Twin Paradox of Special Relativity

Introduction

This page provides unique interactive graphs for the Twin Paradox in Special Relativity. You can see how relativistic effects affect a space traveler in time, distance, velocity and acceleration graphs, and see the equations that produce those graphs.

In the Twin Paradox, there are two twins. One twin (named R for "at rest") remains on the Earth while the other (named M for "moving") travels away in a spaceship, and then returns. The moving twin experiences six phases during her trip, starting with constant acceleration from a standstill, then coasting at constant velocity, and then constant deceleration on the trip out before reaching the turnaround point. She then experiences constant acceleration, then coasting at constant velocity, and then constant deceleration before arriving back on Earth. An animation is shown below.

According to the Theory of Special Relativity, by Albert Einstein, the elapsed time, distance traveled, instantaneous velocity and instantaneous acceleration experienced by the moving twin are different than what the resting twin experiences. The simulation below shows a plot of those quantities, for either or both twins, according to equations worked out on the Twin Paradox Calculations page.

Graphs

In the following graphs, the units are

Measurement Units Used Here Equivalent Units
Time 1 time unit = 1 earth year (about 365 days)
Distance 1 distance unit = 1 light year (about 10 trillion kilometers)
Velocity 1 velocity unit = speed of light (about 300,000 kilometers per second)
Acceleration 1 acceleration unit = 1 earth gravity (about 10 meters per second2)

You may interact with the graphs as follows:

  • Change the time and acceleration values on the left
  • Select values for display using the checkboxes on the right
  • Select timeM or timeR for the x axis using the selection box at the bottom
  • Get values on the curves, and see the current phase of the moving twin's trip, by moving the cursor over the plot
  • Match a vertical axis to its curve and legend entry, by moving the cursor over the vertical axis to reveal the color
Time accelerating or decelerating:
Time coasting:
Acceleration:
[1]The distanceM and velocityM values are not realistic as explained below.
Show:

Why are distanceM and velocityM not realistic?

The distanceM and velocityM values are not always realistic since they are calculated with the "normal" kinematic equations which ignore relativistic effects. Although the distanceM and velocityM predictions are very accurate at slower speeds, and certainly easier to compute, they fail miserably as the moving twin's velocity increases to nearly the speed of light. In fact, if we enter large enough input values then the velocityM predictions will exceed the speed of light (which is 1 in the system of units we are using). In the real world it is impossible to measure a velocity greater than the speed of light, so this proves the velocityM prediction is not always reliable.

The distanceM and velocityM values are provided here to show how the normal kinematic equations work well at slower speeds but fail at faster speeds, and to show how the relativistic equations depart from the normal kinematic equations at faster speeds.

The calculations involving timeM and accelerationM account for special relativity so those values are realistic, once somebody can invent a rocket that can accelerate continuously for months at a time.

Links

The Twin Paradox may be considered an advanced topic. It may be helpful to start with easier topics: