Inertia
Inertia is the resistance of a body to any change in its state of motion. It is the content of Newton's first law — a body continues at rest or in uniform straight-line motion unless a force acts on it — and it is quantified by inertial mass through the second law, F = ma.
Newton set out the law in the Principia (1687) against a background of absolute space, and the difficulty has been there ever since: acceleration relative to what? Newton's rotating-bucket argument treated the answer as absolute space itself. Ernst Mach rejected that in the 1880s and argued that the inertia of a body is determined by its relation to all the other matter in the universe — the water in the bucket knows it is rotating because the fixed stars are there. Einstein named this Mach's principle in 1918 and took it as one of the motivations for general relativity, but the theory as built realises it only partially: it does predict frame dragging by rotating matter, an effect measured by Gravity Probe B and by satellite laser ranging, but it does not derive local inertial frames from the distant mass distribution, and Einstein himself came to regard the principle as not fully implemented.
The equality of inertial and gravitational mass is one of the best-tested facts in physics, confirmed to about one part in 1015. What no accepted theory provides is a mechanism: neither general relativity nor the Standard Model says what physically resists acceleration, or why. This is an acknowledged gap and not a manufactured one.
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Filling that gap is one of the most active themes in the literature collected here, and the proposed answers divide sharply into local and cosmic explanations.
Inertia as an electrodynamic reaction
The largest local programme makes the inertial force an ordinary electromagnetic force acting on the charges inside an accelerated body. Thomas G Barnes stated it early in Electric Explanation Of Inertial Mass (1983): the backward reaction force on an accelerated body is a magnetically induced electric force on its constituent charges, and inertial mass is simply the ratio of that force to the acceleration. David L Bergman's Origin of Inertial Mass (1999) derives the same conclusion from the spinning charged ring of Common Sense Science, with the consequence he emphasises: inertia is not an intrinsic property of matter, inertial mass is a derived quantity rather than a fundamental one, and force is therefore the primitive concept of physics.
Charles William Lucas has developed the most extensive version, across The Electrodynamic Origin of the Force of Inertia, Part 1, Part 2 and Part 3 (2007). He derives inertia, including the centrifugal force, as the average residual of the acceleration terms in his universal electrodynamic force between vibrating neutral dipoles — atomic electrons vibrating with respect to the nuclear protons. The derivation uses relative coordinates only, which he presents as delivering what Mach wanted and general relativity did not; it yields inertial mass equal to gravitational mass; and it contains an additional non-radial R × (R × A) term which he argues accounts for observed non-Newtonian gyroscopic motions that F = ma does not describe. The same mechanism implies that both masses decay slowly over time.
Inertia as an interaction with the distant universe
The Machian answer is argued here just as strongly. Andre K T Assis's Relational Mechanics builds a mechanics in which inertia is a genuine gravitational interaction with distant matter, using a Weber-type force law; his note with Jorge Guala-Valverde, Mass in Relational Mechanics (2000), makes the quantitative claim explicit — doubling the average gravitational mass density of the distant galaxies, holding local quantities fixed, would halve the acceleration of free fall at the Earth's surface. Peter Graneau presses the same case in Mach's Principle & Nonlocal Mass Interactions (2009), condensed from In the Grip of the Distant Universe: The Science of Inertia, written with Neal Graneau: the net Newtonian force from all the bodies in the universe acting on an object should be measurable, and Mach and others identified it with the inertial force. Domina Eberle Spencer, Uma Y Shama and Philip J Mann addressed the question formally in "Holors, Tensors, and the Mystery of Inertia" (1996). See Category:Mach's Principle for this literature.
Inertia as an effect of the medium
A third group locates inertia in the structure of space itself. Stoyan Sarg's A New Approach for Study of Gravity and Inertia from the Point of View of BSM-Supergravitation Unified Theory (2009) derives both gravity and inertia from a Cosmic Lattice of sub-elementary nodes filling the physical vacuum, and predicts that disturbing the nodes' synchronisation around an object will alter its gravitational mass — a gravito-inertial effect he reports testing in small-scale experiments and develops in Gravito-inertial Propulsion Effect Predicted by the BSM - Supergravitation Unified Theory (2008). Robert Guy Grantham's The Fabric of Space as an Electron-Positron Lattice and Implications for GRT (2010), arguing for Menahem Simhony's epola, treats inertia along with gravitation and de Broglie waves as an electromagnetic effect of a lattice of bound electrons and positrons.
In The Four Universal Motions in Physics, Bob de Hilster and David de Hilster reinterpret inertia within their Particle Model: a moving body keeps moving because the random flux of fast small bodies filling space sustains it, the same flux that produces gravity by shadowing. Glenn Borchardt treats inertia within the infinite-universe neomechanics of Universal Cycle Theory: Neomechanics of the Hierarchically Infinite Universe.
Finally, Tom Van Flandern's Does Gravity Have Inertia? (2003) turns the question round, starting from the oldest anomaly in the subject — that all bodies fall at the same rate, so that gravity alone among forces meets no resistance proportional to what it moves.