To explain how going faster than light would allow for time travel and challenge the theory of relativity, we must delve into the Lorentz transformations, which are at the heart of special relativity. These transformations describe how coordinates in space and time change when moving from one inertial frame to another.

Lorentz Transformations

For simplicity, consider two inertial reference frames, S and S’, where S’ is moving at a constant velocity ( v ) relative to S along the x-axis. The Lorentz transformations are given by:

[ t’ = \gamma \left( t – \frac{vx}{c^2} \right) ]
[ x’ = \gamma \left( x – vt \right) ]

where:

• ( t ) and ( t’ ) are the time coordinates in frames S and S’, respectively,
• ( x ) and ( x’ ) are the spatial coordinates in frames S and S’, respectively,
• ( c ) is the speed of light,
• ( \gamma ) is the Lorentz factor, defined as:
  [ \gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}} ]

Time Dilation and Length Contraction

These transformations lead to the phenomena of time dilation and length contraction:

• Time Dilation: A moving clock ticks slower than a stationary one.
  [ \Delta t’ = \gamma \Delta t ]
• Length Contraction: A moving object’s length contracts along the direction of motion.
  [ L’ = \frac{L}{\gamma} ]

Hypothetical Faster-than-Light Travel

For speeds greater than the speed of light (( v > c )), the Lorentz factor ( \gamma ) becomes imaginary because the term ( \sqrt{1 – \frac{v^2}{c^2}} ) is negative. Mathematically:
[ \gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}} ]

If ( v > c ), then:
[ 1 – \frac{v^2}{c^2} < 0 ]
[ \gamma = \frac{1}{\sqrt{-(\frac{v^2}{c^2} – 1)}} ]
[ \gamma = \frac{1}{i\sqrt{\frac{v^2}{c^2} – 1}} ]
[ \gamma = \frac{i}{\sqrt{\frac{v^2}{c^2} – 1}} ]

where ( i ) is the imaginary unit (( i^2 = -1 )).

Causality and Time Reversal

To see how this leads to time travel, consider a spacetime interval ( \Delta s^2 ) between two events. In special relativity, the interval is given by:
[ \Delta s^2 = c^2 \Delta t^2 – \Delta x^2 ]

For an object moving faster than light:
[ \Delta x > c \Delta t ]
[ \Delta s^2 = c^2 \Delta t^2 – \Delta x^2 < 0 ]

This implies a spacelike interval, which can be interpreted as the events having a cause-effect relationship that could be reversed in some reference frames. To illustrate this with a hypothetical example, consider a tachyon (a hypothetical particle that travels faster than light).

Tachyon and Causal Paradoxes

Suppose a tachyon travels from point A at time ( t_A ) to point B at time ( t_B ). In the rest frame, if it moves with velocity ( v > c ):
[ t_B – t_A = \frac{\Delta x}{v} ]

In another frame moving relative to the first, the time coordinate transformation could yield:
[ t_B’ < t_A’ ]

This means that in this new frame, the event at point B (reception of the tachyon) happens before the event at point A (emission of the tachyon), suggesting backward time travel.

Making Relativity Useless

These results are fundamentally incompatible with the causality principle of special relativity. If superluminal travel allows backward time travel, it introduces paradoxes like the “grandfather paradox,” where an effect can precede its cause. Thus, the theory of relativity, which assumes causality and a universal speed limit, would be rendered inadequate to describe such phenomena.

Conclusion

In summary, the equations of special relativity show that exceeding the speed of light leads to imaginary time intervals and potential causality violations, suggesting time travel. This fundamentally contradicts relativity’s core principles, indicating that if faster-than-light travel were possible, a new theoretical framework beyond relativity would be necessary.