# FedSpectral+: Spectral Clustering using Federated Learning

Janvi Thakkar*, Devvrat Joshi

###### Abstract

Clustering in graphs has been a well-known research problem, particularly
because most Internet and social network data is in the form of graphs.
Organizations widely use spectral clustering algorithms to find clustering in
graph datasets. However, applying spectral clustering to a large dataset is
challenging due to computational overhead. While the distributed spectral
clustering algorithm exists, they face the problem of data privacy and
increased communication costs between the clients. Thus, in this paper, we
propose a spectral clustering algorithm using federated learning (FL) to
overcome these issues. FL is a privacy-protecting algorithm that accumulates
model parameters from each local learner rather than collecting users’ raw
data, thus providing both scalability and data privacy. We developed two
approaches: FedSpectral and FedSpectral+. FedSpectral is a baseline approach
that uses local spectral clustering labels to aggregate the global spectral
clustering by creating a similarity graph. FedSpectral+, a state-of-the-art
approach, uses the power iteration method to learn the global spectral
embedding by incorporating the entire graph data without access to the raw
information distributed among the clients. We further designed our own
similarity metric to check the clustering quality of the distributed approach
to that of the original/non-FL clustering. The proposed approach FedSpectral+
obtained a similarity of 98.85% and 99.8%, comparable to that of global
clustering on the ego-Facebook and email-Eu-core dataset.

## Introduction

Organizations worldwide actively use graph clustering algorithms to map data
into communities to infer their relationships. Particularly, spectral
clustering (Ng, Jordan, and Weiss 2001) (Von Luxburg 2007) is one of the
widely used algorithms to cluster graphs. It employs graph eigenvector
embedding by preserving node relationships in euclidean space. However,
applying spectral clustering on a graph with billions of nodes is challenging
as it requires lots of computational cost and storage space. Researchers in
the past had tried to distribute the graph and train the model in a
distributed manner by achieving comparable accuracy. Nevertheless, in this
case, the communication cost has increased exponentially (Macey and Zomaya
1998). In addition, (Chen et al. 2010) also developed parallel spectral
clustering based on the sparse similarity matrix. However, the major drawback
of this work is that it assumes that any data in the parallel system can be
accessed by any client. But, this raises concerns for data privacy if we share
the information with different clients. In this work, we remove this
assumption by only sharing the representations learned in the local client to
the global server instead of directly communicating with neighboring clients.
111* Authors contributed equally Data privacy in distributed settings was the
very reason for the introduction of federated learning. The new machine
learning algorithms were becoming complex and required a large amount of data
to generalize on a task. However, in reality, data is distributed over many
organizations, and sharing them is difficult as it concerns data privacy.
Thus, the ‘Federated Learning (FL)’ term coined by Google (McMahan et al.
2017) was the first to propose an algorithm that combines intelligence across
the clients without compromising data security. There are two key advantages
of using FL over traditional machine learning algorithms. First, it allows
training the models on local devices and then sending the trained parameters
to the central server; this provides privacy to local learners as there is no
exchange of raw data among clients. Secondly, models are trained over
different local learners and thus do not need a large dataset to be present in
the central cloud.

Researchers around the globe have tried to propose several federated learning
algorithms for different machine learning models. They have tried to integrate
the differential privacy-based approaches in the distributed setting. Some of
the past literature works includes logistic regression (Chen, Rezapour, and
Tzeng 2018) (Nikolaenko et al. 2013), deep neural networks (McMahan et al.
2017) (Yang et al. 2019) (Bonawitz et al. 2019), support vector machines
(Smith et al. 2017) and gradient boosted decision trees (Cheng et al. 2021)
(Li, Wen, and He 2020).

Thus, we propose FedSpectral+, a novel algorithm for federated spectral
clustering. The approach integrates the power iteration method (Booth 2006)
and federated learning to enhance spectral clustering by decreasing the
communication cost between clients and providing data privacy. Our algorithm
takes advantage of the FL technique by learning the representations locally
and aggregating them over the server. Firstly, the server creates a single
matrix and sends it to all the clients parallelly. The clients run power
iterations over this matrix using their own data and send them back to the
server. This is repeated for a fixed number of rounds till the matrix
converges to the global graph embedding. The proposed approach was tested on
the ego-facebook and email-Eu-core datasets. We obtained a similarity of
98.85% and 99.8%, comparable to that of global clustering.

Our main contribution includes:

  1. 1.

We propose the state-of-the-art FL framework for spectral clustering, namely,
FedSpectral+. The approach uses the power iteration method to iteratively
learn the eigenvector embedding while maintaining data privacy.

  2. 2.

We experimentally prove the effectiveness of our algorithm on ego-Facebook and
email-Eu-core datasets and give a detailed study of factors influencing the
results of FedSpectral+.

## Related Work

The concept of federated learning(FL) has aroused a plethora of
interdisciplinary studies, primarily due to its broad applicability in
privacy-constrained scenarios (Li et al. 2020). Federated Averaging (FedAvg)
algorithm (McMahan et al. 2017), a simple baseline, which aggregates the local
client updates to obtain the global model without exchanging the raw data.
(Lalitha et al. 2019) proposed employing peer-to-peer FL for graph-structured
data. Further, for the link prediction task, (Chen et al. 2021) proposed FedE
to exploit centralized aggregation to utilize FL over knowledge graphs.
However, they lack in training the multiple knowledge graphs together.
Moreover, several works have also explored the FL on graph neural networks -
(Wang et al. 2020), (Zhang et al. 2021). But, the use of FL in spectral
clustering still remains unexplored. The nearest work includes that of (Chen
et al. 2010) that proposes parallel spectral clustering, which revolves around
the idea of using the sparse similarity matrix. However, the major drawback of
this work is that it assumes that any data in the parallel system can be
accessed from any node. But, if data is shared among nodes, this compromises
individual information, raising concerns about data privacy.

Thus, in this work, we propose FedSpectral+, a federated spectral clustering
approach that globally clusters the data shared across different clients
without sharing raw data. This approach handles the problem of data privacy by
using the advantage of the FL technique, which only aggregates the
representations learned by the clients and, thus, no direct exchange of raw
data between the clients and server. In addition, these federated spectral
settings also allow one to use large datasets without worrying about the
computational cost, as the data is distributed over several clients.

## Problem Statement and Real World

This section explains the rationale for recommending the FedSpectral+ strategy
and provides a concrete example of how it can be applied in practice.

Let there be different social network platforms. Each platform has information
about the unique phone number or email address of every person registered in
the network, which is the primary key in their database. For every
person(represented by a node), there will be a relationship between pair of
nodes(represented by an edge) within the master set of nodes. Here, the master
set of nodes refers to all the phone numbers/unique ids available worldwide.
If a number is not registered on a particular platform, the social network
platform still takes it into consideration by making zero edges of that id
with other people/ids. Furthermore, we can consider edges to hold the weight
equivalent to the amount of interaction between two nodes (number of messages
exchanged). There is a possibility that a person interacting with another
person on a particular social network might not interact with the same person
on another. If a person interacts with a large number of people and has more
than average information exchange, capturing that person’s attention to a
product can help reduce advertisement costs. It is because if that person
likes a product, he/she will do the free advertisement to his/her friends.
Each social network platform privately stores the interaction information due
to privacy concerns. However, without the knowledge of the entire graph, it
will be hard to learn the amount of interaction among individuals over the
global graph. Thus, our algorithm solves this problem using the federated
learning approach.

## Proposed Approach

In this paper, we eliminate the drawback of privacy concerns in distributed
spectral clustering by generating the eigenvector embedding of the global
graph without revealing raw data to the server. Our first approach is the
baseline - FedSpectral algorithm. Here, the clients create the spectral
clustering labels of the local graph data. The server receives these
clustering labels, which further creates the similarity graph by aggregating
the clusterings. While there is no exchange of raw information between any
clients, the algorithm doesn’t consider the relation of particular pair of
nodes within two different clients. Due to this, the output clustering
significantly deviates from the global clustering. To overcome this, we
proposed the FedSpectral+ approach that uses the power iteration method to
generate the client’s eigenvector embeddings, which also considers the
information from all other local clients by utilizing the aggregated result of
the server. This global aggregation helps us to learn the overall graph
structure. Thus this approach preserves privacy and produces significantly
better clustering than the baseline.

### Preliminaries

  1. 1.

global clustering: It is the spectral clustering of the entire graph.

  2. 2.

global aggregated clustering: It is the clustering resulting from the
federated server-side algorithms described in this section.

  3. 3.

Power iteration: A method to iteratively find the approximate eigen embedding
of a matrix.

  4. 4.

Similarity Graph: The similarity graph is constructed by assigning a value to
each pair of nodes by using equation 1.

  5. 5.

Overlap of graph: The overlap of the graph is the fraction of times the same
edge is repeated over the clients.

  6. 6.

global embedding: Actual eigenvector embedding constructed using spectral
clustering on the entire graph.

### Setting

Here, we discuss how the graph is distributed among clients for experimental
purposes. We create a null adjacency matrix for each client with the size of
the total number of nodes in the graph. Then each edge is randomly assigned to
the client uniformly. To regulate the overlap of a graph, i.e., the number of
clients that will hold a particular edge, we assign the edge to a fixed number
of random clients. Thus, creating our dataset for experimental purposes.

Input: client: client identifier

numOfClusters: Number of clusters to be formed in clustering the graph data

Output: S: List of labels of N nodes after clustering

Internal Variables inside client’s memory:

numOfNodes: Number of nodes in graph

L: Normalized Laplacian matrix of size $N^{2}$ where $N$ is number of nodes in
graph

$eigenVectors$ = $computeEigenvectors(L)$

Sort eigenVectors based on eigenvalues

$vectorEmbedding$ = Transpose of $eigenVectors$ matrix

$labels$ = $KMeansClustering(vectorEmbedding)$

Return: $labels$

function name: getClientLabels

Algorithm 1 FedSpectral, Client Code

function name: getClientLabels

Input: clientList: client list

numOfClients: Number of clients

numOfNodes: Number of nodes in graph

numOfClusters: Number of clusters

Output: S: List of labels of N nodes after global clustering

similarityGraph: initialize a similarity matrix of size $N^{2}$ with zeros as
the similarity matrix of graph

for  _client in clientList_ do

labels = $getClientLabels(client,numOfClusters)$

for  _i in range(N)_ do

for  _j in range(N)_ do

if $labels[i]==labels[j]$

$similarityGraph[i][j]$

$+=$ $1/numOfClients$

end for

end for

end for

$L$ = $createLaplacian(similarityGraph)$

$eigenVectors$ = $computeEigenvectors(L)$

Sort eigenVectors based on eigenvalues

$vectorEmbedding$ = Transpose of $eigenVectors$ matrix

$globalLabels$ = $KMeansClustering(vectorEmbedding)$

Return: $globalLabels$

Algorithm 2 FedSpectral, Server Code

###  FedSpectral Algorithm (Baseline)

Spectral clustering in distributed settings faces the problem of data privacy
because it requires sharing the edge data of graphs to create the embedding.
As a result, it reveals the relationship between nodes in the graph of a
particular client. We propose our baseline - FedSpectral approach, which does
not share the graph’s edge information between the clients or the server.

In this approach, each client calculates the spectral clustering of its local
graph. The labels of all nodes are then delivered to the server by each
client. The server then creates a similarity graph using the clustering
information received from all clients. The similarity graph is constructed by
assigning a value to each pair of nodes using equation 1. Finally, the server
performs spectral clustering on the resulting similarity graph to calculate
global clustering.

Algorithm 1 provides pseudo-code for the client side of this approach. It
calculates the cluster labels of the graph data that is local to the client
using spectral clustering. Algorithm 2 provides pseudo-code for the server
side of this approach. It gets the local spectral clustering labels from each
client in the list clientList. It then uses the labels received from the
client to create a similarity graph using the rule :

$similarityGraph[node_{i}][node_{j}]=\\\
\frac{\\#\>of\>clients\>implying\>nodes(i,j)\>have\>same\>labels}{\\#\>of\>clients}$
(1)

Finally, the server applies spectral clustering on this $similarityGraph$ and
returns the global aggregated clustering.

###  FedSpectral+ Algorithm

Spectral clustering needs the entire graph to be kept in a single memory,
which limits the method’s use. However, sharing raw graph data with the global
server and other clients raises privacy problems. The baseline - FedSpectral
technique ignores the primary idea of spectral clustering that an embedding
should be constructed using the knowledge of the entire graph rather than
sections of graphs within individual clients. This is because a particular
node in two separate client datasets can have a different vector embedding. As
a result, we are unable to employ the method of calculating spectral embedding
individually for each client. We need to find a way to aggregate the
eigenvector embedding of clients to learn the representation of the entire
graph. In addition, we also need the approach to achieve the former without
sharing information about the presence of an edge within the data of a local
client. Therefore, in this approach, we use the power iteration method to
iteratively learn the eigenvector embedding of the global graph while
preventing the exchange of edge data amongst the clients and server.

Input: client: client identifier

iters: Number of power iterations

eigenVectors: A list of smallest $K$ approximate eigenvectors of global graph,
dimension: $N\times K$

Output: eigenVectors: Modified list of smallest $K$ approximate eigenvectors
of global graph, dimension: $N\times K$

Internal Variables inside client’s memory:

numOfNodes: Number of nodes in graph

L: Normalized Laplacian matrix of size $numOfNodes^{2}$ of client

I: Identity matrix of size $numOfNodes^{2}$

$L\>=\>I\>-\>L$

for  _iter in range( $iters$)_ do

$eigenVectors$ = $L\times eigenVectors$

end for

Return: $eigenVectors$

function name: getPowerIterationClient

Algorithm 3 FedSpectral+, Client Code

function name: getPowerIterationClient

In this approach, we want to learn the embedding as close as possible to the
global embedding. The clients initially learn the local embedding using the
power iteration method over their Laplacian matrix and send it to the server.
Now, the server aggregates(averages) all the embeddings from the local clients
and calls this the approximate embedding. The server then sends this
approximate embedding to the clients. The client repeats the power iteration
method using this learned approximate embedding to incorporate the local graph
structure. Here, each node relates to the neighboring nodes of the other
clients using approximate embedding. Thus, it gives the impression that we are
learning the embedding by incorporating the features of the entire graph. As
the number of aggregation rounds increases, the approximate embedding becomes
nearer to the global embedding since the global graph structure gets clearer
at every step. In the proposed approach, the only exchange of information
includes sending the approximate embedding between the server and the clients.
Therefore, there is no direct sharing of raw information about the local graph
data, maintaining privacy in our federated approach.

Input: clientList: client list

numOfClients: Number of clients

numOfNodes: Number of nodes in graph

numOfClusters: Number of clusters

iters: Number of power iteration per client

globalRounds: Number of global aggregation rounds

Output: S: List of labels of N nodes after global clustering

v: initialize a $numOfNodes\times numOfCluster$ matrix randomly

for  _round in range( $globalRounds$)_ do

p = $numOfClients$ copies of $v$

for  _client in range(numOfClients)_ do

$clientId$ = $clientList[client]$

$p[client]$ = $getPowerIterationClient($

$clientId,numOfClusters,iters,p[client])$

end for

$v$ = average of all matrices in $p$

$v,r$ = $qrDecomposition(v)$

end for

$globalLabels$ = $KMeansClustering(v)$

Return: $globalLabels$

Algorithm 4 FedSpectral+, Server Code

Algorithmically, we randomly initialize a matrix $v$ of size $N\times K$.
Given the nature of spectral clustering, if there are $K$ clusters, then the
first $K$ eigenvectors of the graph’s Laplacian matrix should be used to
generate the nodes’ vector embeddings. Therefore, after completion of the
algorithm, each of the columns of $v$ will represent one of the first $K$
eigenvectors of the global graph. Let $C$ be the number of clients
contributing to the clustering. We create $C$ copies of $v$ and call it $p$.
Then we pass these matrices to their corresponding clients. Let $p[i]$ be the
$v$ matrix of client $i$. Clients will now run the power iteration approach to
calculate the first $K$ eigenvectors simultaneously on their local graphs.
After completing the power iterations, all clients will send their $p[i]$
matrix to the global server. The server will compute the average of all the
$p[i]$ matrices of clients and then orthogonalize the resulting matrix using a
QR decomposition. It will assign the $q$ of the QR decomposition to $v$. Again
the server will share $C$ copies of this aggregated matrix $v$ to clients, and
then the same process will be repeated for a number of $globalRounds$. After
completing all the global rounds, we get the approximate first $K$
eigenvectors in the form of the $N\times K$ matrix, with each column
representing one eigenvector. Since $v$ matrix is the eigenvector embedding of
the global graph, we directly apply $KMeans$ clustering algorithm over this
vector embedding and return the $globalLabels$.

Algorithm 3 provides the pseudo-code for the client side of this approach.
Since the power iteration method calculates the eigenvectors corresponding to
the largest eigenvalues, each client applies this function to the normalized
local Laplacian matrix: $L\>=\>I\>-\>L$. It then runs the power iterations
over $eigenVectors$ with $L$ as the matrix multiplier. Finally, it returns the
modified $eigenVectors$. For the pseudo-code of server-side of this algorithm,
refer Algorithm 4. The QR decomposition used in this approach is taken from
the NumPy library of python (numpy.linalg.qr()).

Input: numOfNodes: number of nodes in graph

aggregatedLabels: labels from the federated algorithm

globalLabels: labels from non distributed setting

Output: $clusterSimilarity$

$misMatch$ = 0

for  _i in range(N)_ do

for  _j in range(N)_ do

if $globalLabels[i]==globalLabels[j]$

if $aggregatedLabels[i]!=aggregatedLabels[j]$

$misMatch$ $+=$ $1$

end for

end for

$clusterSimilarity$ = $1-\frac{misMatch}{numOfNodes^{2}}$

$Return:$ $clusterSimilarity$

Algorithm 5 clusterSimilarityMetric Figure 1: The figure a) on the left, shows
the plot of clustering similarity Vs iteration per client on two datasets. The
figure b) on the right, shows the plot of clustering similarity Vs global
rounds on two datasets.

## Evaluation

### Metric

We have created our own metric for comparing the similarity between two
different clustering of a graph. Our metric algorithm penalizes the similarity
score for each pair of nodes if the pair lie in the same cluster in global
clustering and lie in different clusters in global aggregated clustering. The
similarity score ranges from 0 to 1. A higher similarity score represents that
both the clustering are more similar. Refer to algorithm 5 for the pseudo-code
of similarity score. Please Note: The y-axis in the plots 1, 2 and 3 is the
similarity metric defined above.

### Datasets

  1. 1.

ego-Facebook (Leskovec and Krevl 2014): It is an undirected social network
graph dataset consisting of 4039 nodes and 88234 edges. The data is made up of
anonymized Facebook social circles.

  2. 2.

email-Eu-core (Leskovec and Krevl 2014): It is a directed network graph
dataset consisting of 1005 nodes and 25571 edges. It is an email communication
link dataset among the members of an organization. We converted this graph to
undirected for experimental purpose.

Figure 2: The figure a) on the left, shows the plot of how the clustering
similarity Vs iteration per client changes with respect to global rounds on
ego-Facebook dataset. The figure b) on the right, shows the plot of how the
clustering similarity Vs iteration per client changes with respect to the
number of clusters on ego-Facebook dataset.

### Experimental Setting

In FedSpectral approach:

  * •

We used 5 clients and 10 clusters as the parameters.

In FedSpectral+ approach:

  * •

For both dataset, we used 5 clients and 10 clusters as the parameters.

  * •

For the ego-Facebook dataset, we set $iters$ as 6 and $globalRounds$ as 20 for
the best results.

  * •

For the email-Eu-core dataset, we set $iters$ as 1 and $globalRounds$ as 1 for
the best results.

### Results and Analysis

We experimented with our FedSpectral and FedSpectral+ approach on ego-Facebook
and email-Eu-core datasets.

  * •

The FedSpectral approach achieved a clustering similarity of 77.63% on ego-
Facebook dataset and 87.05% on email-Eu-core dataset respectively.

  * •

The FedSpectral+ approach achieved a clustering similarity of 98.85% on ego-
Facebook and 99.8% on email-Eu-core dataset respectively.

In Figure 1, a) shows the variation in similarity with the varying number of
iterations for ego-Facebook and email-Eu-core dataset with constant 20 and 1
global rounds respectively on the FedSpectral+ algorithm with graph
distributed using 40% overlap.

  * •

We can observe from the plot that as the number of iterations increases for
ego-Facebook, the similarity increases and then again drops. The increase in
similarity is because the number of iterations required to converge the
eigenvectors is not sufficient. However, the similarity drops because
aggregation of eigenvectors is done after a larger number of power iterations.
Thus, it focuses more on the local graph embedding instead of the global
graph.

  * •

We can observe from the plot that as the number of iterations increases for
email-Eu-core dataset, the similarity remains almost constant. Here, the
number of edges in the graph is very high in comparison to ego-Facebook the
number of nodes. However, it is a small dataset, so it takes lesser iterations
to converge. Since, for each client, the local graph requires only one
iteration to converge to the local embedding, increasing the iterations does
not affect the similarity as the number of global rounds is fixed to 1.

Figure 3: The figure a) on the left, shows the plot of how the clustering
similarity Vs iteration per client changes with respect to the overlap of the
graph on ego-Facebook dataset. The figure b) on the right, shows the plot of
how the FedSpectral and FedSpectral+ approach performs in terms of clustering
similarity when the number of clients is varied on ego-Facebook dataset.

In Figure 1 b) shows the variation in similarity with the varying number of
global rounds for ego-Facebook and email-Eu-core dataset with constant 10
iterations per client on the FedSpectral+ algorithm with graph distributed
using 40% overlap.

  * •

For the ego-Facebook dataset, we can observe from the plot that as the number
of global rounds increases, the similarity increases. It is because the higher
the number of global rounds, the higher the convergence of FedSpectral+
method.

  * •

For the email-Eu-core dataset, we can observe from the plot that as the number
of global rounds increases, similarity remains constant since the algorithm
converges in the first global round.

Figure 2 a) shows how the similarity plot changes with the number of global
rounds for FedSpectral+. We can observe that for a fixed number of iterations,
as the number of global rounds increases, the similarity increases. This is
because the method converges better with more global rounds.
Figure 2 b) shows how the similarity plot changes with the number of clusters
for the FedSpectral+ algorithm. Since the ego-Facebook dataset has 10
communities, we can observe that for a fixed number of iterations, the
similarity is highest for 10 clusters in general.

Figure 3 a) shows how similarity plot changes with the overlap of graph data
distribution for the FedSpectral+ algorithm. We can observe that with higher
graph overlap, the similarity increases in general. It is because the local
embedding becomes more informative because of the availability of more
information about the global graph. Therefore, in the aggregation step at the
global server, the deviation in the local embeddings from different clients is
smaller, resulting in better similarity.

In Figure 3 b) shows the comparison of FedSpectral approach with the
FedSpectral+ method. We can see that as the number of clients increases, and
if we keep the overlap of graph data within the clients to be fixed, then the
similarity drops. It is because the graph structure present with a particular
client becomes sparsed as the edges are distributed over more clients. We can
also observe that FedSpectral+ approach works significantly better than the
baseline - FedSpectral.

Figure 4: Visual representation of ego-Facebook dataset using FedSpectral
approach on right, and the global clustering on left. Figure 5: Visual
representation of ego-Facebook dataset using FedSpectral+ approach on right,
and the global clustering on left.

For the visual comparison, we used the ego-Facebook dataset. We can observe
that in Figure 4, the FedSpectral approach cannot distinguish all the clusters
in comparison to the global clustering. It is because the FedSpectral approach
disregards that a particular node in a client may not have information about
all the neighboring nodes which are spread over other clients. However, in
Figure 5, the FedSpectral+ approach is able to get the highest detailing in
its clustering of the ego-Facebook dataset. This is because the embedding is
constructed using the power iteration method on each client and then
aggregated by the server to learn the embedding of the global graph.

## Conclusion

To conclude, in this work, we proposed a state-of-the-art approach -
FedSpectral+, a novel algorithm for federated spectral clustering. We used the
power iteration method to iteratively learn the eigenvector embedding of the
global graph. As there is no exchange of raw data between clients and the
server, the proposed approach overcomes the main issue of spectral clustering
in the distributed setting by providing data privacy. We also give a detailed
analysis of factors impacting the FedSpectral+. As validated from the
experiments, the similarity of clustering obtained using the proposed approach
was comparable to that of global clustering. Thus, proving the efficacy of our
approach.

## Future Work

The suggested approach was only evaluated for its performance. In the future,
we plan to test how resilient our algorithm is against different adversarial
attacks, such as inference and poisoning attacks (Shen et al. 2020). We also
intend to assess the possibility of a data breach while transmitting local
updates to the server. To further strengthen data privacy, we will improve our
method by combining it with differentially private algorithms and secured
multiparty computation.

## Acknowledgement

We would like to thank Prof. Anirban Dasgupta (IIT Gandhinagar) for his
continuous support and guidance throughout the research.

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  *

# The advance of Mercury’s perihelion

Bertrand Berche1 , Ernesto Medina2 0000-0002-4254-807X 0000-0002-1566-0170 1
Laboratoire de Physique et Chimie Théoriques, Université de Lorraine - CNRS,
Nancy, France  2 Departamento de Física, Colegio de Ciencias e Ingeniería,
Universidad San Francisco de Quito, Diego de Robles y Vía Interoceánica,
Quito, 170901, Ecuador<EMAIL_ADDRESS>
###### Abstract

A very famous “test” of the General Theory of Relativity (GTR) is the advance
of Mercury’s perihelion (and of other planets too). To be more precise, this
is not a prediction of General Relativity, since the anomaly was known in the
XIXth century, but no consistent explanation had been found yet at the time
GTR was elaborated. Einstein came up with a solution to the problem in 1914.
In the case of Mercury, the closest planet to the Sun, the effect is more
pronounced than for other planets, and observed from Earth; there is an
advance of the perihelion of Mercury of about 5550 arc seconds per century
(as/cy). Among these, about $5000$ are due to the equinox precession (the
precise value is $5025.645$ as/cy) and about $500$ ($531.54$) to the influence
of the external planets. The remaining, about $50$ as/cy ($42.56$), are not
understood within Newtonian mechanics. Here, we revisit the problem in some
detail for a presentation at the undergraduate level.

††: Eur. J. Phys.

Keywords: Mercury, perihelion, Kepler problem, General Relativity

##  1 Introduction

The problem of the advance of Mercury’s perihelion is a well-known example of
a phenomenon that remained unexplained for a long time and which attracted the
attention of many physicists in the 19th century before finding an
interpretation within the framework of the theory of General Relativity. It is
now a textbook case that played an important role in the acceptance of this
theory and which is taught to illustrate the successes of General Relativity.
The study of this problem is very rich because it allows us to illustrate the
usefulness of perturbative calculations in classical Newtonian dynamics to
understand the most important part of the effect observed as being due to the
influence of external planets on the movement of Mercury, before devoting
oneself to the relativistic approach of the gravitational effect due to the
Sun which fully explains the residual advance, extremely tenuous, with a
remarkable precision. However, the literature does not generally present what
could be considered as the ultimate verification: how the influence of
external planets, when treated in a relativistic framework, does not bring an
additional correction, which could compete with the exceptional agreement
between General Relativity and observations. The purpose of this article is to
present this entire approach at an undergraduate level.

##  2 The Kepler problem in Newton dynamics and the statement of the problem

As a starting point, let us consider the Newtonian approach to planets’ motion
around the Sun. This is an undergrad problem that can be found in all
textbooks on classical mechanics [1, 2]. We look for bound states in the
Newtonian gravitational potential of the Sun. The gravitational force is
central, therefore the angular momentum ${\bm{\sigma}}={\bf r}\times m{\bf
v}=mr^{2}\dot{\theta}\ \\!{\bf u}_{\varphi}-mr^{2}\sin\theta\dot{\varphi}\
\\!{\bf u}_{\theta}$ is conserved (we use spherical coordinates, and the dot
denotes a derivative w.r.t. time). The planet motion is thus confined to stay
in a plane perpendicular to ${\bm{\sigma}}$ w.r.t. which the angle $\theta$ is
measured, ($\theta$ is fixed to $\pi/2$) and
${\bm{\sigma}}=-mr^{2}\dot{\varphi}{\bf
u}_{\theta}(\pi/2)=+mr^{2}\dot{\varphi}{\bf u}_{z}$ while ${\bf v}=\dot{r}{\bf
u}_{r}+r\dot{\varphi}{\bf u}_{\varphi}$. The expression of the square of the
velocity $|{\bf v}|^{2}=\dot{r}^{2}+|{\bm{\sigma}}|^{2}/(m^{2}r^{2})$ follows
and is used in the second constant of motion, the total energy,

$E={\textstyle\frac{1}{2}}m|{\bf
v}|^{2}-\frac{GM_{\odot}m}{r}=\frac{|{\bm{\sigma}}|^{2}}{2{m}r^{2}}\Bigl{(}\frac{1}{r^{2}}\Bigl{(}\frac{dr}{d\varphi}\Bigr{)}^{2}+1\Bigr{)}-\frac{GM_{\odot}m}{r}$
(1)

where we also used $\dot{r}=\dot{\varphi}\ \\!dr/d\varphi$. Here,
$G=6.67430\times 10^{-11}{\rm m}^{3}\ \\!{\rm kg}^{-1}{\rm s}^{-2}$ is
Newton’s gravity constant and $M_{\odot}=1.9891\times 10^{30}$kg the mass of
the Sun, while $m=M_{\hbox{\mercury}}=3.285\times 10^{23}$kg is the mass of
Mercury (data for the planets of the solar system are listed in table 1). We
rewrite the energy equation in the form

$\frac{1}{r^{2}}\Bigl{(}\frac{dr}{d\varphi}\Bigr{)}^{2}+1=\frac{2mr^{2}}{|{\bm{\sigma}}|^{2}}\Bigl{(}E+\frac{GM_{\odot}m}{r}\Bigr{)}$
(2)

and perform a change of variable $u=1/r$ to get

$\Bigl{(}\frac{du}{d\varphi}\Bigr{)}^{2}+u^{2}=\frac{2m}{|{\bm{\sigma}}|^{2}}(E+{GM_{\odot}m}u)$
(3)

which is then differentiated w.r.t. $\varphi$ to lead to a harmonic equation:

$\frac{d^{2}u}{d\varphi^{2}}+u=\frac{GM_{\odot}m^{2}}{|{\bm{\sigma}}|^{2}}.$
(4)

Adding a particular solution of the complete equation,
$u_{0}=\frac{GM_{\odot}m^{2}}{|{\bm{\sigma}}|^{2}}$, to a general solution
$A\cos(\varphi-\alpha)$ of the homogeneous equation, and fixing the initial
conditions to have $\alpha=0$, we obtain the solution

$u(\varphi)=\frac{GM_{\odot}m^{2}}{|{\bm{\sigma}}|^{2}}(1+e\cos\varphi)$ (5)

where $e$ is called the eccentricity. The value of $e$ is fixed in terms of
the constants by the first equation of motion (3) where the insertion of
$u(0)=u_{0}(1+e)$, $u^{\prime}(0)=0$ provides the second order equation
$u_{0}^{2}(1+e)^{2}-2u_{0}(1+e)-2mE/|{\bm{\sigma}}|^{2}=0.$ We get closed
ellipses when $E<0$ and the parameters of the trajectory are given by

$\frac{1}{r}=\frac{1}{p}(1+e\cos\varphi),\quad\frac{1}{p}=u_{0}=\frac{GM_{\odot}m^{2}}{|{\bm{\sigma}}|^{2}},\quad
e=\sqrt{1+\frac{2E|{\bm{\sigma}}|^{2}}{G^{2}M_{\odot}^{2}m^{3}}}.$ (6)

In the case of Mercury, $e=0.205630$ and $p=a(1-e^{2})=55.46\times 10^{9}$m
with $a=57.91\times 10^{9}$m the semi-major axis. Since $E~{}{\propto}~{}m$
and $|{\bm{\sigma}}|~{}{\propto}~{}m$, $p$ and $e$ do not depend on $m$; so
the motion is completely fixed by the initial conditions, but doesn’t depend
on the planet mass $m$ (hence the name universal gravitation theory).

Planet | Mass (kg) | perihelion (km) | aphelion (km)
---|---|---|---
Mercury, ☿ | $3.302\times 10^{23}$ | $46.0\times 10^{6}$ | $69.8\times 10^{6}$
Venus, ♀ | $4.868\times 10^{24}$ | $107.5\times 10^{6}$ | $108.9\times 10^{6}$
Earth, ♁ | $5.974\times 10^{24}$ | $147.1\times 10^{6}$ | $152.1\times 10^{6}$
Mars, ♂ | $6.418\times 10^{23}$ | $206.6\times 10^{6}$ | $249.2\times 10^{6}$
Jupiter, ♃ | $1.899\times 10^{27}$ | $740.7\times 10^{6}$ | $816.1\times 10^{6}$
Saturn, ♄ | $5.685\times 10^{26}$ | $1.349\times 10^{9}$ | $1.504\times 10^{9}$
Uranus, ⛢ | $8.683\times 10^{25}$ | $2.735\times 10^{9}$ | $3.006\times 10^{9}$
Neptune, ♆ | $1.024\times 10^{26}$ | $4.459\times 10^{9}$ | $4.537\times 10^{9}$
Sun, ☉ | $1.989\times 10^{30}$ |  |
Table 1: Masses and distances in the solar system.

In conclusion, the previous approach assumes point-like masses describing
Mercury and the Sun with Newtonian central forces operating; the elliptical
orbit of Mercury is closed; thus, the perihelion always occurs at the same
place in the orbital motion. In the literature this fixed orientation of the
orbit is characterized by the conserved (fixed) Runge-Lenz vector ${\bf
A}={\bf p}\times{\bf L}-mk\hat{\bf r}$ .

Now, since we know that the observations show that the motion indeed stays in
a plane, but the ellipse is not closed in reality, its perihelion, the closest
distance to the sun, rotates (actually, it precesses). In the case of Mercury,
the closest planet to the Sun, this is more pronounced than for other planets.
Observed from the Earth, there is a known advance of the perihelion of Mercury
of $5599.745$ arc seconds per century (as/cy) (the data are from Ref. [3]),
less than two degrees per century, which doesn’t seem to be a very large
amount, but astronomical observations were already very accurate long ago! Of
these, the equinox precession is responsible for most of the observed value,
$5025.645$ as/cy. This is the first and largest contribution to the perihelion
advance, and it is because the Earth (from where we observe the phenomenon) is
not perfectly spherical, and the combined effect of the Sun and the Moon
exerts a torque on the Earth which produces a precession of the daily rotation
axis of the Earth with a period of about $26000$ years, first observed by
Hipparcos in 125 BC. The order of magnitude of these $\approx 26000$ years is
thus $360\times 3600/260\simeq 5000$ as/cy. As said above, this is an order of
magnitude, and the measurement is $5025.645$. This is not an anomaly of
Mercury’s motion but a relative effect due to the motion of the observer
located on Earth.

The question we address now is about the remaining differences and the
importance of the various causes we may invoke.

##  3 The influence of external planets within the Newtonian theory of
gravitation

The second effect is more difficult to analyze, even in Newtonian dynamics. We
consider the effect on Mercury’s motion of an external planet P of mass
$M_{\rm P}$, the gravitational influence of which is assimilated to that of a
ring of matter of radius $R_{\rm P}$, i.e., the mean radius of the orbit of P
[4, 5], centered on the position of the Sun. This is obviously an
approximation, but we will see that the predictions are consistent in terms of
orders of magnitude with more sophisticated calculations found in the
literature. This model introduces a non-Newtonian ($1/r^{2}$) force effect,
which breaks the conservation of the Runge-Lenz vector and makes it rotate,
producing a precession of the perihelion. This circular ring of matter carries
a linear mass density of $\lambda=M_{\rm P}/(2\pi R_{\rm P})$. An important
property here is that all the planets have their orbits almost on the same
plane, so Mercury’s position is in the plane of planet P’s orbit. If one
considers two symmetric positions on the “ring”, they both carry a mass
element $dm=\lambda R_{\rm P}d\theta$ and produce on Mercury’s position a
gravitational field contribution which, after being projected on the direction
from the Sun to Mercury, is written as

$|d{\bf G}_{\rm P}|=G\lambda R_{\rm
P}d\theta\Bigl{(}\frac{\cos\phi_{1}}{r_{1}^{2}}-\frac{\cos\phi_{2}}{r_{2}^{2}}\Bigr{)}\simeq
G\lambda d\theta\Bigl{(}\frac{1}{r_{1}}-\frac{1}{r_{2}}\Bigr{)}d\theta.$ (7)

The angles and distances are defined in figure 1. The last equality comes from
the approximation $R_{\hbox{\mercury}}\simeq p<R_{\rm P}$ (for example in the
case of Venus, $R_{\rm P}$, of the order of the semi-major axis, is about
$a_{\hbox{\venus}}=108.209\times 10^{9}$km), in which case
$\phi_{1}\simeq\phi_{2}\simeq\theta$. Then, with

$\displaystyle r_{1}=-r\cos\theta+\sqrt{r^{2}\cos^{2}\theta-(r^{2}-R_{\rm
P}^{2})},$ (8) $\displaystyle
r_{2}=+r\cos\theta+\sqrt{r^{2}\cos^{2}\theta-(r^{2}-R_{\rm P}^{2})},$ (9)

we get a (repulsive) central gravitational field after integration over
$\theta$,

$\delta{\bf G}_{\rm P}=+\frac{GM_{\rm P}}{2R_{\rm P}}\frac{r}{R_{\rm
P}^{2}-r^{2}}{\bf u}_{r}\simeq\frac{GM_{\rm P}}{2R_{\rm
P}^{3}}\Bigl{[}r+\frac{r^{3}}{R_{\rm P}^{2}}\Bigr{]}{\bf u}_{r},$ (10)

where the expansion is allowed because the radial distance of Mercury to the
Sun $r<R_{\rm P}$, the radius of the orbits of the external planets
(typically, for Mercury and Venus, the ratio $r/R_{\rm P}$ is about $0.5$ – it
could be worth taking into account the second-order expansion – and the
correction is smaller for the other planets).

Figure 1: Gravitational influence of an external planet P on the motion of
Mercury (internal trajectory) in the solar system.

It is convenient to write the gravitational potential due to the planet P:

$\displaystyle\delta\phi_{\rm P}$ $\displaystyle=$
$\displaystyle-\int\delta{\bf G}_{\rm P}\cdot d{\bf r}$ (11) $\displaystyle=$
$\displaystyle\frac{GM_{\rm P}}{4R_{\rm P}}\ln\Bigl{(}1-\frac{r^{2}}{R_{\rm
P}^{2}}\Bigr{)}\simeq-\frac{GM_{\rm P}}{4R_{\rm
P}^{3}}\Bigl{[}r^{2}+\frac{r^{4}}{2R_{\rm P}^{2}}\Bigr{]}$

and the total gravitational potential energy (Sun plus the Planet P) as a
correction to the simple effect of the Sun,

$\displaystyle V(r)$ $\displaystyle=$
$\displaystyle-\frac{GM_{\odot}m}{r}+\delta V_{\rm P}(r)$ (12)
$\displaystyle\simeq$ $\displaystyle-\frac{GM_{\odot}m}{r}-\frac{GM_{\rm
P}m}{4R_{\rm P}^{3}}\Bigl{[}r^{2}+\frac{r^{4}}{2R_{\rm
P}^{2}}\Bigr{]}\quad\hbox{for $r<R_{\rm P}$}$ $\displaystyle=$
$\displaystyle-\frac{GM_{\odot}m}{r}\Bigl{(}1+\frac{M_{\rm
P}}{M_{\odot}}\frac{r^{3}}{4R_{\rm P}^{3}}\Bigl{[}1+\frac{r^{2}}{2R_{\rm
P}^{2}}\Bigr{]}\Bigr{)}.$

With the analytic form of the correction $\delta V_{\rm P}(r)$ to the purely
Newtonian gravitational potential energy, it is possible to estimate the
effect on the trajectory of Mercury using a perturbation analysis.

In Section 2, we have established the equation of motion (4) for the Kepler
problem, i.e. the motion of $m$ in the gravitational field exerted by the Sun.
For a more general central potential corresponding now to Sun plus planet P,

$V(u)=-GM_{\odot}mu+\delta V_{\rm P}(u),$ (13)

the equation of motion becomes

$\frac{d^{2}u}{d\varphi^{2}}+u=-\frac{m}{|{\bm{\sigma}}|^{2}}\frac{dV(u)}{du}=\frac{GM_{\odot}m^{2}}{|{\bm{\sigma}}|^{2}}-\frac{m}{|{\bm{\sigma}}|^{2}}\frac{d\delta
V_{\rm P}(u)}{du}.$ (14)

If we denote $u_{\rm K}$ the solution of Kepler problem and we seek for a
perturbation solution, setting $u(\varphi)=u_{\rm K}(\varphi)+u_{1}(\varphi)$,
the derivative of $\delta V_{\rm P}(u)$ w.r.t. $u$ at the r.h.s. is expanded
in the vicinity of Kepler solution, and the equation of motion becomes

$\displaystyle\frac{d^{2}u_{\rm K}}{d\varphi^{2}}+u_{\rm
K}+\frac{d^{2}u_{1}}{d\varphi^{2}}+u_{1}=\frac{1}{p}-\frac{m}{|{\bm{\sigma}}|^{2}}\left.\frac{d\delta
V_{\rm P}(u)}{du}\right|_{u_{\rm
K}}-\frac{m}{|{\bm{\sigma}}|^{2}}\left.\frac{d^{2}\delta V_{\rm
P}(u)}{du^{2}}\right|_{u_{\rm K}}u_{1}+\dots$

Simplifying the equation at the Kepler problem level, we obtain an equation
for the correction

$\frac{d^{2}u_{1}}{d\varphi^{2}}+\Bigl{(}1+\frac{m}{|{\bm{\sigma}}|^{2}}\left.\frac{d^{2}\delta
V_{\rm P}(u)}{du^{2}}\right|_{u_{\rm
K}}\Bigr{)}u_{1}=-\frac{m}{|{\bm{\sigma}}|^{2}}\left.\frac{d\delta V_{\rm
P}(u)}{du}\right|_{u_{\rm K}}$ (16)

where we read that the angular frequency of the perturbed motion is

$\Omega^{2}=1+2\omega_{1}=1+\frac{m}{|{\bm{\sigma}}|^{2}}\left.\frac{d^{2}\delta
V_{\rm P}(u)}{du^{2}}\right|_{u_{\rm K}},$ (17)

hence the solution $u_{1}(\varphi)=A\cos(\Omega\varphi)+B\sin(\Omega\varphi)$.

The perihelion (the smallest value of $r(\varphi)=1/u(\varphi)$) corresponds
to the largest value of $u(\varphi)$. It is obtained at $\varphi=0$ and equals
to $u_{\rm max}=(1+e)/p+A$, ($A>0$). The same value is recovered slightly
after a revolution at $\varphi=2\pi+\Delta\varphi$ (it will appear that
$\Delta\varphi>0$) such that $u_{\rm
K}(2\pi+\Delta\varphi)=(1+e)/p+O(\Delta\varphi^{2})$ and
$u_{1}(2\pi+\Delta\varphi)=A+B(2\pi(\Omega-1)+\Omega\Delta\varphi)+O(\Delta\varphi^{2})$.
To linear order in $\Delta\varphi$, the perihelion is recovered if the
coefficient of $B$ vanishes. It follows that the advance (this will appear to
be positive) of the perihelion per revolution due to the gravitational force
exerted by the planet P is written as:

$\Delta\varphi_{\rm
P}=\frac{2\pi(1-\Omega)}{\Omega}=-2\pi\omega_{1}=-2\pi\frac{1}{2}\frac{m}{|{\bm{\sigma}}|^{2}}\left.\frac{d^{2}\delta
V_{\rm P}(u)}{du^{2}}\right|_{u_{\rm K}}.$ (18)

Using the expression of $\delta V_{\rm P}(u)$ we get to leading order

$\Delta\varphi_{\rm P}=2\pi\frac{3M_{\rm P}}{4M_{\odot}}\frac{p^{3}}{R_{\rm
P}^{3}}=\pi\frac{3}{2}\frac{M_{\rm
P}}{M_{\odot}}\Bigl{(}\frac{a(1-e^{2})}{R_{\rm P}}\Bigr{)}^{3}.$ (19)

In the case of the planet Venus (for
$M_{\hbox{\venus}}/M_{\odot}={2.447}\times 10^{-6}$ and
$p/R_{\hbox{\venus}}={0.495}$, where $R_{\hbox{\venus}}$ is taken as the
arithmetic mean of the distances to the aphelion and the perihelion), we get

$\displaystyle\Delta\varphi_{\hbox{\venus}}\simeq{1.399}\times
10^{-6}\Bigl{(}\frac{{\rm rad}}{{\rm
rev}}\Bigr{)}\frac{360}{2\pi}\Bigl{(}\frac{\rm deg}{\rm
rad}\Bigr{)}\frac{3600}{1}\Bigl{(}\frac{\rm sec}{\rm
deg}\Bigr{)}\frac{1}{0.240}\Bigl{(}\frac{\rm rev}{\rm
year}\Bigr{)}\frac{100}{1}\Bigl{(}\frac{\rm year}{\rm century}\Bigr{)}$
$\displaystyle\simeq{120}\hbox{ arcsec}/\hbox{century}.$ (20)

The next term in the potential (12) adds another correction of ${0.571}\times
10^{-7}$rad/rev, or $49.087$ as/cy, hence a total contribution of the
influence of Venus at the first order perturbation expansion of $169$ as/cy.
This is the strongest correction among the planets, the next one being due to
Jupiter, which is more distant but far more massive.

Another method is used, e.g., in Ref. [6], in terms of forces, but we have
adapted it here in terms of potential energies. More precise values have been
determined numerically in an article in Am. J. Phys.[5] where we can find more
accurate data. The largest contributions are from Venus (the closest planet)
and Jupiter (the heaviest planet), and in the previous paper mentioned, Davies
reports the numerical estimates for each planet, e.g., for Venus
$\Delta\varphi_{\hbox{\venus}}={273.30}$ as/cy and for Jupiter
$\Delta\varphi_{\hbox{\jupiter}}={156.75}$ as/cy while for the Earth,
$\Delta\varphi_{\hbox{\earth}}={91.49}$ as/cy and for Uranus
$\Delta\varphi_{\hbox{\uranus}}={0.14}$ as/cy. These results are in better
agreement with those of the specialized literature (see table 2) than ours,
but the method that we employed is suitable for a presentation at the
undergrad level.

Origin |  $\Delta\varphi$ (as/cy) | Ref. [7]
---|---|---
Venus, ♀ | $277.856\pm 0.68$ |
Earth, ♁ | $90.038\pm 0.08$ |
Mars, ♂ | $2.536\pm 0.00$ |
Jupiter, ♃ | $153.584\pm 0.00$ |
Saturn, ♄ | $7.302\pm 0.01$ |
Uranus, ⛢ | $0.141\pm 0.00$ |
Neptune, ♆ | $0.042\pm 0.00$ |
Sun (☉) asphericity | $0.010\pm 0.02$ |
general precession of the equinoxes | $5025.645\pm 0.50$ |
Sum | $5557.18\pm 0.85$ |
observed advance | $5599.74\pm 0.41$ |
remaining difference | $42.56\pm 0.94$ |
GTR effect | $43.03\pm 0.03$ | $42.98$
Table 2: Various contributions to the advance of the perihelion of Mercury,
from G.M. Clemence [3]. The GTR main contribution (final value, according to
Eq. (53)) was later corrected by Nobili and Will [7].

When one sums up all the contributions, as well as a tiny effect due to the
non-exact Newtonian form of the Sun’s gravitational potential due to its non-
perfect sphericity, there remains a very small difference with the
observations. That difference is not explained by the classical theory of
gravitation, as one can read in the results [3] given in table 2.

##  4 Looking for possible explanations of the remaining $43$ arc seconds per
century

This tiny number, about $43$ arc seconds per century (compared to $~{}5599$
as/cy observed), has to find a further explanation. A slight modification to
Newton’s law of gravitation has been proposed as well as the idea of a still
unknown celestial object (which was even given a name, Vulcan), the influence
of which would add to the other planets to produce the desired $43$ arc
seconds per century, but nothing was discovered as a possible candidate.

A similar hypothesis had been put forward earlier to explain the anomalies in
the motion of the planet Uranus. It was all the merit and glory of Le Verrier
to specify by calculation the mass and position of the new planet, named
Neptune, which was observed then by Galle in Berlin111The planet was within
$1^{\circ}$ of where Le Verrier had predicted (and $10^{\circ}$ of where Adams
had sooner predicted).. In the XIXth century, carrying out perturbative
calculations was not an easy task. A page of Le Verrier’s calculations is
given in figure 2.

Figure 2: One of the 200 pages of calculation of Le Verrier. From J.-P.
Verdet, Astronomie et astrophysique, Larousse, Paris 1993, p.731.

It is also instructive to read Le Verrier himself about the motion of Uranus
[8]:

> A few years ago, we had barely begun to suspect that the movement of Uranus
> was modified by some unknown cause when all possible hypotheses were already
> hazarded on the nature of this cause. It is true that everyone simply
> followed the inclination of their imagination without providing any
> consideration to support their assertion. We thought of the resistance of
> the ether, we spoke of a large satellite which accompanied Uranus, or of a
> still unknown planet whose disturbing force should be taken into
> consideration, we even went so far as to suppose that at this enormous
> distance from the Sun, the law of gravitation could lose something of its
> rigour.

A modification of the law of gravitation at large distances is still a very
current debate today. In the context of Mercury’s anomalies, this hypothesis
was proposed as a correction to Newton’s gravitation, considered by Hall [9,
10], who had shown that any $n>2$ in a gravitational force of the form
$GMm/r^{n}$ would result in an advance of the perihelion, and that $n=2.000\
\\!000\ \\!16$ was enough to explain the mysterious $43$ as/cy of Mercury. But
then, the same $n$ was spoiling the results concerning the other planets in
the solar system, which was not acceptable.

The assumptions mentioned by Le Verrier to explain Uranus anomalies are still
among the most popular possible causes introduced in cosmology to explain the
deviations observed in the evolution of the scale parameter of the metric of
the Universe when only observable sources of gravity are considered. Dark
matter is indeed similar to the introduction of a supplementary planet,
unknown at the time, and dark energy can be considered the analogue of a
modification of the law of gravitation.

Eventually, the route of a modification of gravity will appear successful for
Mercury’s anomaly. Indeed, there is no escape and a relativistic approach has
to be used to try to solve the “tiny $43$ arc seconds per century”.

##  5 The resort to Special Relativity

First the contribution of Special Relativity should be considered, and there
has been a controversy on the role of the purely special relativistic
contribution, as one can see in this “ironic” quotation [11]222We keep the
reference numbering of the original quotation.:

> The question arises as to what is the prediction from “Special Relativity”.
> The literature on this is rather erratic.
>
> Early work (1906-1911) by Poincaré [21, 22], Lorentz [23], de Sitter [24]
> and others (…) inferred that the result from Special Relativity for the
> precession of the perihelion of Mercury is only $1/6$ that of the observed
> value. An effort by Nordström in 1912 [25] predicted precession $-1/6$ of
> the observed value.
>
> In 1917, Lodge [26] claimed to be inspired by Special Relativity to consider
> velocity- dependent corrections to the precession of the perihelion, but
> actually reverted to Newton’s analysis of precession in case of a force law
> $1/r^{n}$ for $n$ different than $2$ [27], as extended by [29, 30]. A debate
> followed between Eddington and Lodge [31, 32, 33, 34, 35].
>
> In 1929, Kennedy [36] gave two analyses of Newtonian precession of the
> perihelion, with corrections for retardation and for Special Relativity,
> claiming negligible effects in both cases. It was stated by Goldstein [37]
> (1950) that the result from Special Relativity is $1/6$ that of General
> Relativity (…).
>
> In 1984, Phipps [39] claimed that the result of Special Relativity is the
> same as that of General Relativity.
>
> In 1986, Peters [40] noted that Phipps made a computational error, and
> claimed the correct result of Phipps’ model is $1/2$ that of General
> Relativity (…).
>
> In 1987, Biswas [41] claimed that the result of (his interpretation of)
> Special Relativity is the same as that of General Relativity.
>
> In 1988, Frisch [42] discussed “post-Newtonian” approximations, claiming
> that use of “relativistic momentum” but Newtonian gravity gives the result
> of Goldstein [37], $1/6$ of the observed precession of the perihelion of
> Mercury, while including the gravitation due to gravitational field energy
> doubles the result, to $1/3$ of the observed precession of the perihelion of
> Mercury.
>
> In 1989, Peters [43] argued that Biswas’ calculation was in error.
>
> In 2006, Jefimenko proposed a theory of “cogravitation”, and claimed it
> predicted $1/3$ of observed precession of the perihelion of Mercury (…).
>
> In 2015, Wayne [46] claimed that Special Relativity can explain the
> precession of the perihelion of Mercury.
>
> In 2016, Lemmon and Mondragon [47] argued that Special Relativity predicts
> $1/3$ of the rate of the precession of the perihelion according to General
> Relativity.
>
> In 2020, Corda [48] claimed that Newtonian gravity completely explains the
> precession of the perihelion of Mercury (without consideration of
> relativity), but not that of other planets. Then, he argued that General
> Relativity also explains the precession, but only if one includes the effect
> of “rotational time dilation”.
>
> In 2022, D’Abramo [49] claimed that Corda [48] was wrong.
>
> What is going on here?

To understand thecontroversy, let us first look at the purely kinematic
contribution of Special Relativity (neglecting the influence of external
planets). The Lagrangian of a particle in a potential $V(r)=-GM_{\odot}m/r$ is

$L=-\frac{mc^{2}}{\gamma}+\frac{GM_{\odot}m}{r}$ (21)

where $\gamma^{-1}=\sqrt{1-|{\bf v}|^{2}/c^{2}}$ and $|{\bf
v}|^{2}=\dot{r}^{2}+r^{2}\dot{\theta}^{2}+r^{2}\sin^{2}\theta\dot{\varphi}^{2}$.
Lagrange equations are therefore

$\displaystyle\gamma r\dot{\theta}^{2}+\gamma
r\sin^{2}\theta\dot{\varphi}^{2}-\frac{GM_{\odot}m}{r^{2}}=\gamma\ddot{r}+\dot{\gamma}\dot{r},$
$\displaystyle\gamma r^{2}\sin 2\theta=\frac{d}{dt}(\gamma
r^{2}\dot{\theta}),$ $\displaystyle\frac{d}{dt}(\gamma
r^{2}\sin^{2}\theta\dot{\varphi})=0.$ (22)

Using the Lagrange equations is an option, but we could also use the first
integrals, conservation of the angular momentum, and energy, as we did in
Newtonian mechanics. The same works here, which leads to a more direct
derivation.

The angular momentum is written as ${\bm{\sigma}}={\bf r}\times m\gamma{\bf
v}={\bf r}\times{\bf p}$, so that $\frac{d{\bm{\sigma}}}{dt}={\bf v}\times{\bf
p}+{\bf r}\times\frac{d{\bf p}}{dt}$. The first term obviously vanishes, and
the second vanishes for central potentials for which $\frac{d{\bf p}}{dt}={\bf
F}=F(r){\bf r}/r$. Now, since ${\bm{\sigma}}$ is conserved, we deduce again
that the motion stays within a plane, and we choose $\theta=\pi/2$
($\dot{\theta}=0$), measuring the $\theta$ angle w.r.t. the direction of the
angular momentum, the kinematic factor $\gamma$ takes the form

$\gamma^{2}=\Bigl{[}1-\frac{1}{c^{2}}\left(\frac{dr}{d\varphi}\Bigr{)}^{2}\dot{\varphi}^{2}-\frac{r^{2}}{c^{2}}\dot{\varphi}^{2}\right]^{-1}.$
(23)

A first constant of motion follows from the fact that $L$ doesn’t depend on
$\varphi$, hence

$|{\bm{\sigma}}|=p_{\varphi}=\frac{\partial L}{\partial\dot{\varphi}}=m\gamma
r^{2}\dot{\varphi}.$ (24)

The second constant of motion is obviously the energy,

$E=p_{r}\dot{r}+p_{\varphi}\dot{\varphi}-L=m\gamma\dot{r}^{2}+m\gamma
r^{2}\dot{\varphi}^{2}+mc^{2}\gamma^{-1}-\frac{GM_{\odot}m}{r}=mc^{2}\gamma-\frac{GM_{\odot}m}{r}.$
(25)

Using the definition of the angular momentum, we eliminate $\dot{\varphi}$ in
(23), and factorize out $\gamma^{2}$, leading to

$\gamma^{2}=1+\frac{|{\bm{\sigma}}|^{2}}{m^{2}c^{2}r^{4}}\Bigl{(}\frac{dr}{d\varphi}\Bigr{)}^{2}+\frac{|{\bm{\sigma}}|^{2}}{m^{2}c^{2}r^{2}}=\Bigl{(}\frac{E}{mc^{2}}+\frac{GM_{\odot}}{rc^{2}}\Bigr{)}^{2}.$
(26)

Like in the Newtonian case, the change of variable $u=\frac{1}{r}$ simplifies
the equation into

$1+\frac{|{\bm{\sigma}}|^{2}}{m^{2}c^{2}}\Bigl{[}\Bigl{(}\frac{du}{d\varphi}\Bigr{)}^{2}+u^{2}\Bigr{]}=\Bigl{(}\frac{E}{mc^{2}}+\frac{GM_{\odot}u}{c^{2}}\Bigr{)}^{2}.$
(27)

A formula analogous to (14) is obtained if we write the equation of motion
which follows from the derivative of (27) w.r.t. $\varphi$

$\frac{d^{2}u}{d\varphi^{2}}+\left(1-\frac{G^{2}M_{\odot}^{2}m^{2}}{|{\bm{\sigma}}|^{2}c^{2}}\right)u=\frac{GM_{\odot}m^{2}}{|{\bm{\sigma}}|^{2}}\left(\frac{E}{mc^{2}}\right).$
(28)

This equation qualitatively differs from the classical case by the value of
the angular velocity, which is now

$\Omega^{2}=1-\frac{G^{2}M_{\odot}^{2}m^{2}}{|{\bm{\sigma}}|^{2}c^{2}}\simeq
1+2\omega_{1}$ (29)

that differs from unity, inducing a shift of the perihelion. We can estimate
this shift as we did for the influence of external planets.

$\Delta\varphi_{\rm
SR}\simeq\frac{2\pi(1-\Omega)}{\Omega}=-2\pi\omega_{1}=2\pi\frac{1}{2}\frac{G^{2}M_{\odot}^{2}m^{2}}{|{\bm{\sigma}}|^{2}c^{2}}=\frac{\pi
GM_{\odot}}{a(1-e^{2})c^{2}}.$ (30)

The numerical value is estimated by inserting the classical parameters of the
ellipse, $|{\bm{\sigma}}|^{2}=GM_{\odot}m^{2}p$ and $p=a(1-e^{2})$. We get

$\Delta\varphi_{\rm SR}=8.649\times 10^{-8}\hbox{rad}/\hbox{rev}=7.433\
\\!\hbox{as/cy}.$ (31)

To link with the quotation of McDonald’s, it appears to be 6 times smaller
than the observed one of 43. So we have to conclude that Special Relativity is
not enough to explain the whole effect observed. However, we can notice that
in the controversy reported by McDonald [11], the pioneers Poincaré, Lorentz
or de Sitter were right!

##  6 How General Relativity solves the problem

Now, let us follow the same lines of reasoning in full General Relativity.
General Relativity encodes gravitational energy in the metric,

$ds^{2}=(1-(2GM_{\odot}/(rc^{2}))c^{2}dt^{2}-(1-(2GM_{\odot}/(rc^{2}))^{-1}dr^{2}-r^{2}d\Omega^{2},$
(32)

and the zero-mass limit recovers the case of Special Relativity, therefore
there will be no need to add the result (31) to the present calculation.

The Lagrangian333The action of a free particle in Special Relativity
$S=-mc\int ds$. $L=-mc\frac{ds}{dt}$ now reads as

$L=-mc^{2}\frac{d\tau}{dt}=-mc^{2}\Bigl{[}g_{00}(r)-\frac{1}{g_{00}(r)c^{2}}\Bigl{(}\frac{dr}{dt}\Bigr{)}^{2}-\frac{r^{2}}{c^{2}}\Bigl{(}\frac{d\varphi}{dt}\Bigr{)}^{2}\Bigr{]}^{1/2}$
(33)

in terms of the proper time $\tau$, and the potential term is hidden in the
metric tensor component. The argument for planar motion still works. Hence we
have already simplified the problem considering fixed $\theta=\pi/2$, and we
have assumed Schwarzchild metric (32)
$g_{00}(r)=1-\frac{2GM_{\odot}}{rc^{2}}=-1/g_{11}(r)$. Like in the special
relativistic case, we have two constants of motion. The angular momentum is
the first

$|{\bm{\sigma}}|=\frac{\partial
L}{\partial\dot{\varphi}}=mr^{2}\Bigl{(}\frac{d\varphi}{d\tau}\Bigr{)}.$ (34)

Here, $\dot{\varphi}=d\varphi/dt$ shouldn’t be confused with $d\varphi/d\tau$.
The momentum associated to the radial coordinate is equal to

$\frac{\partial
L}{\partial\dot{r}}=\frac{m}{g_{00}(r)}\Bigl{(}\frac{dr}{d\tau}\Bigr{)},$ (35)

with the same notation $\dot{r}=dr/dt$. The energy follows

$E=\frac{d\tau}{dt}\Bigl{[}\frac{m}{g_{00}(r)}\Bigl{(}\frac{dr}{d\tau}\Bigr{)}^{2}+\frac{|{\bm{\sigma}}|^{2}}{mr^{2}}+mc^{2}\Bigr{]}.$
(36)

The square of the interval provides an alternative identity,

$\displaystyle c^{2}$ $\displaystyle=$ $\displaystyle
g_{00}(r)\Bigl{(}\frac{cdt}{d\tau}\Bigr{)}^{2}-\frac{1}{g_{00}(r)}\Bigl{(}\frac{dr}{d\tau}\Bigr{)}^{2}-r^{2}\Bigl{(}\frac{d\varphi}{d\tau}\Bigr{)}^{2},$
(37)

which leads to the relation

$\frac{1}{g_{00}(r)}\Bigl{(}\frac{dr}{d\tau}\Bigr{)}^{2}=g_{00}(r)c^{2}\Bigl{(}\frac{dt}{d\tau}\Bigr{)}^{2}-c^{2}-\frac{|{\bm{\sigma}}|^{2}}{m^{2}r^{2}}.$
(38)

This latter expression is now inserted in (36) and one obtains the simple form

$E=g_{00}(r)mc^{2}\Bigl{(}\frac{dt}{d\tau}\Bigr{)}.$ (39)

Now, equations (34) and (39) inserted in (37) lead to

$\displaystyle g_{00}(r)m^{2}c^{2}$ $\displaystyle=$
$\displaystyle\frac{E^{2}}{c^{2}}-\frac{|{\bm{\sigma}}|^{2}}{r^{4}}\Bigl{(}\frac{dr}{d\varphi}\Bigr{)}^{2}-g_{00}(r)\frac{|{\bm{\sigma}}|^{2}}{r^{2}}$
(40) $\displaystyle g_{00}(u)m^{2}c^{2}$ $\displaystyle=$
$\displaystyle\frac{E^{2}}{c^{2}}-|{\bm{\sigma}}|^{2}\Bigl{(}\frac{du}{d\varphi}\Bigr{)}^{2}-g_{00}(u)|{\bm{\sigma}}|^{2}u^{2}$
(41)

where the second line follows from the previous one by the usual change of
variable $u=\frac{1}{r}$. We next take another derivative w.r.t. $\varphi$ to
get the relativistic equation of motion

$\displaystyle\frac{d^{2}u}{d\varphi^{2}}+g_{00}(u)u$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{dg_{00}(u)}{du}\Bigl{(}\frac{m^{2}c^{2}}{|{\bm{\sigma}}|^{2}}+u^{2}\Bigr{)},$
(42) $\displaystyle\frac{d^{2}u}{d\varphi^{2}}+u$ $\displaystyle=$
$\displaystyle\frac{GM_{\odot}m^{2}}{|{\bm{\sigma}}|^{2}}+\frac{3GM_{\odot}}{c^{2}}u^{2}.$
(43)

It is instructive to compare with the equation of motion in the Kepler
approximation (4) or in the special relativistic case (28), both of them being
linear. We rewrite these equations (with labels K for Kepler and SR for
Special Relativity) here for the purpose of comparison,

$\displaystyle\frac{d^{2}u_{\rm K}}{d\varphi^{2}}+u_{\rm K}$ $\displaystyle=$
$\displaystyle\frac{GM_{\odot}m^{2}}{|{\bm{\sigma}}|^{2}},$ (44)
$\displaystyle\frac{d^{2}u_{\rm SR}}{d\varphi^{2}}+u_{\rm SR}$
$\displaystyle=$
$\displaystyle\frac{GM_{\odot}m^{2}}{|{\bm{\sigma}}|^{2}}+\frac{G^{2}M_{\odot}^{2}m^{2}}{|{\bm{\sigma}}|^{2}c^{2}}u_{\rm
SR}$ (45)

and clearly the main difference is that in General Relativity we get a non
linear equation. This latter equation in (43) is solved perturbatively around
the classical case, allowing the periodic solution and harmonics at multiple
frequencies, together with a possible shift of the fundamental frequency of
the Kepler solution. We thus allow

$\displaystyle\phi=\Omega\varphi=(1+\omega_{1}+\dots)\varphi,$ (46)
$\displaystyle u(\varphi)=u_{\rm K}(\phi)+u_{1}(\phi)+\dots,$ (47)

where $\omega_{1}$ and $u_{1}$ are small perturbations, with $u_{\rm
K}(\phi)\sim\cos\phi$ and $u_{1}(\phi)\sim\cos 2\phi$, …Inserting these
expansions in (43) leads to

$\displaystyle\Omega^{2}\frac{d^{2}u}{d\phi^{2}}+u=\frac{1}{p}+\frac{3GM_{\odot}}{c^{2}}u^{2},$
(48) $\displaystyle 0\hbox{th order\quad}$ $\displaystyle\frac{d^{2}u_{\rm
K}}{d\varphi^{2}}+u_{\rm K}=\frac{1}{p}$ (50) $\displaystyle 1\hbox{st order
}$
$\displaystyle\frac{d^{2}u_{1}}{d\varphi^{2}}+u_{1}=\hbox{const}+\frac{2e}{p}\Bigl{(}\frac{3GM_{\odot}}{pc^{2}}+\omega_{1}\Bigr{)}\cos\phi$
$\displaystyle\qquad\qquad\qquad\qquad+\frac{3GM_{\odot}e^{2}}{2p^{2}c^{2}}\cos
2\phi.$

The 0th order solution is indeed Kepler solution and, demanding that $u_{1}$
has no dependence at the same frequency leads to the first order correction of
the angular velocity,

$\omega_{1}=-\frac{3GM_{\odot}}{pc^{2}}.$ (51)

It follows an advance of the perihelion

$\Delta\varphi_{0}=-2\pi\omega_{1}=\frac{\pi
6G^{2}M_{\odot}^{2}m^{2}}{c^{2}|{\bm{\sigma}}|^{2}}=\frac{6\pi
GM_{\odot}}{a(1-e^{2})c^{2}},$ (52)

being 6 times larger than in the special relativistic treatment.

The numerical value is

$\Delta\varphi_{0}=5.05\times 10^{-7}\hbox{rad}/\hbox{rev}=43.2\
\\!\hbox{as/cy}$ (53)

for Mercury. This numerical result agrees remarkably with the observations
collected, for the case of Mercury, in table 2. The results for other bodies
in the solar system are given in table 3

Celestial body |  $\Delta\varphi_{0}$ measured (in as/cy) | predicted in GTR (as/cy)
---|---|---
Mercury | $43.11\pm 0.45$ |  $43.03$ [3] (or 42.98 [7])
Venus | $8.4\pm 4.8$ | $8.6$
Earth | $5.0\pm 1.2$ | $3.8$
Icarus | $9.8\pm 0.8$ | $10.3$
Table 3: Advance of the perihelion in the solar system. From Ref. [12].

Resolving a disagreement between theory and observations that had been
misunderstood for years was a major breakthrough that consolidated the General
Theory of Relativity of Einstein. Although numerically minor, solving the
discrepancy was essential on fundamental grounds and this opened the era of
high precision astronomy and astrophysics.

##  7 But what about the influence of external planets in GTR?

For consistency, we must now ensure that taking into account the effect of the
external planets at the level of General Relativity does not ruin the
formidable agreement of the previous calculation. For that, let us remind that
at the level of Newtonian gravitation theory, the equation of motion for
central potentials $V(r)$ reads as

$\frac{d^{2}u}{d\varphi^{2}}+u(\varphi)=-\frac{m}{|{\bm{\sigma}}|^{2}}\frac{dV}{du},$
(54)

while in GTR the corresponding equation is (42)

$\frac{d^{2}u}{d\varphi^{2}}+g_{00}(u)u=-\frac{1}{2}\frac{dg_{00}(u)}{du}\Bigl{(}\frac{m^{2}c^{2}}{|{\bm{\sigma}}|^{2}}+u^{2}\Bigr{)}$
(55)

with

$g_{00}(u)=1+\frac{2\phi(u)}{c^{2}}.$ (56)

Using the potential energy found in Section 3, limited to the leading order
for the planet contribution,

$V(u)=m\phi(u)=-GM_{\odot}mu-\frac{GM_{\rm P}m}{4R_{\rm P}^{3}u^{2}},$ (57)

we get the following equation of motion in the classical case

$\displaystyle\frac{d^{2}u}{d\varphi^{2}}+u(\varphi)$ $\displaystyle=$
$\displaystyle\underbrace{\frac{GM_{\odot}m^{2}}{|{\bm{\sigma}}|^{2}}}_{{\hbox{\footnotesize
Sun influence \vrule
height=0.0pt,depth=3.41432pt,width=0.0pt}}\atop{\hbox{\footnotesize Newtonian
level}}}-\underbrace{\frac{GM_{\rm P}m^{2}}{2R_{\rm
P}^{3}|{\bm{\sigma}}|^{2}}\frac{1}{u^{3}}}_{{\hbox{\footnotesize External
planet influence \vrule
height=0.0pt,depth=3.41432pt,width=0.0pt}}\atop{\hbox{\footnotesize at the
Newtonian level}}}=\frac{1}{p}-\frac{M_{\rm P}}{2M_{\odot}}\frac{1}{pR_{\rm
P}^{3}}\frac{1}{u^{3}}.$

This is the equation that we analyzed earlier.

Within General Relativity, we have seen that the equation of motion takes the
form of Eq. (42) with now

$\displaystyle g_{00}(u)=1-\frac{2GM_{\odot}}{c^{2}}u-\frac{GM_{\rm
P}}{2R_{\rm P}^{3}c^{2}}\frac{1}{u^{2}}.$ (59)

The equation of motion follows in the form

$\displaystyle\frac{d^{2}u}{d\varphi^{2}}+u(\varphi)$ $\displaystyle=$
$\displaystyle\underbrace{\frac{GM_{\odot}m^{2}}{|{\bm{\sigma}}|^{2}}}_{{\hbox{\footnotesize
Sun influence \vrule
height=0.0pt,depth=3.41432pt,width=0.0pt}}\atop{\hbox{\footnotesize Newtonian
level}}}+\underbrace{\frac{3GM_{\odot}}{c^{2}}u^{2}}_{{\hbox{\footnotesize Sun
correction \vrule
height=0.0pt,depth=3.41432pt,width=0.0pt}}\atop{\hbox{\footnotesize GTR
level}}}$ $\displaystyle-\underbrace{\frac{M_{\rm
P}}{2M_{\odot}}\frac{1}{pR_{\rm P}^{3}}\frac{1}{u^{3}}}_{{\hbox{\footnotesize
External planet influence \vrule
height=0.0pt,depth=3.41432pt,width=0.0pt}}\atop{\hbox{\footnotesize at the
Newtonian level}}}-\underbrace{\frac{GM_{\rm P}}{4R_{\rm
P}^{5}c^{2}}\frac{1}{u^{3}}}_{{\hbox{\footnotesize External planet correction
\vrule height=0.0pt,depth=3.41432pt,width=0.0pt}}\atop{\hbox{\footnotesize at
the GTR level}}}$ (60)

Comparing Eq. (60) with Eq. (LABEL:Eq63) we see that only the last term
differs in the planet contribution. When we compare the order of magnitude of
the two contributions due to the planet, the GTR vs the classical
contributions, the ratio is about $10^{8}$ times smaller. This proves that
there is no need for further analysis of this latter effect which leads, at
the experimental accuracy, to the same results as in Newtonian gravity. The 43
as/cy are indeed due to the sole correction of the influence of the Sun in
General Relativity.

##  8 A problem recently revisited

C.M. Will is a specialist of the experimental verifications of the General
Theory of Relativity. He recently revisited the problem of the advance of
Mercury’s perihelion [13] and found a new contribution arising from the
interaction between Mercury’s motion and the gravitomagnetic field of the
moving planets. The numerical contribution is very small, with a few parts in
a million of the GTR main contribution to precession, but considers that this
should be detectable experimentally. It has to be noticed that the GTR
correction to the planets’ contribution that we have estimated is still a
factor 10 smaller, but it might be accessible in the future.

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