Light frequency change is proportional to gravitational potential change
========================================================================

When light is propagating in a gravitational field, its frequency changes proportionally to the
change in the potential of the gravitational field.

Let us consider an infinitesimally small section :math:`d \vec r` of the light's path, such that
the frequency of light is constant within that section. In that case we can obtain a dependency
between the change in light's frequency and the change in the gravitational potential.

**Notes:**

    .. _gravitational potential:

#. The *gravitational potential* :math:`\varphi` is defined as a scalar quantity such that the equation
   :math:`\vec g = - \nabla \varphi` holds where :math:`\vec g` is the vector of acceleration due to gravity
   and :math:`\nabla` is the nabla operator.

#. :math:`d \varphi = - \left( \vec g, d \vec r \right)` where :math:`\left( \vec a_1, \vec a_2 \right)` is
   the dot product between :math:`\vec a_1` and :math:`\vec a_2`.

**Links:**

#. Formula 72.4 on p. 378 of "General Course of Physics" (Obschiy kurs fiziki), vol. 1 by Sivukhin D.V. (1979).

.. py:currentmodule:: symplyphysics.laws.waves.light_frequency_change_is_proportional_to_gravitational_potential_change

.. py:data:: frequency_change

    The infinitesimal change in :attr:`~symplyphysics.symbols.classical_mechanics.temporal_frequency` after passing an infinitesimal section
    :math:`d \vec r`.

Symbol:
    :code:`d(f)`

Latex:
    :math:`df`

Dimension:
    :code:`frequency`


.. py:data:: frequency

    The :attr:`~symplyphysics.symbols.classical_mechanics.temporal_frequency` of light within an infinitesimal section :math:`d \vec r`.

Symbol:
    :code:`f`

Latex:
    :math:`f`

Dimension:
    :code:`frequency`


.. py:data:: gravitational_potential_change

    The infinitesimal change in :ref:`gravitational potential <gravitational potential>` after passing an
    infinitesimal section :math:`d \vec r`.

Symbol:
    :code:`d(phi)`

Latex:
    :math:`d \phi`

Dimension:
    :code:`velocity**2`


.. py:data:: law

    :code:`d(f) / f = -d(phi) / c^2`


    Latex:
        .. math::
            \frac{df}{f} = - \frac{d \phi}{c^{2}}