Question

Can a body travel faster than light?
Please give equations as proof

Answer 1

Nothing can travel faster than teh speed of light.

As you increase your speed (exerting energy to do so, or in the parlance of physicists "doing work"), your mass increases, meaning more work is needed to accelerate it faster, which increases your mass more, meaning even more work is needed to accelerate you and so on. It is probably best illustrated using the equation for relativistic kinetic energy.

\[E_k=\frac{mc^2}{\sqrt{1−v^2/c^2}}-mc^2\]

Recall that the work done in accelerating a particle of mass m from a velocity v1 to v2 is the difference in kinetic energy i.e. \(W=E_{k2}−E_{k1}\). But look at the bottom line of the equation above. When v=c the bottom line collapses to zero, and thus you are dividing the top line by zero. Any number divided by zero results in infinity (mathematicians prefer to say it is undefined). Physically what happens is that as \(v\) increases toward \(c\), the kinetic energy increases rapidly until at v=c the kinetic energy becomes infinite. Remember we said that the work done was \(W=E_{k2}−E_{k1}\). Well you can see that to push an object from \(v_1\) to \(v_2\) requires an infinite amount of work (and hence an infinite amount of energy applied to the particle) since \(W=\infty−E_{k1}=\infty\), since \(E_{k1}\) will always be finite, and infinity minus a finite number will always leave you with infinity.

Thus you cannot push a particle with mass to the speed of light, meaning you cannot accelerate beyond the speed of light. Photons can travel at the speed of light, because they do not have mass, and indeed they will only ever travel at the speed of light, as any energy you give them to change their speed will just change their frequency instead. (light slowing in an optical medium is a different phenomena).

Answer 2

Einstein's equations of special relativity postulate that the speed of light in a vacuum is invariant in inertial frames. That is, it will be the same from any frame of reference moving at a constant speed. The equations do not specify any particular value for the speed of the light, which is an experimentally determined quantity for a fixed unit of length. Since 1983, the unit of length (the meter) has been defined using the speed of light.

The experimental determination has been made in vacuum. However, the vacuum we know is not the only possible vacuum which can exist. The vacuum has energy associated with it, called the vacuum energy. This vacuum energy can perhaps be changed in certain cases. When vacuum energy is lowered, light itself has been predicted to go faster than the standard value 'c'. This is known as the Scharnhorst effect. Such a vacuum can be produced by bringing two perfectly smooth metal plates together at near atomic diameter spacing. It is called a Casimir vacuum. Calculations imply that light will go faster in such a vacuum by a minuscule amount: a photon traveling between two plates that are 1 micrometer apart would increase the photon's speed by only about one part in 1036. Accordingly there has as yet been no experimental verification of the prediction. A recent analysis argued that the Scharnhorst effect cannot be used to send information backwards in time with a single set of plates since the plates' rest frame would define a "preferred frame" for FTL signaling. However, with multiple pairs of plates in motion relative to one another the authors noted that they had no arguments that could "guarantee the total absence of causality violations", and invoked Hawking's speculative chronology protection conjecture which suggests that feedback loops of virtual particles would create "uncontrollable singularities in the renormalized quantum stress-energy" on the boundary of any potential time machine, and thus would require a theory of quantum gravity to fully analyze. Other authors argue that Scharnhorst's original analysis which seemed to show the possibility of faster-than-c signals involved approximations which may be incorrect, so that it is not clear whether this effect could actually increase signal speed at all.

The physicists Günter Nimtz and Alfons Stahlhofen, of the University of Koblenz, claim to have violated relativity experimentally by transmitting photons faster than the speed of light. They say they have conducted an experiment in which microwave photons - relatively low energy packets of light - travelled "instantaneously" between a pair of prisms that had been moved up to 3 ft apart, using a phenomenon known as quantum tunneling. Nimtz told New Scientist magazine: "For the time being, this is the only violation of special relativity that I know of." However, other physicists say that this phenomenon does not allow information to be transmitted faster than light. Aephraim Steinberg, a quantum optics expert at the University of Toronto, Canada, uses the analogy of a train traveling from Chicago to New York, but dropping off train cars at each station along the way, so that the center of the train moves forward at each stop; in this way, the speed of the center of the train exceeds the speed of any of the individual cars