Abstract. Assuming a general timelike congruence of worldlines as a reference frame, we derive a covariant general formalism of inertial forces in General Relativity. Inspired by the works of Abramowicz et. al. (see e.g. Abramowicz and Lasota 1997 Class. Quantum Grav. 14 A23-30), we also study conformal rescalings of spacetime and investigate how these affect the inertial force formalism. While many ways of describing spatial curvature of a trajectory has been discussed in papers prior to this, one particular prescription (which differs from the standard projected curvature when the reference is shearing) appears novel. For the particular case of a hypersurface-forming congruence, using a suitable rescaling of spacetime, we show that a geodesic photon is always following a line that is spatially straight with respect to the new curvature measure. This fact is intimately connected to Fermat's principle, and allows for a certain generalization of the optical geometry as will be further pursued in a companion paper (Jonsson and Westman 2006 Class. Quantum Grav. 23 61). For the particular case when the shear-tensor vanishes, we present the inertial force equation in threedimensional form (using the bold face vector notation), and note how similar it is to its Newtonian counterpart. From the spatial curvature measures that we introduce, we derive corresponding covariant differentiations of a vector defined along a spacetime trajectory. This allows us to connect the formalism of this paper to that of Jantzen et. al. (see e.g. Bini et. al. 1997 Int. J. Mod. Phys. D 6 143-98).

As the velocity of a rocket in a circular orbit near a black hole increases, the outwardly directed rocket thrust must increase to keep the rocket in its orbit. This feature might appear paradoxical from a Newtonian viewpoint, but we show that it follows naturally from the equivalence principle together with special relativity and a few general features of black holes. We also derive a general relativistic formalism of inertial forces for reference frames with acceleration and rotation. The resulting equation relates the real experienced forces to the time derivative of the speed and the spatial curvature of the particle trajectory relative to the reference frame. We show that an observer who follows the path taken by a free (geodesic) photon will experience a force perpendicular to the direction of motion that is independent of the observers velocity. We apply our approach to resolve the submarine paradox, which regards whether a submerged submarine in a balanced state of rest will sink or float when given a horizontal velocity if we take relativistic effects into account. We extend earlier treatments of this topic to include spherical oceans and show that for the case of the Earth the submarine floats upward if we take the curvature of the ocean into account.

Abstract. In special relativity a gyroscope that is suspended in a torque-free manner will precess as it is moved along a curved path relative to an inertial frame S. We explain this effect, which is known as Thomas precession, by considering a real grid that moves along with the gyroscope, and that by definition is not rotating as observed from its own momentary inertial rest frame. From the basic properties of the Lorentz transformation we deduce how the form and rotation of the grid (and hence the gyroscope) will evolve relative to S. As an intermediate step we consider how the grid would appear if it were not length contracted along the direction of motion. We show that the uncontracted grid obeys a simple law of rotation. This law simplifies the analysis of spin precession compared to more traditional approaches based on Fermi transport. We also consider gyroscope precession relative to an accelerated reference frame and show that there are extra precession effects that can be explained in a way analogous to the Thomas precession. Although fully relativistically correct, the entire analysis is carried out using three-vectors. By using the equivalence principle the formalism can also be applied to static spacetimes in general relativity. As an example, we calculate the precession of a gyroscope orbiting a static black hole.

We show that by employing the standard projected curvature as a measure of spatial curvature, we can make a certain generalization of optical geometry (Abramowicz and Lasota 1997 Class. Quantum Grav. 14 A23). This generalization applies to any spacetime that admits a hypersurface orthogonal shearfree congruence of worldlines. This is a somewhat larger class of spacetimes than the conformally static spacetimes assumed in standard optical geometry. In the generalized optical geometry, which in the generic case is time dependent, photons move with unit speed along spatial geodesics and the sideways force experienced by a particle following a spatially straight line is independent of the velocity. Also gyroscopes moving along spatial geodesics do not precess (relative to the forward direction). Gyroscopes that follow a curved spatial trajectory precess according to a very simple law of three-rotation. We also present an inertial force formalism in coordinate representation for this generalization Furthermore, we show that by employing a new sense of spatial curvature (Jonsson 2006 Class. Quantum Grav. 23 1) closely connected to Fermat's principle, we can make a more extensive generalization of optical geometry that applies to arbitrary spacetimes. In general this optical geometry will be time dependent, but still geodesic photons move with unit speed and follow lines that are spatially straight in the new sense. Also, the sideways experienced (comoving) force on a test particle following a line that is straight in the new sense will be independent of the velocity.PACS numbers: 04.20.-q, 95.30.Sf

I present a way to visualize the concept of curved spacetime. The result is a curved surface with local coordinate systems (Minkowski Systems) living on it, giving the local directions of space and time. Relative to these systems, special relativity holds. The method can be used to visualize gravitational time dilation, the horizon of black holes, and cosmological models. The idea underlying the illustrations is first to specify a field of timelike four-velocities. Then, at every point, one performs a coordinate transformation to a local Minkowski system comoving with the given four-velocity. In the local system, the sign of the spatial part of the metric is flipped to create a new metric of Euclidean signature. The new positive definite metric, called the absolute metric, can be covariantly related to the original Lorentzian metric. For the special case of a 2-dimensional original metric, the absolute metric may be embedded in 3-dimensional Euclidean space as a curved surface.Comment: 15 pages, 20 figure