Lorentz transformation of time
==============================

Let us consider two inertial reference frames :math:`S` (lab frame) and :math:`S'` (proper frame). The proper frame moves
with speed :math:`v` relative to the lab frame. Both frames contain identical fixed (relative to their own frame)
clocks that are synchronized according to the `Einstein rule <https://en.wikipedia.org/wiki/Einstein_synchronisation>`__.
Let :math:`x, y, z, t` be the coordinates and time of some event in frame :math:`S`, and :math:`x', y', z', t'` be the coordinates
and time of the same event in frame :math:`S'`. Assuming that the space is uniform and isotropic and that the time is
uniform, there exists a linear dependence between :math:`x, y, z, t` and :math:`x', y', z', t'`, which is called the **Lorentz
transformation** of space and time.

**Notation:**

#. :math:`c` (:code:`c`) is :attr:`~symplyphysics.quantities.speed_of_light`.

**Notes**

#. Lab frame :math:`S` is usually thought as stationary, and proper frame :math:`S'` is the one that is considered to be moving
   relative to lab frame and the moving object in question is at rest in the proper frame.
#. In this law, the Lorentz transformation from the lab frame :math:`S` into the proper frame :math:`S'` is described. In order to get
   an opposite transformation (from the proper frame :math:`S'` into the lab frame :math:`S`), replace all primed variables with unprimed
   ones and vice verce, and replace :math:`v` with :math:`-v`. This is consistent with the fact that frame :math:`S` can be viewed as moving
   with speed :math:`-v` relative to frame :math:`S'`, and hence the same Lorentz transformation can be applied.
#. In the limit :math:`v/c \ll 1` the formula reduces to the classical Galilean transformation :math:`t' = t`.

**Conditions:**

#. Space is uniform and isotropic.
#. Time is uniform.
#. The relative frame velocity is parallel to the :math:`x`-axis.

**Links:**

#. `Wikipedia, first formula in box <https://en.wikipedia.org/wiki/Lorentz_transformation#Coordinate_transformation>`__.

.. py:currentmodule:: symplyphysics.laws.relativistic.lorentz_transformation_of_time

.. py:data:: time_in_proper_frame

    :attr:`~symplyphysics.symbols.basic.time` in proper frame :math:`S'`.

Symbol:
    :code:`t'`

Latex:
    :math:`t'`

Dimension:
    :code:`time`


.. py:data:: time_in_lab_frame

    :attr:`~symplyphysics.symbols.basic.time` in lab frame :math:`S`.

Symbol:
    :code:`t`

Latex:
    :math:`t`

Dimension:
    :code:`time`


.. py:data:: position_in_lab_frame

    :attr:`~symplyphysics.symbols.classical_mechanics.position` in lab frame :math:`S`.

Symbol:
    :code:`x`

Latex:
    :math:`x`

Dimension:
    :code:`length`


.. py:data:: proper_frame_speed_in_lab_frame

    :attr:`~symplyphysics.symbols.classical_mechanics.speed` of proper frame :math:`S'` relative to lab frame :math:`S`.

Symbol:
    :code:`v`

Latex:
    :math:`v`

Dimension:
    :code:`velocity`


.. py:data:: law

    :code:`t' = (t - v * x / c^2) / sqrt(1 - (v / c)^2)`


    Latex:
        .. math::
            t' = \frac{t - \frac{v x}{c^{2}}}{\sqrt{1 - \left(\frac{v}{c}\right)^{2}}}