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Just as the five-dimensional metric comprises the 4-D metric framed by the electromagnetic vector potential, the 5-dimensional stress—energy tensor comprises the 4-D stress—energy tensor framed by the vector 4-current. Kaluza's original hypothesis was purely classical and extended discoveries of general relativity. The quantization of electric charge could then be nicely understood in terms of integer multiples of fifth-dimensional momentum.

It recapitulated much of the classical theory of Kaluza described above, and then departed into Klein's quantum interpretation.

Klein solved a Schroedinger-like wave equation using an expansion in terms of fifth-dimensional waves resonating in the closed, compact fifth dimension. Kaluza published his derivation in as an attempt to unify electromagnetism with Einstein's general relativity. In , Oskar Klein proposed that the fourth spatial dimension is curled up in a circle of a very small radius , so that a particle moving a short distance along that axis would return to where it began.

The distance a particle can travel before reaching its initial position is said to be the size of the dimension. This extra dimension is a compact set , and construction of this compact dimension is referred to as compactification.

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In modern geometry, the extra fifth dimension can be understood to be the circle group U 1 , as electromagnetism can essentially be formulated as a gauge theory on a fiber bundle , the circle bundle , with gauge group U 1. In Kaluza—Klein theory this group suggests that gauge symmetry is the symmetry of circular compact dimensions. Once this geometrical interpretation is understood, it is relatively straightforward to replace U 1 by a general Lie group.

Such generalizations are often called Yang—Mills theories. If a distinction is drawn, then it is that Yang—Mills theories occur on a flat spacetime, whereas Kaluza—Klein treats the more general case of curved spacetime. The base space of Kaluza—Klein theory need not be four-dimensional spacetime; it can be any pseudo- Riemannian manifold , or even a supersymmetric manifold or orbifold or even a noncommutative space. The construction can be outlined, roughly, as follows. Given a connection on the bundle, and a metric on the base manifold, and a gauge invariant metric on the tangent of each fiber, one can construct a bundle metric defined on the entire bundle.

Computing the scalar curvature of this bundle metric, one finds that it is constant on each fiber: this is the "Kaluza miracle". One did not have to explicitly impose a cylinder condition, or to compactify: by assumption, the gauge group is already compact. Next, one takes this scalar curvature as the Lagrangian density , and, from this, constructs the Einstein—Hilbert action for the bundle, as a whole.

The equations of motion, the Euler—Lagrange equations , can be then obtained by considering where the action is stationary with respect to variations of either the metric on the base manifold, or of the gauge connection. Variations with respect to the base metric gives the Einstein field equations on the base manifold, with the energy—momentum tensor given by the curvature field strength of the gauge connection. On the flip side, the action is stationary against variations of the gauge connection precisely when the gauge connection solves the Yang—Mills equations.

Thus, by applying a single idea: the principle of least action , to a single quantity: the scalar curvature on the bundle as a whole , one obtains simultaneously all of the needed field equations, for both the spacetime and the gauge field. However, an attempt to convert this interesting geometrical construction into a bona-fide model of reality flounders on a number of issues, including the fact that the fermions must be introduced in an artificial way in nonsupersymmetric models.

Nonetheless, KK remains an important touchstone in theoretical physics and is often embedded in more sophisticated theories. It is studied in its own right as an object of geometric interest in K-theory. Even in the absence of a completely satisfying theoretical physics framework, the idea of exploring extra, compactified, dimensions is of considerable interest in the experimental physics and astrophysics communities.

A variety of predictions, with real experimental consequences, can be made in the case of large extra dimensions and warped models. For example, on the simplest of principles, one might expect to have standing waves in the extra compactified dimension s.

This set of possible mass values is often called the Kaluza—Klein tower. Similarly, in Thermal quantum field theory a compactification of the euclidean time dimension leads to the Matsubara frequencies and thus to a discretized thermal energy spectrum.

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However, Klein's approach to a quantum theory is flawed [ citation needed ] and, for example, leads to a calculated electron mass in the order of magnitude of the Planck mass. Brandenberger and Vafa have speculated that in the early universe, cosmic inflation causes three of the space dimensions to expand to cosmological size while the remaining dimensions of space remained microscopic. One particular variant of Kaluza—Klein theory is space—time—matter theory or induced matter theory , chiefly promulgated by Paul Wesson and other members of the Space—Time—Matter Consortium.

In other words, the cylinder condition of the previous development is dropped, and the stress—energy now comes from the derivatives of the 5D metric with respect to the fifth coordinate. Because the energy—momentum tensor is normally understood to be due to concentrations of matter in four-dimensional space, the above result is interpreted as saying that four-dimensional matter is induced from geometry in five-dimensional space.

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The general equations can be shown to be sufficiently consistent with classical tests of general relativity to be acceptable on physical principles, while still leaving considerable freedom to also provide interesting cosmological models. The Kaluza—Klein theory has a particularly elegant presentation in terms of geometry. In a certain sense, it looks just like ordinary gravity in free space , except that it is phrased in five dimensions instead of four. The equations governing ordinary gravity in free space can be obtained from an action , by applying the variational principle to a certain action.


Let M be a pseudo- Riemannian manifold , which may be taken as the spacetime of general relativity. If g is the metric on this manifold, one defines the action S g as. By applying the variational principle to the action. Here, R ij is the Ricci tensor. In the absence of charges and currents, the free-field Maxwell equations are.

To build the Kaluza—Klein theory, one picks an invariant metric on the circle S 1 that is the fiber of the U 1 -bundle of electromagnetism. In this discussion, an invariant metric is simply one that is invariant under rotations of the circle. The consistency conditions are:. The connection A on the fiber bundle is related to the electromagnetic field strength as.

That there always exists such a connection, even for fiber bundles of arbitrarily complex topology, is a result from homology and specifically, K-theory. Applying Fubini's theorem and integrating on the fiber, one gets. Varying the action with respect to the component A , one regains the Maxwell equations. Applying the variational principle to the base metric g , one gets the Einstein equations.

In this case, the coupling between gravity and the electromagnetic field is not constant, but has its own dynamical field, the radion. If the base manifold is four-dimensional, the Kaluza—Klein manifold P is five-dimensional. The fifth dimension is a compact space , and is called the compact dimension. The technique of introducing compact dimensions to obtain a higher-dimensional manifold is referred to as compactification.

Compactification does not produce group actions on chiral fermions except in very specific cases: the dimension of the total space must be 2 mod 8 and the G-index of the Dirac operator of the compact space must be nonzero. The above development generalizes in a more-or-less straightforward fashion to general principal G -bundles for some arbitrary Lie group G taking the place of U 1.

In such a case, the theory is often referred to as a Yang—Mills theory , and is sometimes taken to be synonymous. If the underlying manifold is supersymmetric , the resulting theory is a super-symmetric Yang—Mills theory. Thus, Hume and his intellectual progenies suppose all the laws of nature are known, definite and predictable. But as will be shown later, this position is both subjective and fallacious. Again, this is predicated on a worldview of a world-machine. But two questions raised by Moltmann are immediately crucial and relevant. First, is the scientific claim on the knowability of nature valid? Second, are the laws of nature really fixed and eternal decrees for temporal events or are they themselves part of temporal happenings? Interestingly, development in modern science itself provides challenging insights in the light of these questions. As Hefling correctly observes, The mechanistic determinism that was once thought inseparable from scientific knowledge as such has begun to be dismantled by science itself.

It is interesting how science has changed its own terms in the last century. For instance— and of some serious implications—scientific development has now shifted grounds in its understanding of the nature of physical law, from classical physics which is universal in form and states what must happen to a quantum physics which is statistical in form and states what might probably happen. This is a point firmly argued by Worthing who also plausibly concludes that quantum theory, therefore, creates a flexibility which makes it difficult to speak with certainty as to future happenings with nature.

This recent understanding of physical laws is of serious implications for the discussion on the possibility or otherwise of miracles. First, it reveals that the laws of nature are themselves part of temporal happenings and not unalterable orders for cosmological events. They are not fixed, universal and unalterable laws, but are part of the created order. There is, therefore, a continuous unveiling of God's supreme and sovereign will and acts in nature, contrary to the views of deists.

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The bottom line is that what scientists call laws of nature may not necessarily be laws in the true sense of it, but mere generalizations of future expectations based on past empirical observations and their consequent inferences. They are conclusions on observations made in the past, and so cannot be used as a basis for full proof judgment or prediction of the future.

In other words, there is no logical conclusiveness in asserting that because an occurrence habitually happened in the past it therefore follows that it must happen in the same manner or, better put, cannot happen otherwise in the future. This reveals a few facts about the Newtonian World-machine worldview of the critics. First, it demonstrates that such worldview lends credence to deism, and presupposes that God finished his work on and with nature, setting in place the laws that would govern the affairs of nature, without further active involvement from him.

Nothing can be more fallacious. Truth is God is still at work in and with nature up to this day. Second, it also demonstrates that the claims on the knowability of nature by some scientists and philosophers have been the climax of intellectual arrogance. Theirs was, and remains, a developing knowledge built on statistical probabilities based on past empirical observations. It is, therefore, clear that the definition of miracle that speaks of violation or transgression of laws of nature is misleading. A miracle is not a violation of any law—natural or otherwise.

This by far reflects what miracle truly is because it shows that what scientists term laws of nature are not a comprehensive body of knowledge but only as has been explored and accessed by science. It also reveals that science is yet to unveil a full or comprehensive understanding of the universe, even though it is constantly making astounding progress.