The Ultimate Minkowski diagram - American History Information Guide and Reference

The Minkowski diagram is a graphical tool used in special relativity to visualize spacetime with regard to an inertial reference frame. They are also referred to as "spacetime diagrams".

Contents

1 Basic Minkowski Diagram

2 The Light Cone

3 Examples

3.1 The Twin Paradox
3.2 Hyperbolic Motion: The Uniformly Accelerating Object

4 Superposition of Inertial Reference Frames

4.1 Constructing the Minkowski Diagram
4.2 The Mechanics and Meaning of Superposition

4.2.1 Time Axis, or Time-like Lines
4.2.2 Space Axis, or Space-like Lines

4.3 Superposition and the Lorentz Transformation
4.4 Superposition and Time Dilation
4.5 Superposition and Length Contraction
4.6 Superposition and Tachyons

Basic Minkowski Diagram

A Minkowski diagram is a representation of Minkowski space in an inertial reference frame. The simplest Minkowski diagram involves two dimensions, a time dimension and one space dimension. The time dimension is drawn vertically while the space dimension is drawn horizontally. Additionally, the scales of the axes can be adjusted such that a photon ( $γ$ ) passing through the origin describes a line of slope 1, or rather $c / c$ (the slope is with respect to the time axis). $c$ is the observer invariant speed, or the speed of light.

The Light Cone

Extending the Minkowski diagram to include a second space dimension then rotating the light path around the time axis creates a light cone.

Examples

The Twin Paradox

See the twin paradox.

Hyperbolic Motion: The Uniformly Accelerating Object

See hyperbolic motion.

Superposition of Inertial Reference Frames

The superposition of inertial reference frames using a Minkowski diagram is an easy way to visualize the relationship of the two frames with regard to an event or a series of events.

Constructing the Minkowski Diagram

To begin, each observer, say $S$ and $S^\prime$ , constructs a Minkowski diagram with a single space dimension and time dimension with the observer at the origin. To differentiate the two sets of variables for each observer, observer $S$ uses $x$ and $t$ for the space and time dimensions while observer $S^\prime$ uses $x^\prime$ and $t^\prime$ .

The trivial case is when $S$ and $S^\prime$ occupy the same reference frame. This involves a direct superposition of one Minkowski diagram on top of the other.

Furthermore, if we synchronize the two sets of coordinates such that when one observer stands behind the other observer, as far as the other space dimensions are concerned, both observers set their clocks to zero, then the trivial case results in both coordinate systems being identical.

The non-trivial case involves one observer, say $S^\prime$ , moving with a speed $v$ in relation to $S$ (and vise versa through symmetry). The resulting superposition has the axes of $S^\prime$ tilted such that the slope of the time axis is $v / c$ and the slope of the space axis is $c / v$ (the slopes are with respect to the $t$ axis). The light lines themselves become equivalent in the superposition.

The Mechanics and Meaning of Superposition

Time Axis, or Time-like Lines

A way to understand how the time axis of one coordinate system, $S^\prime$ , is superimposed on another coordinate system, $S$ , is to imagine the observer $S^\prime$ holding a flashing light source. In relation to $S^\prime$ the light source remains stationary as it flashes on and off. However in relation to $S$ the light source is in motion in the x-dimension. Graphing the events results in a tilted line in $S$ and a perfectly vertical line in $S^\prime$ . From this we can conclude that not only does the tilted line in $S$ represent the path, or worldline of $S^\prime$ but also the time axis of $S^\prime$ itself. Such a line is referred to as a time-like line .

Space Axis, or Space-like Lines

A way to understand how the space axis of one coordinate system, $S^\prime$ , is superimposed on another coordinate system, $S$ , is to imagine a series of light sources flashing simultaneously with respect to $S^\prime$ . Because of the relative nature of simultaneity, $S$ observes a series of flashing light sources going off in sequence one after the other. The result is a tilted line in $S$ and a perfectly horizontal line in $S^\prime$ . From this we can conclude that the tilted line in $S$ represents the line of simultaneity for $S^\prime$ which is its space axis. Such a line is referred to as a space-like line .

Superposition and the Lorentz Transformation

[To be written]

Superposition and Time Dilation

[To be written]

Superposition and Length Contraction

Minkowski diagram showing relativistic length contraction.

An examination a Minkowski diagram shows this affect is caused by a tilting relationship between two reference frames in space-time. The yellow area is a light cone. The black line are the space (x) and time (t) axis of observer in frame S. The blue line is the time axis (t') of frame S'. The purple line is the space axis (x') of a ship in frame S'. Note that in frame S' the light cone reaches both ends of the ship at the time while, they don't in frame S. That because when observer S is looking at both ends of the ship is looking a different points in the frame S' time lined. So the forward end of ship S' seen by observer S has not moved as far as the aft end, this results in ship S' being forshortened to observer S.

Superposition and Tachyons

Tachyons are theoretical particles with speeds greater than $c$ . The path described by a tachyon $τ$ corresponds to a space-like line. Drawing the path of a tachyon in a Minkowski diagram results in the conclusion that tachyons travel forwards or backwards in time depending on the observer.

Furthermore, drawing the path of a tachyon in a Minkowski diagram results in the conclusion that tachyons also travel at infinite speed, depending on the observer.

Categories: Special relativity

Your American History Reference Guide! - Minkowski diagram

Minkowski diagram