quickly. In 1909 it was noticed that beta particles (electrons) emitted by different radioactive substances possessed different charge to mass ratios. The various par-ticles were travelling at significant fractions of the speed of light. Further-more, the greater the speed of the beta particle, the smaller was its charge:mass ratio. When the effect of mass dilation was accounted for, the beta particles were all found to have the same charge:mass ratio.
Modern particle accelerators, however, demonstrate mass dilation every time they are used. As they accelerate particles, such as electrons or protons, to relativistic speeds, ever greater forces are required as the particles’ masses progressively increase.
mA
---Mass dilation is the increase in the mass of an object as observed from a reference frame in relative motion.
Figure 5.14 Mass as a function of velocity
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