Many textbooks dealing with general relativity do not demonstrate the derivation of forces in enough detail. The analyses presented herein demonstrate straightforward methods for computing forces by way of general relativity. Covariant divergence of the stress–energy–momentum tensor is used to derive a general expression of the force experienced by an observer in general coordinates. The general force is then applied to the local co-moving coordinate system of a uniformly accelerating observer, leading to an expression of the inertial force experienced by the observer. Next, applying the general force in Schwarzschild coordinates is shown to lead to familiar expressions of the gravitational force. As a more complex demonstration, the general force is applied to an observer in Boyer–Lindquist coordinates near a rotating, Kerr black hole. It is then shown that when the angular momentum of the black hole goes to zero, the force on the observer reduces to the force on an observer held stationary in Schwarzschild coordinates. As a final consideration, the force on an observer moving in rotating coordinates is derived. Expressing the force in terms of Christoffel symbols in rotating coordinates leads to familiar expressions of the centrifugal and Coriolis forces on the observer. It is envisioned that the techniques presented herein will be most useful to graduate level students, as well as those undergraduate students having experience with general relativity and tensor analysis.

When the covariant form of Maxwell’s equations are applied to a rotating reference frame, a choice must be made to work with either a covariant electromagnetic tensor Fαβ or a contravariant electromagnetic tensor Fαβ. We argue that which tensor one chooses is ultimately dictated by whether one chooses to express the electric and magnetic fields in terms of a vector basis or in terms of a one-form basis, dual to the vector basis. We explain that when fields are expressed as one-forms, the covariant electromagnetic tensor is used; and when fields are expressed as vectors, the contravariant tensor is used. Using this formalism, we derive general field equations expressed in terms of vector and one-form fields in the rotating and laboratory frames when matter is present. Fields in the presence of matter are then related to those in a vacuum by using a covariant form of Minkowski’s constitutive equations, generalized to noninertial frames. Both vector and one-form field equations are used to derive the fields observed in the reference frame of a polarizable, permeable cylinder that rotates within an axially directed magnetic field. We find that the vector and one-form field equations both lead to predictions consistent with experimental results. We conclude that the choice between working with a covariant or contravariant electromagnetic tensor depends upon whether one chooses to express fields as vectors or as one-forms.

We apply relativistic electrodynamics to a rotating linear medium. Covariant field equations are used to derive general field equations in a rotating coordinate system. We argue that the relation between fields in the presence of matter and those in a vacuum is necessarily dependent upon the coordinate system used. Constitutive equations are then derived in the rotating and laboratory reference frames. We find that our constitutive equations in the laboratory frame agree with Minkowski’s constitutive equations, derived on the basis of special relativity in 1908. Thus we conclude that special relativity can be used in the analysis of experiments involving rotational motion. To exemplify the use of special relativity, we derive an experimentally observed result of a 1913 experiment performed by Wilson and Wilson in which a polarizable, permeable cylinder was rotated in a uniform, axially directed magnetic field.

Archimedes' principle is well known to state that a body submerged in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the body. Herein, Archimedes' principle is derived from first principles by using conservation of the stress–energy–momentum tensor in general coordinates. The resulting expression for the force is applied in Schwarzschild coordinates and in rotating coordinates. Using Schwarzschild coordinates for the case of a spherical mass suspended within a perfect fluid leads to the familiar expression of Archimedes' principle. Using rotating coordinates produces an expression for a centrifugal buoyancy force that agrees with accepted theory. It is then argued that Archimedes' principle ought to be applicable to non-gravitational phenomena, as well. Conservation of the energy–momentum tensor is then applied to electromagnetic phenomena. It is shown that a charged body submerged in a charged medium experiences a buoyancy force in accordance with an electromagnetic analogue of Archimedes' principle.

A recent assertion that inertial and gravitational forces are entropic forces is discussed. A more conventional approach is stressed herein, whereby entropy is treated as a result of relative motion between observers in different frames of reference. It is demonstrated that the entropy associated with inertial and gravitational forces is dependent upon the well known lapse function of general relativity. An interpretation of the temperature and entropy of an accelerating body is then developed, and used to relate the entropic force to Newton's second law of motion. The entropic force is also derived in general coordinates. An expression of the gravitational entropy of in-falling matter is then derived by way of Schwarzschild coordinates. As a final consideration, the entropy of a weakly gravitating matter distribution is shown to be proportional to the self-energy and the stress-energy-momentum content of the matter distribution.