Lorentz factor

The Lorentz factor features in the Theory of Relativity to predict changes in the passage of time, apparent length, and energy, among other uses. It is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}}

Derivation

A common relativistic derivation of the Lorentz factor proceeds as follows.

First, imagine an ideal clock, at least as according to the special theory of relativity wherein the speed of light $c$ is an impermeable, universal constant. This clock consists of two connected, parallel mirrors. Call the distance between them Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} . Somehow, a light beam has started bouncing between the two mirrors at the speed of light. We can define the time it takes to complete a round trip - a distance of $2d$ - at the speed of light as $t$, according to someone moving with the clock.

So, according to such an observer, the time taken for the light to complete a round trip is given by the typical equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = \frac{2d}{c}} .

Imagine someone else is watching the clock from elsewhere, but not moving with the clock. Label the time they see passing as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t'} . This may or may not be equal to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} , the time seen by the observer moving with the clock.

Now, suppose the clock - and its attached observer - begin to move with a velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} relative to the external observer. In other words, the external observer thinks they are stationary, and sees the clock and attached observer as moving with velocity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v} . The movement of the clock is perpendicular to the beam of light bouncing between mirrors. The clock starts moving the instant that the light bounces off the "bottom" mirror.

The distance the clock travels to the right before the light reaches the "top" mirror and bounces back to the "bottom" is given by the usual equation of motion

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Distance = vt'}

according to the external observer.

Since the speed of light is constant, it will move half this distance by the time the light reaches the top mirror, for a "halfway" distance of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{vt'}{2}} .

Drawing it out, we can effectively create an isoceles triangle, where the base is the distance of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle vt'} and the center, height line from the middle of the base to the tip is the distance between the mirrors: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} . There are two equivalent right triangles within; their hypotenuses are given by the Pythagorean Theorem:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\sqrt{\frac{v^2t'^2}{4} + d^2} = D} where D is the total distance, i.e. the two hypotenuses.

However, the speed of light Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} is constant, so the distance it travelled can also be written as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D = ct'} . Thus the total distance that the light travels is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\sqrt{\frac{v^2t'^2}{4} + d^2} = ct'} .

From here, it's algebra: logical but allowable manipulations to get into a useful form. First, square both sides:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4(\frac{v^2t'^2}{4} + d^2) = v^2t'^2 + 4d^2 = c^2t'^2} after distributing the 4; it cancels with the one in the fraction denominator:

Then we can subtract from both sides to get:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4d^2 = c^2t'^2 - v^2t'^2 = t'^2(c^2 - v^2)} after factoring out the time variable.

Divide both sides by the speed terms in parentheses.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4d^2}{c^2 - v^2} = t'^2}

Take the square root.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2d}{\sqrt{c^2 - v^2}} = \frac{2d}{c\sqrt{1 - \frac{v^2}{c^2}}} = t'} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c^2} was factored out of the square root's interior, where it then reduced to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .

The final step is to remember that the distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d} between mirrors was also given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d = \frac{ct}{2}} . Substitute that into the above to get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2ct}{2c\sqrt{1 - \frac{v^2}{c^2}}} = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} = t'}

Thus, the time that passes for the external observer is given by the time passed for someone moving with the clock multiplied - or "scaled" - by a factor of

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \gamma} : the Lorentz factor.

Implications

The general behavior of the Lorentz factor can be seen by experimenting with different hypothetical speeds as a fraction of light. For instance, say a starship travels at about 86.6% the speed of light. This can be represented in decimal form when plugging into the Lorentz factor as

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{1}{\sqrt{1 - (.866)^2}} = \frac{1}{\sqrt{1 - .75}} = \frac{1}{\sqrt{.25}} = \frac{1}{.5} = 2}

From the time dilation equation found as part of the above derivation, this gives us

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t' = 2t}

which means that for every arbitrary unit of time - say, one second - passing for someone moving with the clock, two seconds pass for the external, stationary observer.

Additionally, Einstein's energy-mass equivalence is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = mc^2 \gamma = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}}

Notice that the denominator of the Lorentz factor grows with speeds approaching the speed of light; for example, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1}{10}} the speed of light gives a Lorentz factor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = 1.005} , while Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{9}{10}} gives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = 2.3} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{999}{1000}} gives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = 22.366} . It grows exponentially as the speed of the moving observer approaches the speed of light; in fact, the speed of light is an asymptote, and trying to plug it into the Lorentz factor directly gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \frac{1}{\sqrt{1 - \frac{c^2}{c^2}}} = \frac{1}{\sqrt{1 - 1}} = \frac{1}{0} = \infty}

Using that in the energy-mass equation, one finds there would be infinite energy in a particle of any mass going the speed of light. Since energy is conserved, the energy must come from somewhere, so an infinite amount of energy would have to be put into an observer to move them to the speed of light. (This is a key example of light speed being an insurmountable barrier in space travel.) Appropriately, since a photon is known to be without mass, it is the only particle capable of avoiding this limitation and thus travel at the speed of light.

Similarly, if one were to plug larger and larger Lorentz factors into the time dilation equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t' = t\gamma}

one would find the time passing for an external, stationary observer growing greater and greater for a given unit of time, say one second, for the moving observer. In other words, time moves slower the closer one moves to the speed of light. If a hypothetical, massless being were to move at the speed of light exactly, the Lorentz factor would give

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t' = \infty}

or time stands still for this hypothetical observer; an inifite amount of external, stationary time passes for every bit of time passage they experience.

There is also a length contraction expression, which gives the length of a body moving along the path of motion from the perspective of the external observer

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l' = \frac{l}{\gamma} = l\sqrt{1 - \frac{v^2}{c^2}}} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l} is the "proper length" of the body according to someone moving with it and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l'} is the length of the body as seen by an external, stationary observer

Since we are dividing by the Lorentz factor, the perceived length will always be less than that of the undilated, proper length. For a body moving at the speed of light, subtlety requires one to directly reexamine the Lorentz contraction

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l' = l\sqrt{1 - \frac{c^2}{c^2}} = 1\sqrt{1 - 1} = 0}

In other words, a body moving towards the speed of light appears shorter and shorter, until at the speed of light it seems to disappear entirely.

Note that, for physically tangible, sublight situations, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma > 1} always. So multiplying something by the Lorentz factor will increase the quantity while dividing by it will decrease the quantity.

Faster Than Light

Conventionally, faster than light travel is regarded as impossible. The reasoning is the insurmountable barrier that is the speed of light in the Lorentz factor, as detailed above.

However, this does not preclude faster than light travel per se. It is - hypothetically - possible that something could be "born" on the other side of the light speed barrier to begin with. Such hypothetical particles are called tachyons.

What the resulting Lorentz factor would yield physically is pure speculation. Notice that faster than light speed would give a negative number in the denominator of the Lorentz factor

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \frac{1}{\sqrt{1 - a}}} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a > 1} .

The result makes the matter more of a mystery.

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 - a = b < 0 } . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = \frac{1}{\sqrt{b}} = \frac{1}{z}} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = x + iy} is an arbitrary complex number, with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = \sqrt{-1}} , i.e. the "imaginary number."

In other words, the Lorentz factor would involve an imaginary number.

Physically, the resulting relativistic equations - for time, length, energy, etc. - would have to either have something imaginary about them with which to cancel the imaginary number, or the resulting relativistic phenomenon would simply be "imaginary," whatever that may mean. (The label "imaginary" is per se just a name given to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{-1}} .)

A common interpretation using time dilation is that a tachyon would travel backwards in time. The basis for this logic is in the kinematic equations called the Lorentz transformations. There are several; the relevant one is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = \gamma (t' - x \frac{v}{c^2}) = \gamma t'(1 - \frac{x}{t'} \frac{v}{c^2})} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is the change in position.

Consider an observer moving at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 86.6%} the speed of light again, observing the movement tachyons.

Plugging in values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma = 2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{v}{c} = .866} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t' = 1} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{x}{t'} = 2c} (or twice the speed of light) gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = 2(1 - \frac{2c}{1} \frac{.866}{c}) = 2 - 4(.866) = -1.464}

so negative time has passed; the tachyon seems to have arrived before it left, and so has moved back in time according to one of the available reference frames. This creates obvious potential for paradox.

This possibility of effect precluding cause indicates that, if tachyons truly existed, then there may be no free will, and actions would always be pre-determined. Otherwise, one observer may have an "early warning" of something that will happen in another observer's frame and somehow change it; but then special relativity has it that observers in all reference frames must agree on whether a spacetime event occurs at all or not: no in-between.

This would create a time travel paradox, so it is thought that one would not be allowed to change the past, regardless of wishes; one's actions would be pre-determined, not freely chosen at will. Dr. Richard Muller vividly illustrates this in his "tachyon murder paradox," commenting that it gives him personal reason to wish faster than light travel impossible.