Someone on this site recently posed the following thought experiment questioning the postulate of special relativity that no object can travel faster than the speed of light - c. Imagine a one dimensional problem in which two travelers, move close to the speed of light, but in opposite directions - one moving close the speed of light in the positive direction and the other traveler moving close to the speed of light in the negative direction.

Wouldn't these observers perceive the relative velocity of the respective other as exceeding the speed of light (i.e., approx. 2c)?

Special relativity posits the following:

1) There exist an infinite set of reference frames moving at constant velocities with respect to one another in which the laws of physics are identical (these are called inertial frames).

2) The speed of light is independent of the motion of its source and is finite.

3) There is a finite limiting speed for all physical phenomena - the speed of light - c.

This is a good problem to explore some basic concepts in special relativity.

Imagine an observer moving at a constant velocity with respect to a stationary observer. How are the coordinates of the stationary observer related to the moving observer?

One naive way to convert coordinates between two frames is:

x'=x-vt

y'=y

z'=z

t'=t

This is called a Galilean transformation. Note that the time coordinate runs at the same rate between frames (t'=t). The Galilean transformation preserves the form of the equations of classical mechanics (i.e., Newton's equation, for example). However, Maxwell's equations, those equations that govern the behavior of electrodynamics are not invariant under Galilean transformation. This is a major problem.

Is there an invariant quantity that preserves physical laws like the form of Maxwell's equations upon a general rotation of coordinates? The answer is what is called the Minkowski metric.

In a first inertial frame, an observer would see a spherical shell of radiation governed by the following:

c^2*t^2-x^2-y^2--z^2=0

While in a second inertial frame, an observer would find:

c^2-*t'^2-x'^2-y'^z-z'^2=0

Assuming a linear relation between coordinates in two inertial frames (if space is homogenous and isotopic):

c^2-*t'^2-x'^2-y'^z-z'^2=k(c^2*t^2-x^2-y^2--z^2) where k is a constant

x'=gamma*(x-beta*ct)

y'=y

z'=z

ct'=gamma*(ct-beta*x)

where beta=v/c

gamma=(1-beta^2)^-(1/2)

The relativistically correct transformation is called a Lorentz transformation. This shows that time and space are intrinsically wrapped into a single object, whose components mix together when moving from one inertial frame to another. This object is called a 4-vector and appears to be inextricably required by the constancy of the speed of light, independent of reference frame.

The Lorentz transformation can be viewed as an abstract type of rotation of 4-vectors. What property of an ordinary vector is preserved upon a rotation in three-dimensions? Answer: The magnitude or length of the vector. This type of transformation, an ordinary rotation, is called a unitary transformation because of the fact that it preserves the magnitude. What is the metric? It is the Euclidean metric |v|=sqrt(x^2+y^2+z^2).

With the abstract type of rotation, the Lorentz transformation preserves the Minkowski metric:

ds^2=c^2*dt^2-dx^2-dy^2-dz^2

where ds, dx, dy and dz refer to infinitesimal changes in those respective variables.

With this framework, the rule for adding velocities relativistically can be derived by using the Lorentz transformation rules and the following shortcut. Taking the first equation and considering an infinitesimal change:

dx'=gamma*(dx-beta*c*dt)

Now taking the last equation above:

c*dt'=gamma*(c*dt-beta*dx)

Now divide the first by the second to obtain:

dx'/dt'=u'=c*(dx-beta*c*dt)/(c*dt-beta*dx)

Divide the right side through by dt:

u'=c*(u-beta*c)/(c-beta*u)

Now plugging in for beta=v/c

u'=c*(u-v)/(c-v*u/c)

Finally, dividing by c:

u'=(u-v)/(1-uv/c^2)

And, u=(v+u')/(1+vu'/c^2)

Considering u, v -> c, it can be seen that u<c.

REFERENCES: John David Jackson, Classical Electrodynamics

Relativistic Addition Of Velocities

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