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Despite being a major component in the teaching of special relativity, the twin `paradox' is generally not examined in courses on general relativity. Due to the complexity of analytical solutions to the problem, the paradox is often neglected entirely, and students are left with an incomplete understanding of the relativistic behaviour of time. This article outlines a project, undertaken by undergraduate physics students at the University of Sydney, in which a novel computational method was derived in order to predict the time experienced by a twin following a number of paths between two given spacetime coordinates. By utilising this method, it is possible to make clear to students that following a geodesic in curved spacetime does not always result in the greatest experienced proper time.

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Several physical problems such as the `twin paradox' in curved spacetimes
have purely geometrical nature and may be reduced to studying properties of
bundles of timelike geodesics. The paper is a general introduction to
systematic investigations of the geodesic structure of physically relevant
spacetimes. The investigations are focussed on the search of locally and
globally maximal timelike geodesics. The method of dealing with the local
problem is in a sense algorithmic and is based on the geodesic deviation
equation. Yet the search for globally maximal geodesics is non-algorithmic and
cannot be treated analytically by solving a differential equation. Here one
must apply a mixture of methods: spacetime symmetries (we have effectively
employed the spherical symmetry), the use of the comoving coordinates adapted
to the given congruence of timelike geodesics and the conjugate points on these
geodesics. All these methods have been effectively applied in both the local
and global problems in a number of simple and important spacetimes and their
outcomes have already been published in three papers. Our approach shows that
even in Schwarzschild spacetime (as well as in other static spherically
symetric ones) one can find a new unexpected geometrical feature: instead of
one there are three different infinite sets of conjugate points on each stable
circular timelike geodesic curve. Due to problems with solving differential
equations we are dealing solely with radial and circular geodesics.

Motivated by a conjecture put forward by Abramowicz and Bajtlik we reconsider
the twin paradox in static spacetimes. According to a well known theorem in
Lorentzian geometry the longest timelike worldline between two given points is
the unique geodesic line without points conjugate to the initial point on the
segment joining the two points. We calculate the proper times for static twins,
for twins moving on a circular orbit (if it is a geodesic) around a centre of
symmetry and for twins travelling on outgoing and ingoing radial timelike
geodesics. We show that the twins on the radial geodesic worldlines are always
the oldest ones and we explicitly find the conjugate points (if they exist)
outside the relevant segments. As it is of its own mathematical interest, we
find general Jacobi vector fields on the geodesic lines under consideration. In
the first part of the work we investigate Schwarzschild geometry.

Recently Abramowicz and Bajtlik [ArXiv: 0905.2428 (2009)] have studied the
twin paradox in Schwarzschild spacetime. Considering circular motion they
showed that the twin with a non-vanishing 4-acceleration is older than his
brother at the reunion and argued that in spaces that are asymptotically
Minkowskian there exists an absolute standard of rest determining which twin is
oldest at the reunion. Here we show that with vertical motion in Schwarzschild
spacetime the result is opposite: The twin with a non-vanishing 4-acceleration
is younger. We also deduce the existence of a new relativistic time effect,
that there is either a time dilation or an increased rate of time associated
with a clock moving in a rotating frame. This is in fact a first order effect
in the velocity of the clock, and must be taken into account if the situation
presented by Abramowicz and Bajtlik is described from the rotating rest frame
of one of the twins. Our analysis shows that this effect has a Machian
character since the rotating state of a frame depends upon the motion of the
cosmic matter due to the inertial dragging it causes. We argue that a
consistent formulation and resolution of the twin paradox makes use of the
general principle of relativity and requires the introduction of an extended
model of the Minkowski spacetime. In the extended model Minkowski spacetime is
supplied with a cosmic shell of matter with radius equal to its own
Schwarzschild radius, so that there is perfect inertial dragging inside the
shell.

Two ideal standard clocks, effectively isolated from interaction with other physical systems, and in a region of the universe free of gravitational fields, are assumed to move in any arbitrary manner so that they coincide on at least two occasions. In general, the reading of one of them will become retarded relative to the other in the interval between successive coincidences. This relative retardation is predicted by the restricted theory of relativity, taken together with the assumption that the ‘rate’ of a clock depends only on its velocity and not on its acceleration. The recognized procedure for calculating the retardation in terms of clock ‘rates’ is set out, and is illustrated by its application to a simple hypothetical experiment in which one clock remains at rest and the other travels away from and back to it with constant speed in a straight line.
There is nothing paradoxical in the predicted retardation as such. The so-called 'clock paradox' arises because an alternative, and apparently valid, calculation procedure, also based on clock 'rates', leads to a contradictory result. The term ‘paradox’ is something of a euphemism since the two predictions are contradictories.
It is shown that the paradox does not arise when direct use is made of the Lorentz transformation without introducing the additional, and non-essential, step in reasoning involved in utilizing the clock ‘rates’. It is then shown that the paradox arises only through using these clock ‘rates’ without due regard to the exact significance of the quantities so described; once this is recognized the paradox is resolved completely within the framework of the restricted theory, which then provides a unique and unambiguous prediction of the relative retardation.
Once the paradox is thus resolved, the general theory of relativity can add nothing significant to the analysis. It is shown that the application of the principle of equivalence is essentially trivial; in effect, Einstein and Tolman evaded the real logical issue raised by the contradictory predictions by denying the applicability of the restricted theory and then utilizing, by means of the principle of equivalence, results obtained from it. This tortuous procedure succeeded in evading the paradox rather than in resolving it; it would obviously be quite invalid were the restricted theory indeed inapplicable to the problem.

We discuss the clock paradox for the case of clocks in co-revolving and
counter-revolving circular orbits in the Kerr metric. We derive an exact
formula for the difference in the two clock rates, which is exceedingly
small in physically realizable measurements.

The nature of the clock paradox is discussed and three solutions are referred to: (a) length contraction, (b) Doppler effects, (c) world lines in chronogeometry. The Special Theory of Relativity gives a complete explanation of the problem and it is pointed out how the use of the General Theory provides merely an additional solution with no physically new aspects. Variations in the clock problem are introduced to make clear the essential asymmetry which exists between the two clocks.

Rapid interstellar travel by means of spacetime wormholes is described in a way that is useful for teaching elementary general relativity. The description touches base with Carl Sagan's novel Contact, which, unlike most science fiction novels, treats such travel in a manner that accords with the best 1986 knowledge of the laws of physics. Many objections are given against the use of black holes or Schwarzschild wormholes for rapid interstellar travel. A new class of solutions of the Einstein field equations is presented, which describe wormholes that, in principle, could be traversed by human beings. It is essential in these solutions that the wormhole possess a throat at which there is no horizon; and this property, together with the Einstein field equations, places an extreme constraint on the material that generates the wormhole's spacetime curvature: In the wormhole's throat that material must possess a radial tension tauâ with the enormous magnitude tauâapprox. (pressure at the center of the most massive of neutron stars) x (20 km)Â²/(circumference of throat)Â². Moreover, this tension must exceed the material's density of mass-energy, rhoâcÂ². No known material has this tauâ>rhoâcÂ² property, and such material would violate all the ''energy conditions'' that underlie some deeply cherished theorems in general relativity. However, it is not possible today to rule out firmly the existence of such material; and quantum field theory gives tantalizing hints that such material might, in fact, be possible.

The ages of twins who travel along intersecting orbits in a central gravitational field are studied using general relativity. It is shown to order v2 / c2 that even though neither twin experiences acceleration, the aging is asymmetric.

We discuss a rather surprising version of the twin paradox in which (contrary to the familiar classical version) the twin who accelerates is older on the reunion than his never accelerating brother.

We consider a new version of the twin paradox. The twins move along the same circular free photon path around the Schwarzschild center. In this case, despite their different velocities, all twins have the same non-zero acceleration. On the circular photon path, the symmetry between the twins situations is broken not by acceleration (as it is in the case of the classic twin paradox), but by the existence of an absolute standard of rest (timelike Killing vector). The twin with the higher velocity with respect to the standard of rest is younger on reunion. This closely resembles the case of periodic motions in compact (non-trivial topology) 3-D space recently considered in the context of the twin paradox by Barrow and Levin, except that there accelerations of all twins were equal to zero, and that in the case considered here, the 3-D space has trivial topology. Comment: Two pages, submitted to Physical Review A

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