Modeling Google's Search Engine with Einstein's Gravity Field Equations

Last Updated on March 22, 2025

Can Einstein's Theory of Gravity Apply to Google Search Space?

Einstein’s General Relativity (GR) equations explain how mass-energy curves spacetime and how geodesics dictate the motion of objects. As theoretical physicist John Wheeler described it, “matter tells spacetime how to curve, and curved spacetime tells matter how to move.”

When it was published in 1915, Einstein’s theory of gravity represented a major shift from the prevailing physics of the era, which had always considered spacetime to be a static, unchanging backdrop—a flat canvas on which forces acted upon objects. Einstein's GR equations revealed that spacetime is a dynamic landscape, where the presence of mass and energy can influence and shape its very fabric. In a similar way, Google's search environment, much like our Universe, evolves and transforms in response to its elements: mass and energy in the Universe, and websites and queries on the Internet. By drawing these parallels, we can start to conceptualize the online search market through the lens of General Relativity.

It is important to acknowledge that while this exercise offers an interesting application of a physical sciences model to computer science data, it does not represent a comprehensive scientific study. Nonetheless, the model's potential applications could be evaluated using real-world search data to determine its practicality and effectiveness.

Mapping Components of Einstein's Field Equations to Components of Search Space

We map the major components of Einstein's General Relativity tensor equations to the following components of search space.

Websites in Search-Spacetime = Mass-Energy Density in Spacetime

• Popular websites (e.g., Wikipedia, Google, Amazon) act like massive celestial bodies that curve the search landscape more strongly.
• Smaller websites, with fewer backlinks and less authority, create minor perturbations in the search-spacetime fabric.

Search Queries = Particles or Light in Curved Spacetime

• A search query is like a photon or particle moving through a curved manifold of website information.
• The trajectory (geodesic) of the query is bent by the influence of massive (authority) websites (ranking algorithms).

Search Engine Algorithms = Field Equations Defining the Curvature of Search Spacetime

• Google’s ranking algorithm (PageRank, BERT, etc.) acts like the Einstein field equations, determining how websites affect the flow of queries.
• Changes in search engine policies (e.g., Google named and core updates) correspond to shifts in the metric tensor, altering how queries travel through the information space.

User Behavior (Time Dilation & Relativistic Effects in Information Retrieval)

• The attention economy follows a relativistic principle: the closer a page ranks to the top results, the more time users spend on it, akin to time dilation near a gravitational well.
• Click-through rates (CTR) and dwell time correspond to proper time experienced by a user in different search result landscapes.

General Relativity (GR) describes the bidirectional interaction between mass and energy. In contrast, when applying this concept to the digital realm, we initially consider it in a unidirectional way: how queries move toward websites. The reverse interaction, where websites move toward queries, is less obvious. However, many website owners actively create content to meet consumer search demands, which involves different levels of analysis and effort. As a result, the concept of a search-spacetime landscape could be refined to define the relationship as simply information interacting with information, where websites offer content answers and queries represent user intent.

This supports the application of the model by considering mass-energy as a form of information, particularly when viewed through the lens of quantum field theories and the Standard Model of particle physics. Exploring this further, there might be a generalized form of the GR equations that could be specifically applied to the Von Neumann entropy of the system being studied. Although such investigations are beyond the scope of this publication, they are acknowledged for offering fundamental backing for the modeling of the search-spacetime tensors that will be discussed. While this may not be the perfect application of an established model to a new field of study and data, it provides valuable preliminary insights that can inspire further exploration.

Applying Gravity Equations to Search Engine Models

The Einstein field equations:

can be mapped to search engine behavior as follows:

​ (Curvature of Search-Spacetime)

Describes how search results bend and cluster due to the presence of websites and content on the Internet. 

(Stress-Energy Tensor of a Webpage)

Represents how websites and their content, with underlying signals (e.g., quality, backlinks, and engagement), contribute to the "gravitational pull" of a site. 

​ (Metric Tensor of Search Landscape)

This concept outlines the search engine's ranking space, which evolves over time due to updates and new indexing strategies. Just as a component of the Einstein Field Equations (EFE) was originally introduced to stabilize the Universe's expansion or contraction, this element is included in our search spacetime model. It accounts for changes in Google rankings that can influence a website's ability to attract queries effectively.

Defining the Metric Tensor for Search Spacetime

In GR, the metric tensor describes how distances are measured in curved spacetime. We propose an analogous search spacetime metric, where:

• Webpages act as massive objects that influence the curvature of search-space.
• Search queries follow geodesics (the shortest path through search space).
• Algorithm updates change the curvature, affecting query trajectories.

A simplified metric for search-spacetime can be written as:

where:

• ----- the ranking position (distance from the top result),
• ----- a ranking warping function (depends on authority, backlinks, and search engine factors),
• ----- represent additional contextual factors (e.g., user intent, personalization).

If we assume high-authority pages distort search-space like gravity, we can model  using a Schwarzschild-like term:

where:

• ----- the authority mass of a website (e.g., PageRank score),
• ----- a gravitational-like constant (scaling factor for backlink influence).

This metric tells us that high-authority pages (large M) create deeper ranking wells, affecting how queries move.

Geodesic Equation for Query Trajectories

In GR, the geodesic equation governs how objects move in curved spacetime:

For search queries, this equation models how queries navigate through the search result space. Specifically:

• ----- represents a query's position in ranking-space.
• ----- the user interaction time (how long a user engages beginning with the SERPs and extended through time on the website).
• ​ ----- the Christoffel symbols, determining how ranking curvature affects query motion.

For a radial query trajectory (query moving through rankings), we approximate:

This behaves like Newtonian gravity, meaning:

• Queries naturally fall toward high-authority pages.
• Lower-quality pages require external forces (SEO, backlinks, user engagement) to remain competitive.

A solution to this equation gives:

which implies that over time, a query drifts toward dominant pages unless acted upon by additional ranking forces.

Search Engine Updates as a Field Equation

Google’s core updates change ranking rules, analogous to how mass-energy distribution affects spacetime curvature. We propose a Search Engine Field Equation inspired by Einstein’s field equation:

where:

• ----- the search result curvature tensor (measuring ranking distortions),
• ​ ----- the content stress-energy tensor (describing website authority, relevance, user engagement),
• ----- represents algorithmic dampening (e.g., anti-spam measures preventing sites from dominating).

A simplified version:

suggests that ranking shifts () are driven by content quality () but limited by algorithmic constraints ().

Predicting SEO Strategies Using Perturbation Theory

In GR, perturbation theory studies small fluctuations in spacetime. Similarly, we can model small SEO optimizations as perturbations in search-space curvature.

For small changes in search ranking:

which means:

• A small increase in quality content, backlinks, or engagement () leads to a small ranking shift ().
• If an algorithm update increases (penalization factor), even a high may not result in ranking gains.

Some potential applications of the model is in predicting ranking drops, SEO recovery strategies, and Google search algorithm change modeling. We'll start with a simple simulation of the Search Engine Field Equation as applied to website rankings with changing user metrics.

Running Simulations in Search Space with the Search Engine Field Equations

The plot illustrates how a search query moves through the ranking space over time under the influence of website authority (mass) and search engine ranking forces.

Key Observations:

Query "Falls" Towards High-Authority Sites

• If a site has high authority (), it acts like a "gravitational well," pulling queries towards it.
• As user interaction time () progresses, the ranking position improves.

SEO & Algorithmic Impact

• If an algorithm update (increase in ) occurs, it could act like a repulsive force, preventing certain websites from dominating.
• Adding backlinks or improving content increases, making the site more attractive to queries.

The plot shows the impact of a search engine algorithm update on a website's ranking trajectory.

Key Observations:

Before the Update (blue line)

• The query naturally moves towards better rankings over time.
• This represents a healthy SEO strategy where content quality and backlinks improve ranking.

After the Update (red dashed line, penalized site)

• The ranking trajectory is slower due to a penalty (e.g., Google core update impact).
• The site struggles to improve in ranking, reflecting a dampening effect () from the algorithm update.
• If the penalty is too strong, the site may stagnate or even drop in rankings.

Insights for SEO Recovery Strategies:

If a site is penalized, it needs to increase  (authority mass) by:

• Improving content relevance (increasing engagement, quality signals).
• Earning higher-quality backlinks to counteract the algorithmic dampening effect.
• Optimizing user signals (click-through rates, dwell time).

The plot now includes a green dotted line that represents a website's ranking recovery trajectory after an algorithmic penalty, assuming SEO improvements over time.

Key Observations:

Before the Algorithm Update (blue line)

• The query naturally falls toward better rankings without interference.

After the Algorithm Update (red dashed line, penalized site)

• The ranking trajectory slows down significantly due to a penalty.
• The site struggles to regain ranking momentum.

Recovery Path with SEO Improvements (green dotted line)

• If SEO efforts increase site authority (), the website starts recovering.
• Recovery is gradual and depends on the rate of improvements.
• Eventually, the ranking approaches the pre-penalty trajectory, but this takes time.

Practical SEO Insights from the Model:

Speed of Recovery Depends on SEO Effort Rate

The speed at which a website can recover from a ranking drop is heavily influenced by the intensity and consistency of SEO efforts. When issues are identified, faster and more comprehensive actions in areas such as content optimization, link-building, and user engagement can significantly reduce recovery time. Content optimization involves updating and improving existing content to make it more relevant and valuable to users, which can quickly enhance the site's SEO performance. Link-building, the process of acquiring high-quality backlinks, is crucial as it increases a site's authority and trustworthiness in the eyes of search engines. Enhancing user engagement, by improving site usability and providing a better user experience, can lead to increased traffic and better rankings. A strategic and consistent approach in these areas can help accelerate recovery and regain lost search engine rankings more effectively.

Severe Penalties Need Aggressive SEO Fixes

When a website suffers from severe penalties, often due to violations of search engine guidelines or significant drops in quality, aggressive SEO fixes are required. Such penalties can drastically lower a site's visibility and traffic, necessitating a substantial increase in authority through focused efforts on acquiring high-quality backlinks, enhancing site trust, and improving user engagement metrics. Addressing the root causes of the penalty, such as removing harmful backlinks, rectifying technical issues, or improving content quality, is essential. Moreover, building a robust backlink profile from reputable sources can help restore the site's authority and trustworthiness. This aggressive approach is crucial to signal to search engines that the site is making sincere efforts to comply with best practices and improve its overall quality.

Recovering Past a Certain Threshold is Difficult

When a website's rankings drop significantly beyond a certain threshold, the path to recovery becomes increasingly challenging. This is due to the unfavorable "search-space curvature," a concept suggesting that the effort required to climb back up the rankings exponentially increases as the drop becomes more severe. In such scenarios, the competitive landscape is often more intense, with numerous sites vying for top positions. As a result, regaining lost ground demands a comprehensive and sustained SEO strategy that involves not only rectifying existing issues but also outperforming competitors. This includes continuous content creation, strategic link-building, and maintaining high levels of user engagement. The process requires patience, resources, and a long-term commitment to SEO best practices to gradually rebuild authority and improve rankings over time.

Updated Insights on SEO Recovery Strategies

The simulation tested a faster SEO improvement rate, but even with a higher authority growth rate, the site still doesn't recover within the given timeframe. This suggests that:

The penalty impact () was too severe

• Even aggressive SEO efforts may not be enough to restore rankings quickly.
• Additional strategies are needed beyond just authority growth.

Different recovery strategies should be tested

• Backlink acquisition: Increases authority mass () faster.
• Content quality improvement: Enhances user engagement and click-through rates.
• Reducing spam signals: If the penalty is related to algorithmic trust factors, removing bad links or improving E-E-A-T (Experience, Expertise, Authoritativeness, Trustworthiness) may help rankings recover faster.

Analysis of SEO Recovery Strategies: Backlinks vs. Content Improvements

The graph compares two separate SEO recovery paths:

Backlink Acquisition (orange dash-dot line)

• Assumes authority growth through high-quality backlinks.
• Faster increase in ranking potential.

Content Improvements (cyan dotted line)

• Assumes recovery through better content quality & engagement.
• More gradual improvement in user signals and trust factors.

Impact of Different Penalty Levels

• Mild penalty (blue, dotted line) → Fastest recovery
• Moderate penalty (orange, dashdot line) → Slower but recoverable
• Severe penalty (red, dashed line) → Significant ranking drop, much harder to recover
• Key Insight: The stronger the penalty, the longer it takes to recover—even with SEO improvements. Severe penalties may require aggressive strategies like content overhauls and strong backlink campaigns.

Delayed SEO Recovery

• Immediate Recovery (Green, Solid Line) → Fastest return to rankings
• Delayed Recovery (Purple, Dashdot Line, starts at τ=10\tau = 10τ=10) → Much slower recovery

Key Insight:

• If SEO efforts start late, the site takes longer to recover.
• Waiting too long to react after a penalty means more ranking loss before recovery starts.
• Immediate action is crucial for mitigating penalties and regaining search rankings quickly.

Is Google Using GR Equations in its Search Engine Algorithms? 

Could Google be using Einstein's General Relativity equations to help understand the search landscape or even to help guide their search algorithms? Other fields have either directly applied GR equations or borrowed inspiration, especially in the application of Riemannian geometry (which GR is built on). For example, finance and stock market modeling, sports analytics, and artificial intelligence and machine learning modeling. This integration does not necessarily replace existing components like machine learning models, data analytics, and user behavior studies but rather complements them by providing additional layers of understanding and optimization.

In terms of the search market, integrating GR equations could be part of a broader strategy that includes machine learning for understanding patterns, data analytics for interpreting vast amounts of information, and user behavior studies for aligning with user intent. Together, these elements could create a more sophisticated and responsive search engine, capable of adapting to the ever-changing digital environment. For example, General Relativity offers a mathematical framework for understanding how time and space change in relation to gravity and motion. By leveraging this, Google could model user behavior and data flow across varying time frames and geographical regions to improve localization and personalization results. Relativistic differences could be accounted for between varying types of devices, operating systems, user behaviors, and more—along with calculations for seasonal variances and both sudden and long-term trending impacts. Google’s Knowledge Graph, where entities and their relationships are mapped, could be enhanced by using a GR-inspired curved entity-attribute search space that better supports non-linear, multidimensional mapping of knowledge. This could allow for a better understanding of complex, interrelated topics where information is more dynamically influenced by authoritative sources.

As we move toward a deeper understanding of quantum gravity via breakthroughs in AdS/CFT, the Holographic Universe, and Quantum Extremal Surfaces, the application of GR to quantum computing may become fully realized—where quantum algorithms may learn to prioritize information differently, especially near quantum boundaries such as spacelike or timelike horizons (e.g., the event horizon of a black hole), where the manner in which information is stored and accessed is fundamentally different. Such processes, which have large implications for entropy (hidden information in a system), may also have far-reaching implications for causality and its evolution in any system that stores, protects, processes, and accesses computational data.

While this remains a theoretical exploration, the idea of incorporating GR equations as a guiding framework within Google's algorithms highlights the potential for innovative approaches to managing complex information systems.

About the Author: Daniel James Stoker is a member of the Internet Brands Performance Team and holds degrees of B.S. in Physics, B.S.H.S. in Physiological Sciences, a M.S. in Computer Sciences Database Technologies, and has completed a PGP in Artificial Intelligence and Machine Learning: Business Applications.

LinkedIn