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This is a short review article in which we discuss and summarize the works of various researchers over past four decades on Zeeman topology and Zeeman-like topologies, which occur in special and general theory of relativity. We also discuss various properties and inter-relationship of these topologies.

In special as well as general theory of relativity, space-time models are usually taken as differentiable manifolds. The main reason for representing a space-time as a topological space which is also a differentiable manifold is that we need space-time to have a well-defined topological dimension and we can talk about curves and their tangent vectors, and neighbourhoods to develop a causal theory of space-time. This is achieved by assuming a pseudo-metric structure on a space-time manifold which enables us to define time-like, null and space-like vectors and corresponding curves. In general theory of relativity, metric also determines the geometry and cur- vature of space-time which represents the gravitational field. In special theory of relativity, Minkowski space M is usually given the topology of real 4-dimensional Euclidean space.

According to Zeeman [

Zeeman also proved that the topology on a light ray induced from this fine topology is discrete. This means that every function on the light cone is continuous, as every function will be continuous if the domain space has discrete topology. In quantum field theory also, we face similar difficulties regarding “real” space-time topology, where we talk frequently about continuous wave functions and fields, but we really do not know the meaning of that because “real” topology of space-time is unknown. However, studies have been dedicated to the topological properties of function spaces, such as spaces of quantum fields, but the study of proper space-time topology which is the most important space of all Physics, remains incomplete. Here, we need a topology in which known quantum quantities such as classical paths on which integrations are to be performed in Feynman’s formalism, or Green’s functions are continuous. We note that if a function is continuous on a space with topology T, it will be continuous in any refinement of T.

We also note that Zeeman topology is a refinement of

On mathematical side, M with Zeeman topology is not a normal topological space , as proved by Dossena [

After Zeeman published his paper in 1967, it attracted attention of some of the relativists cum mathematicians and they proved a number of results which are refinements over Zeeman’s work. Modified results about Zeeman- and Zeeman-like topologies were published in the context of both special as well as general theory of relativity. Most remarkable are the results by S. Nanda [

Since Zeeman topology and other fine topologies defined in special and general theory of relativity in above works have many interesting properties, we discuss these properties and also discuss inter-relationships among these topologies. Most important and remarkable of these results are the results proved by R. Göbel and G. Dossena. Göbel proved that the group of all homeomorphisms of a space-time of general relativity with Zeeman-like topology is the group of all homothetic transformations. And Dossena proved that the first homo- topy group of Zeeman topology for Minkowski space is non-trivial and contains uncountably many subgroups isomorphic to Z. In particular, this topology is not simply connected. Lindstrom generalized the results of Göbel and gave a sequence of Zeeman-like topologies which are in the ascending order of fineness.Thus, in Section 2, we describe Zeeman topology and other fine topologies on Minkowski space and discuss their properties. We also discuss t-topology, s-topology and A-topology introduced by Nanda [

We begin this section with definition of Zeeman topology as given in Dossena [

A set U is open in

Physically speaking, the Zeeman topology M^{Z} is defined as the finest topology on a space-time such that its induced topology on world lines of freely falling test particles with positive rest mass, and on space-like hypersurfaces, is locally Euclidean. Zeeman topology is not as nice as manifold topology, e.g. it is not a normal topological space. On the other hand it has many physically interesting properties: The Zeeman topology does not provide any geometric information along a light ray. Mathematically the topology induced by the Zeeman topology on a light cone is discrete. Secondly, there are many unphysical world lines, e.g. bad trips (cf Penrose [

If we interpret continuity of a world line with respect to Zeeman topology, world lines are automatically phy- sically realistic, namely, piecewise geodesics which are future directed and time-like with finitely many edges. Hence a world line is the orbit of a freely falling test particle within the gravitational field with a finite number of collisions. This result is a well known basic assumption for a kinetic theory in general relativity (cf Ehlers [

Moreover if we allow the Zeeman topology to depend on a gravitational field as well as on the Maxwell field, it is possible to derive the corresponding result for charged particles as we discuss below.

In addition to above discussion, we also note that the group of all homeomorphisms of a space-time with its manifold topology is neither of interest for physics nor for mathematics since it is vast and it reflects no information of space-time. However, the group of all homeomorphisms of a space-time M with respect to its Zeeman topology

After Zeeman published his paper in 1967, the first paper by other researcher on this topic was that of S. Nanda [

Let

Light cone or null cone at x :

Time-like cone at

Space-like cone at

Let

Furthermore, let

Then the topology generated by the family ^{s}. Then

Let

Then t-topology is defined as the topology which has the family

Williams [

Here

Topology

He further proves that the group of homeomorphisms of M^{F} is the conformal group of Minkowski space. This is in fact the group generated by the Lorentz group, translations and dilatations, and thus, it is the same as G.

Williams further describes two more fine topologies for M and describes their homeomorphism groups. The first of these topologies is

Following the argument in Nanda [

Second of these topologies is

Definition 2.1. A-topology: The A-topology on M is defined to be the finest topology on M with respect to which the induced topology on every time-like line and light-like line is one-dimensional Euclidean and the in- duced topology on every space-like hyperplane is three-dimensional Euclidean.

Thus A-topology is strictly finer than the Euclidean topology.

The topology

Popvassilev [

S. Nanda and H.K. Panda [

We now discuss the work of Dossena [

As defined in the begining of this section, Dossena presents Zeeman topology

For two dimensional Minkowski space with topologies

Lemma 2.1. A compact subset of

Lemma 2.2. Let X be a Hausdorff topological space and let

Lemma 2.3. Every Zeno sequence admits a subsequence whose image is a non closed, discrete subset of

Theorem 2.4. A compact subset K of

This is true for A-topology also, as proved by Nanda [

Theorem 2.5. For a subset

1) K is compact in

2) K is compact in

3) K is covered by a finite family

We now discuss countability properties of

We choose an orthonormal frame of reference

dinates

Clearly

Then we have the following proposition:

Proposition 2.6. For every orthonormal frame of reference, the above-mentioned set Q is also dense in

Corollary 2.7. The cardinality of the set

Proposition 2.8.

Zeeman [

Theorem 2.9.

For a path-connected topological space X,

Theorem 2.10.

A topological study of the n-dimensional Minkowski space,

t-topology for four dimensional Minkowski space has been defined above. Similar definition follows for

It thus follows that

dean topology, while

basis for the t-topology and the t-topology is strictly finer than the Euclidean topology on M.

s-topology can be defined similarly on

Summarizing, we have the following:

The collection

Other works on Zeeman-like topologies include that of Struchiner and Rosa [

Struchiner and Rosa [

Domiaty [

Finally, we remark that even though Zeeman topology on Minkowski space has several advantages over the standard topology, it has some drawbacks also. These are as follows:

1) A three dimensional section of simultaneity has no meaning in terms of physically possible experiments. Also, the use of straight time like lines in defining

2) The isometry and conformal groups of

3) The set of

4)

Keeping these drawbacks in mind, Hawking, King and Mc Carthy [

Here, we consider a space-time of general relativity which is assumed to be connected, Hausdorff, paracompact,

The path topology

Thus if a set

HKM show that

Let

Proposition 3.1. Sets of the form

Theorem 3.2.

This property has no analogue in the finer topologies

Theorem 3.3. A path

Theorem 3.4.

Furthermore, HKM determine the group of

To begin with, they prove the following:

Proposition 3.5.

This has been proved for strongly causal space-times. It is done by singling out a subclass of

After proving a series of results, HKM prove the following important theorem:

Theorem 3.6. A

Theorem 3.7. The group of

Finally, HKM give an example of a manifold for which the group of smooth conformal diffeomorphisms is strictly larger than the homothecy group. We note here that for Minkowski space, the two groups are equal.

For more details and proofs, we refer the reader to HKM [

Malament [

Main result of this paper is the following:

Suppose we consider two space-times

Brief summary of the proof is as follows:

If f preserves all continuous curves, then f would be continuous. Given any sequence

The idea to overcome this difficulty is as follows:

To show that f is continuous at p, one proves that one may assume that f is continuous over a `nice-looking’ region near p. Then one uses continuous null geodesic segments in this region to characterize the convergence of points to p. This then leads one to the required result because continuous null geodesics in this region are necessarily preserved by f. For technical details, we refer the reader to Malament [

Fullwood [

Then, define

Now, let

Then,

Fullwood proves that if the space-time V is future and past distinguishing, then the topology

Theorem 3.8. The following three conditions are equivalent upon a space-time manifold:

1)

Do-Hyung Kim [

Definition 3.1.

Proposition 3.9. The above family of open sets define a new topology

Proposition 3.10. The topology

Since

Corollary 3.11.

The construction of

Furthermore, Kim studies homeomorphisms with respect to topology

Definition 3.2. A bijection

Proposition 3.12. If V and N are globally hyperbolic and

Theorem 3.13. If

Theorem 3.14. If

Since

Theorem 3.15. A

Also if

Such bijective mappings have also been studied by Domiaty [

More recently Huang [

The physical meaning of the condition used in this theorem is that images and pre-images of paths which photons travel between emission and absorption should again be such paths.

Coming to the topological properties of Zeeman-like topologies on Minkowski space M again, we note the Theorem proved by Dossena, namely, two dimensional Minkowski space is not simply connected. Its first homotopy group contains uncountably many subgroups isomorphic to Z.G. Agrawal and S. Shrivastava [

Theorem 3.16. A space-time V, equipped with the path topology is not simply connected or locally simply connected. Furthermore, no two closed continuous curves in V with distinct images are homotopic.

Proof: Let

Here, it will not be out of place to mention that Sorkin and Woolgar [

We specify closed sets of

In this section, we describe and discuss the work of Göbel [

We start with definitions of Zeeman topologies as given by Göbel [

Let

Let

Then

On Minkowski space this topology coincides with the topology Z defined by Zeeman mentioned above, for two specially chosen systems

Further Göbel defines a Special system

1) If

2) If

3) If

4) We have

With this definition, the following results follow:

Proposition 4.1. Let

(A curve f is called

that if

Proposition 4.2. If

This implies the following:

Proposition 4.3. For a manifold

1) the curve f is a piecewise geodesic i.e. f is a broken geodesic line with a finite number of edges.

2) the 1-1 map

Göbel then restricts Zeeman topology on a space-time and studies Zeeman topology by incorporating electromagnetic fields. To state the results proved by Göbel in this situation, we need to understand certain notations:

Let V denote a space-time for general relativity and F be a given electromagnetic field on V. An electric charge

Proposition 4.4. If V is a space-time with a given external electro-magnetic field F and a world line f, the following statements are equivalent:

1) f is continuous with respect to the Zeeman topology

2) f is a chain of finitely many connected world lines of freely falling charged test particles.

If F = 0, then Z-continuous world lines are future directed time-like piecewise geodesic lines. For simplicity, we denote

A subset Y of V is open with respect to

(I) U is an arbitrary closed space-like hypersurface contained in a simple region of V.

(II) U is the world line of an arbitrary charged test particle p freely falling in the gravitational and the electro- magnetic field within a simple region of V.

If Q = 0, then condition (II) is equivalent to

(II)’ U is an arbitrary time-like geodesic in a simple region of V.

If U is a simple neighbourhood of p then let

Lemma 4.5. The set

Göbel then proves an important result that

Proposition 4.6. The topology induced by

Thus we do not have any geometric information along a light ray.

The main theorem of Göbel [

Theorem 4.7. Let h be a mapping from space-time V onto a space-time

1) h is a homeomorphism with respect to Zeeman topology Z.

2) h is a homothetic transformation.

Unusual property of Zeeman topology is that homeomorphism characteristic of h implies its differentiability as well as its “linearity”, since h is an isometric map “up to scaling”. Thus we can state this property in the following forms:

Theorem 4.8. The space-times V and

Theorem 4.9. The group of all homeomorphisms with respect to the Zeeman topology coincides with the group of all homothetic transformations of space-time V onto itself.

Thus Zeeman topology contains all information about the metric.

We again note here that (locally) causal maps defined by Göbel [

As far as Minkowski space-time is concerned, Zeeman [

Remark 1. The topology

Theorem 4.10. Let

Further, this topology has a physically attractive feature as follows:

If

Hawking, King and Mc Carthy [

Let

A locally one-one Feynman path is then a Feynman track mentioned above.

Let G denote the group of automorphisms of V given by

1) the Lorentz group of all linear maps leaving quadratic form Q invariant

2) translations and

3) dilatations.

Every element of G either preserves or reverses the partial ordering “<” mentioned above. These features have been studied in details by Nanda, Dossena and Kim.

Remark 2. The topology

Remark 3. The topology

1) It is not locally homogeneous and the light cone through any point can be deduced from it.

2) The group of all homeomorphisms with respect to

3) It induces the 3-dimensional Euclidean topology on every space axis and the 1-dimensional Euclidean topology on every time axis.

For the proof of these properties, we refer the reader to Williams [

Ulf Lindstrom [

Finally, we add a comment about the work of Mashford [

In this article, we have given a short review of Zeeman- and Zeeman-like fine topologies on Minkowski space and space-time of general relativity. We have avoided giving detailed proofs of the results mentioned, otherwise the article would have become lengthy. To the best of our knowledge, we have reviewed most of the research work which appeared on this topic since the first paper was published by Zeeman in 1967. To get a consolidated view about definitions and the main properties of these topologies like their homeomorphism groups and topological properties, we give two tables summarizing definitions and their properties:

Definitions and properties of fine topologies on Minkowski space refer

Sr. No | Fine topology | Homeomorphism group | Topological properties |
---|---|---|---|

1 | Zeeman topology | G = Lorentz group with translations and dilatations | Dossena (2007): neither locally compact nor Lindelof, not normal, separable but not first countable, path-connected but not simply connected |

2 | s-topology | G | G.Agrawal and S. Shrivastava (2012): separable, first countable, path-connected, not regular, not metrizable, not second countable, noncompact, and non-Lindelof, not simply connected |

3 | t-topology | G | G.Agrawal and S. Shrivastava (2009): separable, first countable, path-connected, not regular, not metrizable, not second countable, not locally compact, not simply connected |

4 | A-topology | G | G.Agrawal and Soami Pyari Sinha (2014): separable, not first countable, connected and path-connected, not normal, not metrizable, Not comparable with t-topology nor with s-topology |

5 | Fine topologies | Conformal group of Minkowski space whose | Hausdorff, separable, first countable, but not regular and hence not metrizable |

6 | Weaker than |

Sr. No | Fine topology on space-time of GR | Diffeomorphism Group | Topological properties |
---|---|---|---|

1 | HKM-path topology described by Hawking-King-McCarty (1976) | Conformal diffeomorphisms | Hausdorff, path connected and locally path connected, first countable, separable, but not normal or locally compact |

2 | Extended HKM-topology (Kim, 2006) | Conformal isomorphism group | Finer than Alexandrov topology |

3 | S-topology on Lorentz manifolds (Domiaty, 1985) | Conformal | Hausdorff, first countable and separable, not regular and hence not metrizable, path connected and locally path connected |

4 | Zeeman -like fine topology in general relativity described by Göbel (1976) | Homeomorphism group with respect to Zeeman-like topology is the group of all homothetic transformations of V | Strongly causal space-times |

5 | Lindstrom (1978): Finest topology | Group of Conformal | Space-time need not be strongly causal |

realist view of space-time topology. Other philosophical issues about space-time have been discussed by D. Dieks and M. Redel in two volumes [

Ravindra Saraykar,Sujatha Janardhan, (2016) Zeeman-Like Topologies in Special and General Theory of Relativity. Journal of Modern Physics,07,627-641. doi: 10.4236/jmp.2016.77063