### About

#
*Worldline *and special relativity theory

Activities *A* in the real world happen in four dimensions, three
of space *(x, y ,z*) and one of time (*t*):

*A = A (x, y, z, t)*

For their description we need a four dimensional coordinate system. As
our imagination is only capable of grasping three dimensional objects,
we must use two or three dimensional projections to visualize them. Most
often we restrict graphic illustrations to events of a single point
object moving along one coordinate axis (*x*). Then we can
represent it by a two dimensional graph of position over time:

*x = x(t) *

In contrast to the familiar *way-over-time* scheme of allocating
position to the ordinate and time to the abscissa of a plane coordinate
system, in special relativity is has become habitual to allocate
abscissa to time, and ordinate to position (*Minkowski *Diagram).

*t = t(x) *

This kind of presentation is of special interest when objects move at a
speed not small compared to the speed of light (*c = 2.99 792 458 *10 ^{8}
m/s*), measured relative to a resting observer at the origin of the
system. Using

*ct*instead of

*t*for the ordinate, both axes get the same dimension of

__length__.

*ct = ct(x) *

A trajectory (curve) in this scheme is called a *wordline. *For*
t < 0* it shows the entire past of the event, for *t > 0*
its future. Any point on the world line is called an *event. *

To achieve reasonable scaling for fast objects *2.99 792 458 *10 ^{8}
m/s * (1 unit of time) *is used as unit for the x axis. If time is
measured in seconds the

*x-*unit will be

*2.99 792 458 *10*=

^{8}m ≈ 300 000 km*1 lightsecond*.

With this scaling of *space−time geometry* a light signal
passing the origin will appear as a straight line at 45 degrees to the
axes (a* light cone* if one includes two spatial directions).

The simulation will demonstrate the movement of a particle under
constant acceleration at its *worldline*.

Under the laws of classical mechanics there would be no limit to the speed that an object can achieve under constant acceleration, relative to an observer resting at its starting place. It would follow a parabola in space−time.

Special relativity theory tells us that this is not possible. A real, accelerated object can approach the speed of light at most. As it approaches this range, from the standpoint of the resting observer its mass increases while the speed increment decreases.

In the view of an observer moving with the object, speed continues to increase. The resting observer interprets this impression as caused by dilatation of the time scale in the moving object.

The base of special relativity is the experimentally proven fact that
light (a photon, which has no rest mass) travels with constant velocity *c*
in any system. Its world line is a diagonal both for the resting and the
moving observer. A consequence is that no observed object can travel
faster than light. Therefore any events that have a causal connection
lie above the light cone. The "classical" path becomes unreal when it
reaches the gradient of the cone, independent of how great the
acceleration would be.

# Classical and relativistic worldline

With **Play **an object (black: wheel classical, magenta point:
relativistic) starts at the origin in *x* direction with constant
acceleration *b*. Its "classical" worldline is shown in light gray.
It would cross the red light cone at* x = 2 *and achieve speed of
light at *x = 1* (gradient equals that of the light cone).

At the beginning classical and relativistic objects seem to coincide and
to travel along the paraboloid classical worldline, while a red photon
runs ahead on its light cone with the speed of light. When speed is no
longer small compared to that of light, classical and relativistic
worldlines separate. While the "classical", black *wheel*
pursues the gray parabola, the real, magenta colored object draws its
relativistic blue worldline (a hyperbola), which finally runs parallel
to the light cone when the object is approaching the speed of light.

The difference between "classical" and relativistic case becomes obvious
only when the speed is of the order of magnitude of the speed of light.
With *t* scaled in seconds (*x = 1* corresponds to *300 000
km*), all happenings in "normal life" will be restricted to the
immediate neighborhood of the origin.

The gray line is the classical solution for constant acceleration *b:*

*x = 1/2 bt ^{2 }*

derived from the differential equation

*d ^{2}x / d t ^{2}= b *

The relativistiv movement is numerically calculated with the differential equation

*d ^{2}x / d ^{2}t = b sqrt(1-((dx /dt)/c)*

^{2}

*) = b sqrt(1-(v(t)/c)*

^{2}

*)*

The red line is the light cone with *x = ct*

**Play/Pause** starts and stops the animation.** Reset **leads
back to the starting condition.

Acceleration can be changed with the **slider b. **Default value is *b
= 1 *which results in the* *classical object achieving the speed
of light after *1* unit of *ct* at a distance of *1/2 *unit
of* x* km. After *2* *ct* units the object would surpass
the simultaneously started photon.

In scaling it is open which is the unit of time. If time is scaled in
seconds, the unit of the *x* scale is *light seconds*; if it
is scaled in years, it is *light years*.

The real relativistic path (blue) is a hyperbola, which for small speed is not visibly discernible from the classical parabola. At high speed the hyperbola becomes nearly parallel to the light cone. The object can approach the speed of light, but not achieve it.

**E1:** Start the simulation and check when relativistic effect
become noticable. Try this for different accelerations.

**E2: **Observe the light cone originating from the moving object.
How will the observer at rest appraise light signals emitted forward or
backward?

**E3: **Reflect about the scaling of "earthly" happenings. Take
as an example a rocket with constant acceleration of 10 times that of
earthly gravitation *g = 9,8 m/s ^{2}*. How long will
it take till relativistic effects become noticeable? How far away will
the object be at that time?

**E4: **Hint for E3:** **The graphic shows that relativistic
effects are manifest above 1/2 c. Calculate *t* classically for
that limit with *v = bt*. Then *s = b/2 t. ^{2}
*

**E5: **The time resulting will not be extreme. What is the real
limiting resource? Do some calculation!

### Translations

Code | Language | Translator | Run | |
---|---|---|---|---|

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

Dieter Roess - WEH-Foundation; Fremont Teng; Loo Kang Wee

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### Sample Learning Goals

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### For Teachers

## Worldline and Special Relativity Theory JavaScript Simulation Applet HTML5

### Instructions

#### Acceleration Slider

#### Toggling Full Screen

#### Play/Pause and Reset Buttons

Research

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### Video

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### Version:

### Other Resources

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