From 15a630fbc505da3cdc1dcbe0157774c1c5c02d28 Mon Sep 17 00:00:00 2001
From: Jyotirmaya Shivottam <31299799+JeS24@users.noreply.github.com>
Date: Mon, 18 May 2020 13:25:50 +0530
Subject: [PATCH 1/3] Added EPE0004: Addition of RayTracing module

---
 EPE_00004/1-2.png       | Bin 0 -> 8348 bytes
 EPE_00004/3-4.png       | Bin 0 -> 6068 bytes
 EPE_00004/5.png         | Bin 0 -> 2708 bytes
 EPE_00004/6-7.png       | Bin 0 -> 7054 bytes
 EPE_00004/8.png         | Bin 0 -> 3409 bytes
 EPE_00004/EPE_00004.rst | 500 ++++++++++++++++++++++++++++++++++++++++
 6 files changed, 500 insertions(+)
 create mode 100644 EPE_00004/1-2.png
 create mode 100644 EPE_00004/3-4.png
 create mode 100644 EPE_00004/5.png
 create mode 100644 EPE_00004/6-7.png
 create mode 100644 EPE_00004/8.png
 create mode 100644 EPE_00004/EPE_00004.rst

diff --git a/EPE_00004/1-2.png b/EPE_00004/1-2.png
new file mode 100644
index 0000000000000000000000000000000000000000..3fb5c66aa8bcf513bb1ba4381f73f9b3840168ef
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diff --git a/EPE_00004/EPE_00004.rst b/EPE_00004/EPE_00004.rst
new file mode 100644
index 0000000..6f8cbbe
--- /dev/null
+++ b/EPE_00004/EPE_00004.rst
@@ -0,0 +1,500 @@
+Implementation of a Ray-Tracing Module
+--------------------------------------
+
+Author: Jyotirmaya Shivottam (`JeS24 <https://github.com/JeS24>`_)
+
+date-created: 2020 May 18 <replace with the date you submit the EPE>
+
+date-last-revised: 2020 May 18 <keep this up to date anytime something changes>
+
+date-accepted: TBA <replace with accepted date>
+
+type: Standard Track <one of these four: Standard Track, Informational, Process, Governance>
+
+status: Discussion
+
+--------------
+
+Abstract
+--------
+
+.. The abstract should be a short description of what the EPE will achieve.
+
+In this EPE, I have detailed the plan for implementing a Ray-Tracing
+module for Kerr and Schwarzschild spacetimes, in EinsteinPy.
+
+
+--------------
+
+Detailed description
+--------------------
+
+.. This section describes the need for the EPE.  It should describe the existing
+.. problem that it is trying to solve and why this EPE makes the situation better.
+.. It should include examples of how the new functionality would be used and
+.. perhaps some use cases.
+
+Our goal with this EPE is to implement a Ray-Tracing module, for Kerr
+and Schwarzschild spacetimes, that does the following:
+
+#. Helps in visualizing space-time distortion in a strong gravity region
+   (here, a Black Hole or BH)
+#. Obtains BH shadow in both spacetimes
+#. Performs Time-Independent Radiative Transfer:
+
+   #. To visualize Keplerian Disc/Shell with Lensing effect
+   #. To calculate associated Intensities for Gravitational Redshift
+      calculations
+
+While we will be working with the aforementioned spacetimes, we aim to
+keep the module general to other geometries, as well.
+
+--------------
+
+Branches and pull requests (If applicable)
+------------------------------------------
+
+.. Any pull requests or development branches containing work on this EPE 
+.. should be linked to from here. (An EPE does not need to be implemented 
+.. in a single pull request if it makes sense to implement it in discrete phases). 
+.. If no code is yet implemented, just put "N/A"
+
+This section will be updated, as and when features are implemented.
+
+--------------
+
+Implementation
+--------------
+
+.. This section lists the major steps required to implement the EPE.  Where
+.. possible, it should be noted where one step is dependent on another, and which
+.. steps may be optionally omitted.  Where it makes sense, each  step should
+.. include a link related pull requests as the implementation progresses.
+
+
+A. Approaches
+~~~~~~~~~~~~~
+
+There are two main ways to achieve the aforementioned goals:
+
+#. Numerically Integrating the Geodesic Equation (References `1, 2, 3 &
+   4 <????>`__)
+#. Implementing a semi-analytical solution, particular to the spacetime
+   (References `5 & 6 <????>`__)
+
+A brief comparison of these approaches on relevant factors, is given
+below:
+
++----------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------+
+| Numerical Integration                                                            | Analytical Method (Elliptical Functions)                                                                                                                       |
++==================================================================================+================================================================================================================================================================+
+| Computationally slower, as each point on the geodesic has to be evaluated        | Computationally faster, as arbitrary sections of the geodesic can be computed                                                                                  |
++----------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------+
+| Usually less accurate (Adaptive meshing can help here)                           | Spatial accuracy independent of integration step size                                                                                                          |
++----------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------+
+| Usually makes no assumptions on the underlying geometry                          | Makes several assumptions to make the problem analytically tractable and computationally faster                                                                |
++----------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------+
+| Generalizable to most spacetimes, which is useful for perturbative simulations   | Analytical solutions are specific to the spacetime. Since metric and connection must be known in advance, perturbations cannot be handled in a hands-off way   |
++----------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------+
+| Handles 3D accretion flow well                                                   | Analytical formula for 3D seems complex and computationally intensive to implement                                                                             |
++----------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------+
+| Simpler and faster for higher dimensions                                         | Faster in most other computations                                                                                                                              |
++----------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------+
+| Can be massively sped up by using parallel programming                           | Since calculations are implementation (Math) - specific, speed improvements are negligible                                                                     |
++----------------------------------------------------------------------------------+----------------------------------------------------------------------------------------------------------------------------------------------------------------+
+
+Based on these factors and more importantly, since we require a general
+module, we should go with a Numerical Solver. As far as computation
+speed is concerned, parallel programming seems helpful in bridging the
+gap between the two approaches. To these ends, after an extensive
+literature review, I have the narrowed down to these papers in Numerical
+Relativity:
+
+#. `RAPTOR - Bronzwaer et al
+   (2018) <https://doi.org/10.1051/0004-6361/201732149>`__ -
+   `PDF <https://github.com/JeS24/einsteinpy-bleeding-edge/blob/master/Papers%20(Non-Annotated)/RAPTOR%20-%20Bronzwaer%20-%202018.pdf>`__
+#. `Odyssey - Pu et al
+   (2016) <https://iopscience.iop.org/article/10.3847/0004-637X/820/2/105>`__
+   -
+   `PDF <https://github.com/JeS24/einsteinpy-bleeding-edge/blob/master/Papers%20(Non-Annotated)/ODYSSEY%20-%20Pu%20-%202016.pdf>`__
+#. `GRay2 - Chan et al
+   (2018) <https://iopscience.iop.org/article/10.3847/1538-4357/aadfe5>`__
+   -
+   `PDF <https://github.com/JeS24/einsteinpy-bleeding-edge/blob/master/Papers%20(Non-Annotated)/GRay2%20-%20Chan%20-%202018.pdf>`__
+#. `EHT Collab\* - Rummel et al
+   (2019) <https://arxiv.org/abs/2001.00041>`__
+   -
+   `PDF <https://github.com/JeS24/einsteinpy-bleeding-edge/blob/master/Papers%20(Non-Annotated)/Numerical/EHT%20-%202001.00041.pdf>`__
+
+\*Others are names of software packages, supplementing their work.
+``EHT Collab`` is not. I am using it as a label, in this document.
+
+We will be making use of relevant portions of these works, as described
+below and cited accordingly.
+
+B. Numerical Integration Strategies
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Based on the papers, listed above, we have two strategies for the whole
+problem:
+
+1. Integrate geodesics and Time-Independent Radiative Transfer ODEs, at
+   each integration step (simultaneously), like ``RAPTOR`` and
+   ``Odyssey`` do.
+
+-  Each step is computationally complex.
+-  Integration needs to be performed exactly once.
+-  Same step-sizes are used for both calculations, making the overall
+   simulation more accurate.
+-  GPU/CUDA dependency, through ``numba`` or ``cuPy`` may be needed.
+
+2. Integrate geodesics and have a separate sub-module, that integrates
+   Radiative Transfer equations, using the obtained geodesics, like we
+   have done (somewhat) in the current
+   ``einsteinpy.metric.Schwarzschild`` class or as in ``EHT Collab``.
+
+-  Requires another round of integration, but this time, only on the
+   calculated geodesics, instead of the affine parameter space.
+-  Will be easier to troubleshoot.
+-  In case, one requires only the geodesics (e.g. for plotting geodesics
+   or obtaining the BH Shadow), extra computations will be avoided.
+-  ``numba`` alone should be sufficient.
+
+C. Background
+~~~~~~~~~~~~~
+
+Geometries
+^^^^^^^^^^
+
+We will be working mainly with the Kerr Metric, which is a stationary,
+exact vacuum solution of the Einstein Field Equations for empty
+spacetime, around an uncharged, rotating, axisymmetric BH. This is a
+generalization of the Schwarzschild Metric, that is an EFE solution for
+an uncharged, non-rotating BH. The rotation of the BH is accounted for,
+by the Spin Parameter, :math:`a = J/M` (Angular Momentum / Mass). Under proper
+scaling of units (Geometrized Units, for example), :math:`0 \le a \le 1`. 
+In :math:`a \to 0` limit, we recover the Schwarzschild Metric. As such, a
+single module for Kerr Null Geodesics can be extended to Schwarzschild
+Null Geodesics, by simply setting :math:`a = 0`.
+
+The Kerr Metric is usually given in Boyer-Lindquist Coordinates
+(``BL``), but one other useful coordinate system is Modified Kerr-Schild
+Coordinates (``KS``). We already have an implementation for
+Boyer-Lindquist Coordinates in ``einsteinpy.coordinates``. We can
+implement Modified Kerr-Schild similarly.
+
+Geodesic Equation
+^^^^^^^^^^^^^^^^^
+
+The Geodesic Equation is a numerically stiff second order ODE, which can
+be derived by maximizing the action in 4D spacetime, using variational
+methods. It can be written succinctly using Christoffel Symbols
+(:math:`\Gamma_{bc}^{a}`), as follows (N.B: Einstein Summation has been
+used throughout this document):
+
+.. image:: 1-2.png
+    :align: center
+
+Here, :math:`X` denotes 4-position (point on the geodesic), :math:`g` is the
+metric tensor and :math:`\lambda` denotes the affine parameter. To make the
+computation more accurate, easier to follow and modular to implement, we
+can break down Eq. (1) into two coupled first order ODEs, per coordinate
+(8 in total - :math:`X^a, U^a`):
+
+.. image:: 3-4.png
+    :align: center
+
+Here, :math:`U` and :math:`A` are 4-velocity and 4-accelaration, respectively.
+
+Geometrized Unit System
+^^^^^^^^^^^^^^^^^^^^^^^
+
+Using SI in Numerical Relativity is rather cumbersome due to the huge
+scales involved. It is better and more "natural" to shift to Geometrized
+Units, where we set :math:`G = c = 1` and scale other quantities accordingly.
+We will be using these units in all the calculations in this module. For
+ease-of-use, we will let users input the necessary quantities in SI and
+convert them to Geometric Units implicitly. We will store the final
+outputs in SI, as well.
+
+A list of scaling factors for SI :math:`\leftrightarrow` Geometric can be found
+`here <https://en.wikipedia.org/wiki/Geometrized_unit_system>`__.
+
+Ray-Tracing
+^^^^^^^^^^^
+
+In reality, one has to consider all light rays (*Photons*) close to the
+BH and evolve them according to the Geodesic Equation (Eq. (1) or (3) &
+(4)) to see, which *photons* arrive at the observer's position and also,
+from which direction. This is computationally intractable, given the
+huge number of initial conditions to consider. To handle this
+efficiently, we can instead assume, that the observer is located far
+from the BH and that all rays, arriving at the observer's location, are
+parallel (as is the case for cosmological observations) and trace the
+*photon(s)* backwards in time (so-called *Ray-Tracing*), from the
+observer's location to the BH.
+
+D. Implementation Details
+~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Camera/Image Setup
+^^^^^^^^^^^^^^^^^^
+
+For Ray-Tracing, we first set up a virtual *Camera/Image*, which
+represents a consolidated view into what an observer (placed at each
+pixel of the camera/image) will see. It is a 2D, flat grid of pixels
+(using the aforementioned assumption), with the overall dimensions given
+by *Impact Parameters*, :math:`\alpha` or ``alpha`` horizontally, and
+:math:`\beta` or ``beta`` vertically. These form the coordinates in the
+observer's frame. For an observer (at a pixel) who receives the ray
+(photon) at an inclination angle, :math:`\theta_{Obs}` (from the BH Spin
+Axis), and radial position :math:`r_{Obs} \to \infty`, the celestial
+coordinates in the observer’s frame, :math:`(\alpha, \beta)` can be given
+by Eqs. (16) and (17) in ``Odyssey``. So, we require :math:`\theta_{Obs}`
+and :math:`r_{Obs}`, as the inputs for the placement of the null
+geodesic/photon in the image.
+
+Initial Conditions
+^^^^^^^^^^^^^^^^^^
+
+A summary of the required initial conditions is given below:
+
++-----------------+---------------------------------+
+| Parameter       | Use                             |
++=================+=================================+
+| ``r_obs``       | Distance from BH Spin Axis      |
++-----------------+---------------------------------+
+| ``theta_obs``   | Inclination from BH Spin Axis   |
++-----------------+---------------------------------+
+| ``a``           | BH Spin Parameter               |
++-----------------+---------------------------------+
+| ``alpha``       | Horizontal Impact Parameter     |
++-----------------+---------------------------------+
+| ``beta``        | Vertical Impact Parameter       |
++-----------------+---------------------------------+
+
+Mathematically, :math:`r_{Obs} \to \infty`. For practical purposes, we will
+keep ``r_obs`` sufficiently large, such that, any spacetime curvature,
+in the camera/image plane, is less than the numerical tolerance of the
+simulation - in other words, the flat space approximation for the
+observer holds.
+
+These parameters can be used to specify the initial conditions for the
+photon, i.e. its 4-position and 4-momentum (Eqs. (3) & (4)), that we can
+package together in a vector of length 8 - ``initial_photon_vector``.
+Also, we will have to perform a coordinate transformation from the
+observer's frame, at (:math:`\alpha, \beta`) to the BH frame, and then
+perhaps a transformation to our coordinate system of choice, either BL
+or KS. For initial conditions and related tasks, we can use the
+equations in either Section 2.2 of ``Odyssey`` or Section 2.3 of
+``RAPTOR``. At each integration step, we will compute the covariant
+photon momentum and related quantities, and use these values to update a
+``photon_vector``, that is a length-8 vector, storing values needed for
+the next integration step. These momentum values (:math:`P_\alpha` or
+:math:`k_\alpha`) will be used to perform Radiative Transfer calculations,
+as described in below, under "Radiative Transfer".
+
+A point to note here is, that the Kerr Spacetime admits 2 Killing
+Vectors, leading to Energy (:math:`E`) and Angular Momentum (:math:`L_z`)
+Conservation. Another composite conserved quantity or a constant of
+motion is Carter's Constant, :math:`Q`. Using these, we can eliminate the
+calculations, pertaining to time, :math:`t` and azimuthal angle, :math:`\phi`. This
+will reduce the number of differential equations to be solved from 8 to
+6 (or 5, if axisymmetry is assumed throughout the integration period).
+However, this is a Kerr-specific optimization, that is not extensible to
+most spacetimes. Even in Kerr spacetime, realistic models of the BH
+surroundings require a :math:`\phi` dependence. So, we can either account for
+these constants of motion (see ``Odyssey`` or ``EHT Collab``) or be
+completely general (see ``RAPTOR]``).
+
+Boundary Conditions
+^^^^^^^^^^^^^^^^^^^
+
+We also need to set limits on the BH itself, as getting too close to the
+BH would imply photon absorption/capture into the BH. Similarly, some
+photons may miss the BH by a large distance and evolve infinitely far
+away from the BH. The lower (or closer) limit can be set, using the
+equation for Kerr BH Event Horizon, given as follows (in Geometrized
+units):
+
+.. image:: 5.png
+    :align: center
+
+This is replaced with the expression for the Inner Most Circular Orbit
+(ISCO), which is dependent on :math:`a`, while considering Radiative Transfer
+calculations, vis-a-vis a Keplerian Disk. As for the outer (or farther)
+limit, we can set it arbitrarily, depending on the BH features, we wish
+to study. For example, for a Keplerian Disk, the outer limit is much
+larger, than for plotting null geodesics.
+
+Radiative Transfer
+^^^^^^^^^^^^^^^^^^
+
+As ray-traced photons travel close to the BH, their intensities,
+:math:`I_\nu`, change due to frequency shifts caused by the motion of the
+plasma. To account for this frequency shift, a correction from the
+observed frequency :math:`I_{\nu,\:Obs}` to the local frequency :math:`I_\nu`
+is required at every point along the ray, because physical processes
+take place in the local comoving frame. For this portion of the module,
+we will follow the formulation in Section 3 of ``RAPTOR`` or Section 2.3
+of ``Odyssey``. Both papers use the same second order ODE with Optical
+Depth (:math:`\tau`) as a parameter, however ``Odyssey`` decouples them into
+two first order ODEs and ``RAPTOR`` makes some modifications to make the
+integration numerically performant. We will be using these
+modifications. The final set of ODEs, that will be integrated along the
+geodesic (backward in time), are given below (Eqs. 30 & 31 in
+``Odyssey`` or Eqs. 36 & 37 in ``RAPTOR``):
+
+.. image:: 6-7.png
+    :align: center
+
+Here, :math:`j` and :math:`\alpha` are respectively, the Emission and Absorption
+Coefficients, evaluated at a frequency, :math:`\nu`; and :math:`\gamma` is the
+Relative Energy Shift, between radiation emitted from material
+circulating around the black hole, with 4-velocity, :math:`U^\alpha`, and the
+radiation, that is received by a distant observer (at
+:math:`\lambda_{Obs}`). It is given as follows:
+
+.. image:: 8.png
+    :align: center
+
+Since, we calculate the covariant 4-momentum (:math:`P_\alpha` or
+:math:`k_\alpha`) at each integration step for Eqs. (3) and (4), we can use
+these values directly to compute :math:`\gamma`. Eq. (8), along with Eqs. (6)
+and (7) give us the photon intensities.
+
+E. Notes:
+~~~~~~~~~
+
+#. In ``Gray2``, the authors use KS Coordinates, instead of BL. Their
+   claim is that, this helps with the overall stability of the solver,
+   as we do not have to worry about the polar singularities of BL
+   coordinates. In addition to this, they have opened up Christoffel
+   Symbols, :math:`\Gamma`, into metric derivatives and combined the
+   symmetrized terms, claiming performance improvements, from doing so.
+   These claims need to be verified, but nonetheless provide a potential
+   source of optimization for the module.
+#. In this module, we will not include Compton scattering, refraction
+   effects and polarization effects due to the plasma. Optical Depth is
+   the main parameter for our plasma models.
+
+F. Module Structure
+~~~~~~~~~~~~~~~~~~~
+
+I have listed the major structural components of the module below.
+Please note, that module/class/function names are subject to change, as
+the module develops.
+
+1. Null Geodesics Integrator
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+First, we require a ``NullGeodesic`` class, that inherits from (a
+modified version of) ``einsteinpy.Geodesic``, to solve Eqs. (3) & (4)
+and get the geodesics. This will be able to simulate a photon sheet,
+leading to obtaining Black Hole Shadows. A photon sheet is simply a
+bundle of parallel photons, ray-traced to the BH. For this class, we can
+have the following structure:
+
+::
+
+    NullGeodesic(Geodesic)
+    |
+    |--- initialize_photon() : To initialize the length-8 initial_photon_vector, at a particular pixel in the camera/image
+    |
+    |--- _stepsize() : Calculates optimal step-size, for the next integration step (Based on `RAPTOR`'s implementation)
+    |
+    |--- _connection_coef() : To calculate metric derivatives or Christoffel Symbols
+    |
+    |--- integrate() : Moves integration by 1 step
+    |                : Calls all 3 methods
+
+For ``integrate()``, possible pathways are:
+
+1. Implement our own solver. Some preferable ones are:
+
+    - RK45, with adaptive meshing - Higher accuracy, Stabler, Slower
+    - Verlet Integrator - Lower accuracy, Faster
+    - Adam-Bashforth-Moulton (Predictor-Corrector) Method - Medium accuracy, Most stable, Medium speed
+
+2. Use solvers provided in ``scipy.integrate``, like:
+
+    - ``solve_ivp``
+    - ``odeint``
+    - ``RK45``
+
+2. Time-Independent Radiative Transfer
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+For this portion, we will create a ``RadiativeTransfer()`` class, that
+will inherit ``NullGeodesic()`` from earlier. We can have the following
+structure for this class:
+
+***To be updated***
+
+::
+
+    RadiativeTransfer(NullGeodesic)
+    |
+    |--- _initialize_photon() : 
+    |
+
+A noteworthy point is, if we decide to go with a ``RAPTOR``-like
+implementation with simultaneous geodesic and radiative transfer
+integration, ``numba`` optimizations may be insufficient and we may have
+to go with CUDA/GPU code. For this, we can use ``cuPy`` or ``dask``.
+
+
+References:
+^^^^^^^^^^^
+
+#. `Bronzwaer et al; RAPTOR I: Time-dependent radiative transfer in
+   arbitrary spacetimes
+   (2018) <https://www.aanda.org/articles/aa/full_html/2018/05/aa32149-17/aa32149-17.html>`__
+#. `Chan et al; GRay2: A General Purpose Geodesic Integrator for Kerr
+   Spacetimes
+   (2018) <https://iopscience.iop.org/article/10.3847/1538-4357/aadfe5/meta>`__
+#. `Dexter et al; A FAST NEW PUBLIC CODE FOR COMPUTING PHOTON ORBITS IN
+   A KERR SPACETIME
+   (2009) <https://iopscience.iop.org/article/10.1088/0004-637X/696/2/1616/meta>`__
+#. `Pu et al; ODYSSEY: A PUBLIC GPU-BASED CODE FOR GENERAL RELATIVISTIC
+   RADIATIVE TRANSFER IN KERR SPACETIME
+   (2016) <https://iopscience.iop.org/article/10.3847/0004-637X/820/2/105/meta>`__
+#. `Rummel et al; Constraining Fundamental Physics with the Event
+   Horizon Telescope (2019) <https://arxiv.org/abs/2001.00041>`__
+#. `Yang et al; YNOGK: A NEW PUBLIC CODE FOR CALCULATING NULL GEODESICS
+   IN THE KERR SPACETIME
+   (2013) <https://iopscience.iop.org/article/10.1088/0067-0049/207/1/6>`__
+
+For a relevant literature review and runtime profiling of various
+implementations, including the papers cited in this document, please
+check `here <https://github.com/JeS24/einsteinpy-bleeding-edge>`__.
+
+--------------
+
+Backward compatibility
+----------------------
+
+.. This section describes the ways in which the EPE breaks backward compatibility.
+
+This EPE does not break backward compatibility, but it can potentially
+add to installation overheads with new dependencies, like ``cuPy`` or
+``dask``. This is related to the Radiative Transfer portion of the
+project.
+
+--------------
+
+Alternatives
+------------
+
+.. If there were any alternative solutions to solving the same problem, they should
+.. be discussed here, along with a justification for the chosen approach.
+
+Alternatives have been discussed `up here <????>`__
+
+--------------
+
+Decision rationale
+------------------
+<To be filled in by the coordinating committee when the EPE is accepted or rejected>
+
+

From a28f213a4a17ef8d26052f016a8fbfc0389ad975 Mon Sep 17 00:00:00 2001
From: Jyotirmaya Shivottam <31299799+JeS24@users.noreply.github.com>
Date: Mon, 18 May 2020 13:30:12 +0530
Subject: [PATCH 2/3] EPE00004: Added links

---
 EPE_00004/EPE_00004.rst | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/EPE_00004/EPE_00004.rst b/EPE_00004/EPE_00004.rst
index 6f8cbbe..9c7e361 100644
--- a/EPE_00004/EPE_00004.rst
+++ b/EPE_00004/EPE_00004.rst
@@ -78,9 +78,9 @@ A. Approaches
 There are two main ways to achieve the aforementioned goals:
 
 #. Numerically Integrating the Geodesic Equation (References `1, 2, 3 &
-   4 <????>`__)
+   4 <https://github.com/JeS24/EinsteinPy-EPEs/blob/master/EPE_00004/EPE_00004.rst#references>`__)
 #. Implementing a semi-analytical solution, particular to the spacetime
-   (References `5 & 6 <????>`__)
+   (References `5 & 6 <https://github.com/JeS24/EinsteinPy-EPEs/blob/master/EPE_00004/EPE_00004.rst#references>`__)
 
 A brief comparison of these approaches on relevant factors, is given
 below:
@@ -489,7 +489,7 @@ Alternatives
 .. If there were any alternative solutions to solving the same problem, they should
 .. be discussed here, along with a justification for the chosen approach.
 
-Alternatives have been discussed `up here <????>`__
+Alternatives have been discussed `up here <https://github.com/JeS24/EinsteinPy-EPEs/blob/master/EPE_00004/EPE_00004.rst#a-approaches>`__.
 
 --------------
 

From 3476e6ffb5086eca77d6589cc2692f1909330c24 Mon Sep 17 00:00:00 2001
From: Jyotirmaya Shivottam <31299799+JeS24@users.noreply.github.com>
Date: Mon, 18 May 2020 13:31:50 +0530
Subject: [PATCH 3/3] EPE00004: Edited Links

---
 EPE_00004/EPE_00004.rst | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/EPE_00004/EPE_00004.rst b/EPE_00004/EPE_00004.rst
index 9c7e361..182be69 100644
--- a/EPE_00004/EPE_00004.rst
+++ b/EPE_00004/EPE_00004.rst
@@ -78,9 +78,9 @@ A. Approaches
 There are two main ways to achieve the aforementioned goals:
 
 #. Numerically Integrating the Geodesic Equation (References `1, 2, 3 &
-   4 <https://github.com/JeS24/EinsteinPy-EPEs/blob/master/EPE_00004/EPE_00004.rst#references>`__)
+   4 <#references>`__)
 #. Implementing a semi-analytical solution, particular to the spacetime
-   (References `5 & 6 <https://github.com/JeS24/EinsteinPy-EPEs/blob/master/EPE_00004/EPE_00004.rst#references>`__)
+   (References `5 & 6 <#references>`__)
 
 A brief comparison of these approaches on relevant factors, is given
 below:
@@ -489,7 +489,7 @@ Alternatives
 .. If there were any alternative solutions to solving the same problem, they should
 .. be discussed here, along with a justification for the chosen approach.
 
-Alternatives have been discussed `up here <https://github.com/JeS24/EinsteinPy-EPEs/blob/master/EPE_00004/EPE_00004.rst#a-approaches>`__.
+Alternatives have been discussed `up here <#a-approaches>`__.
 
 --------------