The coordinates \((t, x, y, z)\) in flat spacetime.

The coordinates use Planck units. In other words, \(c = 1\).

Treat two coordinates like vectors and add them together.

>>> Coordinate (2, 1, 0, 0) `plus` Coordinate (-1, 1, 0, 0)
Coordinate (1.0,2.0,0.0,0.0)

Treat two coordinates like vectors and subtract the second one from the first.

>>> Coordinate (2, 1, 0, 0) `minus` Coordinate (-1, 1, 0, 0)
Coordinate (3.0,0.0,0.0,0.0)

Scale each component of a coordinate with the given function.

>>> scale (*2) $ Coordinate (1, 2, 0, 0)
Coordinate (2.0,4.0,0.0,0.0)

Combine the corresponding components of two coordinates with the given function.

>>> combine (*) (Coordinate (2, 3, 1, 2)) (Coordinate (1, -1, 2, 3))
Coordinate (2.0,-3.0,2.0,6.0)

Returns whether two coordinates are approximately equal to each other.

>>> Coordinate (1, 1, 0, 0) `approximates` Coordinate (1, 1, 0, 0)
True
>>> Coordinate (1, -1, 0, 0) `approximates` Coordinate (1, 1, 0, 0)
False

The threshold for being approximately equivalent is somewhat arbitrary.

The spacetime interval between the origin and the given coordinates.

This uses the convention where \(ds^2 = dt^2 - dx^2 - dy^2 - dz^2\).

>>> interval $ Coordinate (2, 1, 0, 0)
3.0

Apply the Lorentz transformation.

This transforms coordinates in an original inertial frame to those in a new inertial frame. It is assumed that the origin coincides between the two frames, and that the velocity only has an \(x\) component.

>>> let expected = Coordinate (0.414213562,0.414213562,2.0,3.0)
>>> let actual = transform (sqrt 0.5) $ Coordinate (1, 1, 2, 3)
>>> actual `approximates` expected
True

For reference, look up the Lorentz transformation.

Returns a Lorentz transformation.

It will transform diagrams in an original inertial frame to those in a new inertial frame. It is assumed that the origin coincides between the two frames, and that the velocity only has an \(x\) component.

>>> import Diagrams.Prelude qualified as D
>>> let c = Coordinate (2, 1, 0, 0)
>>> let v = 0.5
>>> let expected = vectorize2D $ transform v c
>>> let actual = D.transform (transformation v) $ vectorize2D c
>>> abs (actual - expected) < 1e-5
True

For reference, look up the Lorentz transformation.

Convert given coordinates into a two-dimensional Diagrams vector.

It will project the \(x\) coordinate to the \(x\) component and the \(t\) coordinate to the \(y\) component of the vector. The other coordinates will be ignored.

>>> vectorize2D $ Coordinate (1, 1, 0, 0)
V2 1.0 1.0

Draw axes for a spacetime diagram with default options.

Both axes will extend up to length 1 from the origin, and the lightcone will also be drawn. Units will be in Planck units, i.e., \(c = 1\).

>>> let _ = axes

Draw axes for a spacetime diagram with given options.

The horizontal axis will be for the \(x\) space coordinate, while the vertical axis will be for the \(t\) space coordinate. Units will be in Planck units, i.e., \(c = 1\).

>>> let _ = axesWith axesOptions { axesLength = 8 }

Default options for drawing axes.

Length of each axis. The default is 1.

>>> axesOptions { axesLength = 4 }
AxesOptions {axesLength = 4.0, axesLightcone = True}

Whether to draw the lightcone. It is drawn by default.

>>> axesOptions { axesLightcone = False }
AxesOptions {axesLength = 1.0, axesLightcone = False}

Stores various options for drawing axes.