Representation of Poincare Transformations

- NAME
- SYNOPSIS
- DESCRIPTION
- CONSTRUCTORS
- new
- rotation_euler
- rotation_x
- rotation_y
- rotation_z
- boost
- boost_x
- boost_y
- boost_z
- parity
- time_reversal
- OTHER METHODS
- ACCESSORS
- SEE ALSO
- AUTHOR
- COPYRIGHT AND LICENSE

Physics::Lorentz::Transformation - Representation of Poincare Transformations

use Physics::Lorentz; my $rotation = Physics::Lorentz::Transformation->rotation_euler( $alpha, $beta, $gamma ); my $vector = Physics::Lorentz::Vector->new([$t, $x, $y, $z]); my $rotated = $rotation->apply($vector); # or: $rotated = $rotation * $vector; ...

This class represents a Poincare transformation. That is a proper or
improper Lorentz transformation plus a shift by some 4-vector.
(`x' = lamda*x + a`

)

Yes, the class name might be misleading, but honestly, when most non-physicists talk about Lorentz transformations, they mean Poincare transformations anyway. (Pun intended.)

To sum this up, the set of Poincare transformations contains, among others

**Boosts**

**Rotations**

**Space Inversions / Parity**

**Time Inversion**

**Shifts by a constant vector**

**Combinations thereof**

None.

Stringification is overloaded with the `stringify`

method.

Multiplication (*) is overloaded with the `merge`

method for other
transformations:
`$t3 = $t1 * $t2`

corresponds to the following application on a vector:
`t1 * ( t2 * vec )`

. (I.e. t2 first, then t1)
Of course, **Poincare transformations do not commute**!

The assignment form of multiplication is supported for merging transformations but its use is discouraged unless you're into obfuscation.

Multiplication is also overloaded for application to vectors, but only if the
vector is on the right of the transformation: `$t * $v`

is okay, but `$v * $t`

is not.

Creates a new `Physics::Lorentz::Transformation`

object.
Defaults to the unity transformation.

If one argument is present, this argument may either be a 4x4 PDL which will be used internally as the PDL representation of the or an equivalent Perl datastructure (4x4 matrix).

If two arguments are present, the second argument will be used as or
converted to a the Physics::Lorentz::Vector manpage. It defaults to `[0, 0, 0, 0]`

This result will be of the form `x' = lamda * x + a`

where `lamda`

is the
matrix and `a`

is the vector. (And `x`

is the 4-vector that's acted upon.)

Alternative constructor to construct a specific type of Lorentz transformation: A 3D rotation with the Euler angles alpha, beta, gamma.

Three arguments: alpha, beta, gamma.

(First rotate about fixed z-axis by alpha, the about fixed y-axis about beta, then about fixed z-axis by gamma.)

Alternative constructor to construct a specific type of Lorentz transformation: A 3D rotation about the x or 1-axis. First argument is the rotation angle.

Alternative constructor to construct a specific type of Lorentz transformation: A 3D rotation about the y or 2-axis. First argument is the rotation angle.

Alternative constructor to construct a specific type of Lorentz transformation: A 3D rotation about the z or 3-axis. First argument is the rotation angle.

Alternative constructor to construct a specific type of Lorentz transformation:
A boost of rapidity `eta`

(`eta = atanh(v/c)`

) and direction `x`

.

Accepts three arguments: The components of the boost's velocity vector
divided by the speed of light: `(v1/c, v2/c, v3/c)`

.

Alternative constructor to construct a specific type of Lorentz transformation:
A boost of rapidity `eta`

(`eta = atanh(v/c)`

) parallel to the x axis.

Accepts one argument: The boost's velocity divided by the speed of light
`v/c`

.

Alternative constructor to construct a specific type of Lorentz transformation:
A boost of rapidity `eta`

(`eta = atanh(v/c)`

) parallel to the y axis.

Accepts one argument: The boost's velocity divided by the speed of light
`v/c`

.

Alternative constructor to construct a specific type of Lorentz transformation:
A boost of rapidity `eta`

(`eta = atanh(v/c)`

) parallel to the z axis.

Accepts one argument: The boost's velocity divided by the speed of light
`v/c`

.

Returns the parity operation, that is, space inversion.

Returns the time reversal operation.

Merge two transformations into one. Mathematically speaking, if
`a`

is the current transformation, `b`

is another, and `v`

is
a Lorentz vector,
`$a->merge($b)`

is the transformation `v' = a x (b x v)`

.
In words, if applied to a vector, the merged transformation is the
subsequent application of tranformation
`b`

and then `a`

.

Apply transformation to a Lorentz vector or something that can be
converted into a `Physics::Lorentz::Vector`

. Returns a vector.

Returns a copy of the object.

Returns a string representation of the object. Currently, this is the string representation of the internal PDL vector/matrix.

Returns the 4x4 PDL that represents the *Lorentz* transformation.

This is the actual object that is used inside the Transformation object, so be aware that modifying it results in action at a distance!

Returns the Physics::Lorentz::Vector object that is used inside the Transformation object. As with the matrix, this is the actual object. Same caveats.

Note that this isn't a PDL but a Physics::Lorentz::Vector object.

*PDL*, the Physics::Lorentz manpage, the Physics::Lorentz::Vector manpage,

Steffen Müller, <smueller@cpan.org>

Copyright (C) 2006 by Steffen Müller

This library is free software; you can redistribute it and/or modify it under the same terms as Perl itself, either Perl version 5.6 or, at your option, any later version of Perl 5 you may have available.

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