Inverse-square law and its significance

The inverse-square law is a fundamental principle that students encounter in their studies. It finds extensive application in various mathematical equations within the field of physics. Newton’s law of gravitation exemplifies the inverse-square law. The equation governing Newton’s law of gravitation is F = [Gm1 m2]/r2, where F represents the gravitational force exerted between two objects with masses m1 and m2, G denotes the gravitational constant, and r signifies the distance separating these two masses. This law elucidates that the gravitational attraction between two objects diminishes as the inverse of the square of the distance (i.e., inversely proportional to 1/r2).Similarly, Coulomb’s law in electrostatics serves as another illustration of the inverse-square law, demonstrating that the electric force between two charged particles is inversely proportional to the square of the distance separating them. Doubling the distance results in a reduction of the force to a quarter.In essence, the inverse-square law establishes a relationship between physical quantities of the form x proportional to 1/y2, where y typically represents a distance. Consider the scenario of spraying water from a hose. Near the nozzle, the water is concentrated, while at a distance, the same quantity of water is spread over a larger area, resulting in a diminished sensation of force. The inverse-square law encapsulates this “spreading out” phenomenon for numerous physical phenomena. Any equation in which a physical quantity varies inversely with the square of the distance from a source exemplifies the inverse-square law. Consequently, it transcends the realm of gravitational and electric forces, encompassing both forces and fluxes. When energy is emitted by a point source, this law holds true. The brightness of a light source diminishes inversely as the square of the distance from the light source. Similarly, the loudness of sound emanating from a point source decreases with the square of the distance. Moving twice as far from a speaker reduces the sound intensity to one-fourth. The intensity of radioactive emissions also diminishes with the square of the distance from the source.While this law is part of the school level physics, its true meaning is not clear to everyone. Why does Inverse-square law hold good? Is it indicating any special feature about our universe? What is the origin of Inverse-square law?The first answer is that the inverse square law is rooted in geometry. All unbounded waves become spherical at distances r, whose surface area is proportional to r2. To understand this, imagine energy radiating from a point source like light or sound. It will spread outward like an expanding sphere or expanding balloon. Its surface area will enlarge like surface of an expanding balloon. Since the surface area of a sphere is 4πr², the same amount of energy is distributed over a larger area as distance increases, reducing the intensity per unit area proportionally to 1/r². This is how the inverse-square law emerges in nature.The inverse square law also indicates that this universe is "Flat" (Euclidean geometry), as it will not work in a non-Flat or curved space geometry. Curved spaces are important in understanding the structure and behavior of the universe. The curvature of space is a fundamental concept in general relativity, which explains how gravity affects the geometry of space-time. In non-Euclidean geometry, the sum of angles in a triangle is not always 180 degrees, and parallel lines may converge, diverge, or never meet. This geometry is essential for describing the universe where space itself is understood to be curved due to the presence of mass and energy, as predicted by Einstein's theory of general relativity.Curved spaces could be flat (zero curvature), hyperbolic (negative curvature), and spherical (positive curvature). Imagine a 2D Flatland like a sheet of paper. With a positive Curvature (closed) shape, the 2D Flatland or the sheet of paper would curve in and become like a sphere. The circumference of a circle drawn on a sphere would be less that 2πr. With a negative Curvature (open) shape, the 2D Flatland or the sheet of paper would curve out like a saddle (hyperbolic). The circumference of a circle on a saddle would be more than 2πr. Based on this analogy, we can imagine the shape of the 3D universe curved in or out in a hyperspace. The surface area of a sphere in such a curved universe would no longer be 4πr² and the inverse square law will not hold good. In such a universe a long-distance light source would appear brighter or dimmer (depending on the positive or negative curvature) than the inverse square law predicts, because the light rays are being refocused by the geometry of space.It is also true that while the inverse square law indicates that our local universe is flat, it cannot predict the flatness of universe over a large scale. The precise measurements of the Cosmic Microwave Background (CMB) show the universe is spatially flat within a 0.4% margin of error. This means that over the large cosmic scale, the universe is almost flat without any significant curvature. This confirms that the inverse square law is valid even across billions of light years.However, it gives rise to even deeper question. From a point source, why energy spreads like a sphere rather than any other shape? The answer is connected to the spatial dimension of our universe. The "2" in the inverse square law 1/r² is not a random number but derived from the number of spatial dimensions (D) minus one. In a single dimension world (like Line), energy cannot spread out. It travels in a straight line without fading and therefore, its intensity will remain constant (proportional to 1/r^D-1 or 1/r⁰. In a two-dimension world (Flatland), energy spreads in a circle whose circumference is proportional to r. Therefore, the intensity fades proportionally to 1/r^D-1 or 1/r. In a three-dimension world (our universe), energy spreads over a sphere, therefore, the intensity fades proportionally to 1/r^D-1 or 1/r². Similarly, in a four-dimension world (Hyperspace), energy would spread over a hypersphere whose surface volume is 2π²r³, therefore, the intensity would fade as 1/r^D-1 or 1/r³.Consequently, the inverse square law strongly suggests that our universe is a flat universe with only three spatial dimensions. It is important to distinguish this from the four dimensions of space-time. In 4D space-time also, spatial dimensions are limited to three only and the fourth dimension represents time. If the universe possessed a different number of dimensions or exhibited significant curvature, the inverse square law would be violated, resulting in differential fading of forces such as gravity and light over distance. While the inverse square law does not preclude the existence of more than three spatial dimensions. However, if more than three spatial dimensions exist, additional dimensions are tightly curled up and only 3 spatial dimensions are flatly spread.(B Purushartha is an IAS officer, working as Joint Secretary in the Department of Economic Affairs, Ministry of Finance.)