The Mathematical Relationship between Inertial Mass, Gravity, and Spacetime Curvature in Classical and Relativistic Mechanics:

Soumendra Nath Thakur

February 08, 2025

In classical mechanics, mass is considered a fundamental property of matter, representing its resistance to changes in motion (inertia), while energy is defined as the capacity to perform work. Gravity, within this framework, is described as an attractive force between objects with mass, where a greater mass results in a stronger gravitational pull, as governed by Newton’s Law of Universal Gravitation. Mathematically, this relationship is commonly expressed as:

Inertial mass (m) ∝ gravitational acceleration (g).

In relativistic mechanics, however, mass is not strictly an invariant quantity. Instead, relativistic mass appears to increase as an object approaches the speed of light. Additionally, as per general relativity, spacetime is curved by the presence of mass and energy, and this curvature dictates the motion of objects, making gravity emerge as a natural consequence of this geometric distortion rather than a classical force.

From a mathematical perspective, relativistic rest mass (m₀) is directly proportional to rest energy (E), leading to the broader expression:

Inertial mass (m) ∝ (rest mass (m₀) + rest energy (E)).

This reflects the conservation of mass and energy in both classical and relativistic mechanics. Extending this concept further in relativistic interpretations:

Inertial mass (m) ∝ curvature in spacetime ∝ relativistic gravity (G),

where inertial mass is fundamentally linked to both rest mass and rest energy. The curvature of spacetime serves as an additional factor in relativistic gravity, establishing a deeper connection between mass and gravitational effects.

If one suppresses 'deeper connection' of the explicit role of spacetime curvature in mathematical representations on the ground of conservation for mass and energy in both relativistic and classical principles, the relationship simplifies in the classical context as:

Classical inertial mass (m) ∝ gravitational acceleration (g).

However, in a relativistic framework, where gravity is a manifestation of spacetime curvature, the equivalent expression is:

Inertial mass (m) ∝ relativistic gravity (G).

This refined formulation highlights the transition from Newtonian gravity to relativistic gravity, emphasizing the fundamental role of spacetime curvature in shaping gravitational interactions at relativistic scales.