arXiv:0704.0077v1  [physics.gen-ph]  1 Apr 2007
UNIVERSAL FORCES AND THE DARK
ENERGY PROBLEM
AFSAR ABBAS
Centre for Theoretical Physics
JMI, Jamia Nagar, New Delhi - 110025, India
(e-mail : safsarabbas@gmail.com)
Abstract
The Dark Energy problem is forcing us to re-examine our models and our
understanding of relativity and space-time. Here a novel idea of Fundamen-
tal Forces is introduced. This allows us to perceive the General Theory of
Relativity and Einstein’s Equation from a new pesrpective. In addition to
providing us with an improved understanding of space and time, it will be
shown how it leads to a resolution of the Dark Energy problem.
http://arxiv.org/abs/0704.0077v1
Dark Energy is certainly the most puzzling problem in physics and as-
tronomy today [1]. All kind of proposals to solve the problem are being put
forward, but we are nowhere near a resolution of the issues involved. In this
paper the author suggests incorporation of a new concept in our understand-
ing of Nature and which helps in explaining the Dark Energy problem and
has the potential of providing us with an improved understanding of Nature.
The new concept is called ”Universal Force”. It was first proposed by
philosopher-scientist Hans Reichenbach [2]. Although this novel concept was
actually suggested in what may be termed as the philosophical context, here
the author would like to emphasize as to how the same can be used as a
powerful tool in the physical context as well. Actually we shall see that
we have to improve upon the original idea of Reichenbach [3] in a significant
manner to be able to use it in physics and astronomy. As to what changes are
necessary would be discussed below after the original idea of the Universal
Forces of Reichenbach is introduced.
To the sceptics who may believe that philosophy of science has no rele-
vance to the actual science itself, it may be pointed out that one never loses
by being open to ideas from any quarters whatsoever. Obviously the only
criteria would be relevance and proper applicability of the idea in whatever
science one is talking about. A recent Conference Proceeding [4] would attest
to this fact. In the case of the Dark Energy conundrum, wherein we do not
even know where we stand, this point becomes all the more pertinent.
Within the discipline of philosophy of science the Universal Force concept
of Reichenbach has had mixed reactions. While a few philosophers have been
supportive of the concept [5,6,7], some others have been critical of it [8,9,10].
We shall not delve too deep into the philosophical issues of the same, as
that would take us farther away from our main purpose here and which is
to see how to properly utilize the concept as scientists. Hence we shall use
only those points as would be found to be relevant for our purpose here. It
would suffice to mention here as to what Rudolf Carnap has stated in the
Introduction of Reichenbach’s book [3]. He called the concept as ” ... of great
interest for the methodology of physics but what has so far not received the
attention it deserves”. In this paper we shall try to rectify for this failure
of appreciating the concept of the Universal Force - albeit in a somewhat
altered and improved form.
Reichenbach defines two kind of forces - Differential Forces and Univer-
sal Forces. It may be pointed out that the term ”force” here should not be
taken strictly as defined in physics but in a broad and general framework.
In fact Carnap has suggested that the term ”effect” instead of ”force’ would
better serve the purpose [5] and which allows it be used in different frame-
works. Hence to conform with the accepted practice, though in this paper
we shall continue to use the term ”Universal Force” the reader may do well
to remember that what we really mean is ”Universal Effect”.
One calls a force Differetial if it acts differently on different substances. It
is called Universal if it is quantitatively the same for all the substances [3,5].
If we heat a rod of initial length l0 from initial temperature T0 to tempetature
T then its length is given as
l = l0[1 + β(T − T0)] (1)
where β the coefficient for thermal expansion is different for different
materials. Hence this is a Differential Force. Now the correction factor due
to the influence of gravitation on the length of the rod is
l = l0[1− C
φ] (2)
Here the rod is placed at a distance r from sun whose mass is m and φ is
the angle of the rod with respect to the the line sun to rod. C is a universal
constant ( in CGS unit C= 3.7 x 10−29 ). As this acts in the same manner
for any material of mass m, gravity is a Universal Force as per the above
definition.
Reichenbach also gives a general definition of the Universal Forces [3,p
12] as: (1) affecting all the materials in the same manner and (2) there are
no insulating walls against it. We saw above that gravity is such a force,
Indeed gravity is a Universal Force par excellance. It affects all matter
in the same manner. The equality of the gravitational and inertial masses is
what ensures this physically. If the gravitational and inertial masses were not
found to be equal, then one would not have been able to visualize of the paths
of freely falling mass points as geodesics in the four dimentional space-time.
In that case different geodesics would have resulted from different materials
of mass points [3].
Therefore the universal effect of gravitation on different kinds of measur-
ing instruments is to define a single geometry for all of them. Viewed this
way, one may say that gravity is geometerized. ”It is not theory of gravitation
that becomes geometry, but it is geometry that becomes the experience of the
gravitational field” [3, p 256]. Why does the planet follow the curved path?
Not because it is acted upon by a force but because the curved space-time
manifold leaves it with no other choice!
So as per Einstein’s theory of relativity, one does not speak of a change
produced by the gravitational field in the measuring instruments, but regard
the measuring instruments as free from any deforming forces. Gravity being a
Universal Force, in the Einstein’s Theory of Relativity, it basically disappears
and is replaced by geometry.
In fact Reichenbach [3, p 22] shows how one can give a consistent defi-
nition of a rigid rod - the same rigid rods which are needed in relativity to
measure all lengths. ”Rigid rods are solid bodies which are not affected by
Differential Forces, or concerning which the influence of Differential Forces
has been eliminated by corrections; Universal Forces are disregarded. We do
not neglect Universal Forces. We set them to zero by definition. Without
such a rule a rigid body cannot be defined.” In fact this rule also helps in
defining a closed system as well.
All this was formalized in terms of a theorem by Reichenbach [3, p 33]
THEOREM θ :
Given the geometry G0 to which the measuring instruments conform,
we can imagine a Universal Force F which affects the instruments in such
a way that the actual geometry is an arbitrary geometry G, while the ob-
served deviation from G is due to universal deformation of the measuring
instruments.”
G0 + F = G (3)
Hence only the combination G0+F is testable. As per Reichenbach’s prin-
ciple one prefers the theory wherein we put F=0. If we accept Reichenbach
principle of putting the Universal Force of gravity to zero, then the arbitrari-
ness in the choice of the measuring procedure is avoided and the question of
the geometrical structure of the physical space has a unique answer deter-
mined by physical measurement. It is this principle which Carnap praises
highly [5, p 171], ” Whenever there is a system of physics in which a certain
universal effect is asserted by a law that specifies under what conditions in
what amount the effect occurs, then the theory should be transformed so that
the amount of effect would be reduced to zero. This is what Einstein did
in regard to contraction and expansion of bodies in gravitational field.” The
left hand side of Einstein’s equation (below) gives the relevant non-Euclideon
geometry
Gµν = 8πG〈φ|Tµν|φ〉 (4)
In the case of gravity, and in as much as Einsteins’s Theory of Relativ-
ity has been well tested experimentally, we treat the above concept as well
placed empirically. But from this single success Reichenbach generalizes this
as a fundamental principle for all cases where Universal forces may arise. As
Carnap states [5, p 171], ” Whenever universal effects are found in physics,
Reichenbach maintained that it is always possible to eliminate them by suit-
able transformation of theory; such a transformation should be made because
of the overall simplicity that would result. This is a useful general principle,
deserving more attention than it has received. It applies not only to relativ-
ity theory, but also to situations that may arise in the future in which other
universal effects may be observed. Without the adoption of this rule there
is no way to give unique answer to the question - what is the structure of
space?”.
As such Reichenbach goes ahead and tries to apply this principle of elimi-
nation of Universal Forces to another universal effect that he finds and which
arises from considerations of topology ( as an additional consideration over
and above that of geometry ) of space-time of the universe.
The Theorem θ is limited to talking about the geometry of space-time
only. It does not take account of specific topological issues that may arise.
To take account of topology of the space-time we shall have to extend the
said theorem appropriately.
What would one experience if space had different topological properties.
To make the point home Reichenbach considers a torus-space [3, p 63]. This
is quite detailed and extensive. However for the purpose of simplifying the
and shortening the discussion here we shall talk of a two dimensional being
who lives on the surface of a sphere. His measurements tell him so. But in
spite of this he insists that he lives on a plane. He may actually do so as per
our discussion above if he confines himself to metrical relations only. With
an appropriate Universal Force he can he can justify living on a plane. But
the surface of a sphere is topologically different from that of a plane. On a
sphere if he starts at a point X and goes on a world tour he may come back
to the same point X. But this is impossible on a plane. And hence to account
for coming back to the ”same point” he has to maintain that on the plane
he actually has come back to a different point Y - which though is identical
to X in all other respects. One option for him is to accept that he is actually
living on a sphere. However if he still wants to maintain his position that he
is living on a plane then he has to explain as to how point Y is physically
identical to point X in spite of the fact that X and Y are different and distinct
points of space. Indeed he can do so by visualizing a fictitious force as an
effect of some kind of ”pre-established harmony” [3, p 65] by proposing that
everything that occurs at X also occurs at the point Y. As it would affect all
matter in the same manner this corresponds to a Universal Force/Effect as
per Reichenbach’s definition.
This interdependence of corresponding points which is essential in this
”pre-established” harmony cannot be interpreted as ordinary causality, as
it does not require ordinary time to transmit it and also does not spread
continuously through intervening space. Hence there is no mysterious causal
connection between the points X and point Y. Thus this necessarily entails
proposing a ”causal anomaly” [3, p 65]. In short connecting different topolo-
gies through a fictitious Universal Effect of ”pre-established harmony” neces-
sarly calls for introduction of ”causal anomalies”. Call this new hypothesize
Universal Force as A and the Theorem θ be extended to read
G0 + F + A = G (5)
where on the right had side we have given a different capital G which
reduces to G of the original Theorem θ when A is set equal to zero.
Now as per Reichenbach’s law of preferring that physical reality wherein
all Universal Forces are put to zero, he advocates of putting A to zero. He
pointed out that this has the advantage of retaining physical ”causality ”
in our science, This he takes as a success of his methodology. As per Re-
ichenbach [3, p 65] ” The principle of causality is one of its (physics) sacred
laws, which it will not abandon lightly; pre-established harmony, however is
incompatible with this law”.
However, as the said ’causal anomaly” is of topological origin we cannot
be sure in what manner it will manifest itself physically. In addition will not
the Universal Force/Effect of ”pre-established harmony” compensate for it in
some manner? So what one is saying is that it is possible that Reichenbach
was wrong in putting all Universal Forces to zero. It was OK to put F to
zero which justified the geometrical interpretation of gravity. But in the case
of this new topological Universal Force we really do not know enough and
let us not be governed by any theoretical prejudice and let the Nature decide
as to what is happening. So to say, let us look at modern cosmology to see
if it is throwing up any new Universal Forces which may be identified with
our ”pre-established harmony” here.
To understand this let us look at the Einstein’s Equation given above.
Harvey and Schucking [11] correcting for Einstein’s error in understanding
the role of the cosmological term λ have derived the most general equation
of motion to be
Gµν + λgµν = 8πG〈φ|Tµν|φ〉 (6)
They showed that [11] the Cosmological Constant λ above provides a
new repulsive force proportional to mass m, repelling every particle of mass
m with a force
F = mc2
x (7)
Recent data [1] on λ is what leads to the crisis of Dark Energy.
Quite clearly this repulsive force is a new Universal Force as per our
definition and hence conforms to the ”pre-established harmony” aspect of
the ”causal anomaly”. Thus we see that indeed as per the recent data on
accelerating universe we have stumbled upon this new Universal Force which
is of topological origin. Hence the source of dark energy is due to ”causal
anomaly” arising from the unique topological structure of our universe. This
solves the mystery of the origin of Dark Energy.
So we would like to emphasize that it is the accelerating universe ( and
hence the Dark Energy ) which is forcing us to accept the incorporation of
this ”causal anomaly” of topological origin. Implications of this new concept
in physics have now to be explored.
Note that as per Theorem θ when one puts F to be zero then one obtains
the proper non-Euclidean Geometry of Einstein’s equation. But now we
know that full structure is the sum of this non-Euclidean geometry plus A
, the new Universal Force ( as per the modified theorem above ) and this is
what the accelerating universe is forcing us to accept. This is what we called
capital G above. We feel that the DASI data on Ω0 being close to one and
thus showing that the Universe is flat [1] is consistent with capital G being
equal to G+ A. In principle just as per the original Theorem θ one may
add a Universal Force F to Einstein’s non-Euclidean geometry to obtain a
physically relevant Euclidean geometry, so in the same manner given a non-
Euclidean geometry of Einstein on can add an appropriate Universal Force
A to provide a flat universe. And this is exactly what capital G is telling us.
Thus the observed flatness of the universe may be treated as a success of the
new idea proposed here.
One would like to ask as to in what other manner incorporation of this
new ”causal anomaly” may help us in understanding Nature better? Will
it provide new perspectives as answers to quantum mechanical puzzles of
quantum jumps, non-locality etc. These are open questions to be tackled in
future.
REFERENCES
1. M S Turner, ”Making sense of the new cosmology”, Int J Mod Phys,
A17S1 (2002) 180-196
2. Hans Reichenbach (1891-1953) can properly be called a philosopher-
scientist. As a leading philosopher of science he was founder of the Berlin
Circle and a proponent of logical positivism. Among his teachers were David
Hilbert, Max Planck, Max Born and Albert Einstein. He wrote extensively
on the theory of probability, theory of relativity and quantum mechanics.
His philosophical writings have a definite scientific touch in them, very much
akin to that of Descartes, Leibniz and Huygens.
3. H Reichenbach, ”The philosophy of space and time”, Dover, New York
(1957) (Original German edition in 1928)
4. C Callender and N Huggett, ” Physics meets philosophy at the Planck
scale”, Cambridge University Press, UK (2001)
5. R Carnap, ”An introduction to the philosophy of science”, Basic Books,
New York (1966)
6. E Nagel, ”The structure of science”, Routledge and Kegan Paul, Lon-
don (1961)
7. D Dieks, ”Gravitation as a Universal Force”, Synthese, 73 (1987)
381-397
8. B Ellis, ”Universal and Differential Forces”, Brit J Phil Sc, 14 (1963)
177-194
9. A Gruenbaum, ”Philosophical problems of space and time”, Dordrecht,
Holland; D Reidel (1973) or Alfred A Knopf, New York (1963)
10. R Torretti, ”Relativity and geometry”, Pergamon Press (1983)
11. A Harvey and E Schucking, ”Einstein’s mistake and the cosmological
constant”, Am J Phys, 68 (2000) 723-727

  The Dark Energy problem is forcing us to re-examine our models and our
understanding of relativity and space-time. Here a novel idea of Fundamental
Forces is introduced. This allows us to perceive the General Theory of
Relativity and Einstein's Equation from a new pesrpective. In addition to
providing us with an improved understanding of space and time, it will be shown
how it leads to a resolution of the Dark Energy problem.

Introduction of Reichenbach’s book [3]. He called the concept as ” ... of great
interest for the methodology of physics but what has so far not received the
attention it deserves”. In this paper we shall try to rectify for this failure
of appreciating the concept of the Universal Force - albeit in a somewhat
altered and improved form.
Reichenbach defines two kind of forces - Differential Forces and Univer-
sal Forces. It may be pointed out that the term ”force” here should not be
taken strictly as defined in physics but in a broad and general framework.
In fact Carnap has suggested that the term ”effect” instead of ”force’ would
better serve the purpose [5] and which allows it be used in different frame-
works. Hence to conform with the accepted practice, though in this paper
we shall continue to use the term ”Universal Force” the reader may do well
to remember that what we really mean is ”Universal Effect”.
One calls a force Differetial if it acts differently on different substances. It
is called Universal if it is quantitatively the same for all the substances [3,5].
If we heat a rod of initial length l0 from initial temperature T0 to tempetature
T then its length is given as
l = l0[1 + β(T − T0)] (1)
where β the coefficient for thermal expansion is different for different
materials. Hence this is a Differential Force. Now the correction factor due
to the influence of gravitation on the length of the rod is
l = l0[1− C
φ] (2)
Here the rod is placed at a distance r from sun whose mass is m and φ is
the angle of the rod with respect to the the line sun to rod. C is a universal
constant ( in CGS unit C= 3.7 x 10−29 ). As this acts in the same manner
for any material of mass m, gravity is a Universal Force as per the above
definition.
Reichenbach also gives a general definition of the Universal Forces [3,p
12] as: (1) affecting all the materials in the same manner and (2) there are
no insulating walls against it. We saw above that gravity is such a force,
Indeed gravity is a Universal Force par excellance. It affects all matter
in the same manner. The equality of the gravitational and inertial masses is
what ensures this physically. If the gravitational and inertial masses were not
found to be equal, then one would not have been able to visualize of the paths
of freely falling mass points as geodesics in the four dimentional space-time.
In that case different geodesics would have resulted from different materials
of mass points [3].
Therefore the universal effect of gravitation on different kinds of measur-
ing instruments is to define a single geometry for all of them. Viewed this
way, one may say that gravity is geometerized. ”It is not theory of gravitation
that becomes geometry, but it is geometry that becomes the experience of the
gravitational field” [3, p 256]. Why does the planet follow the curved path?
Not because it is acted upon by a force but because the curved space-time
manifold leaves it with no other choice!
So as per Einstein’s theory of relativity, one does not speak of a change
produced by the gravitational field in the measuring instruments, but regard
the measuring instruments as free from any deforming forces. Gravity being a
Universal Force, in the Einstein’s Theory of Relativity, it basically disappears
and is replaced by geometry.
In fact Reichenbach [3, p 22] shows how one can give a consistent defi-
nition of a rigid rod - the same rigid rods which are needed in relativity to
measure all lengths. ”Rigid rods are solid bodies which are not affected by
Differential Forces, or concerning which the influence of Differential Forces
has been eliminated by corrections; Universal Forces are disregarded. We do
not neglect Universal Forces. We set them to zero by definition. Without
such a rule a rigid body cannot be defined.” In fact this rule also helps in
defining a closed system as well.
All this was formalized in terms of a theorem by Reichenbach [3, p 33]
THEOREM θ :
Given the geometry G0 to which the measuring instruments conform,
we can imagine a Universal Force F which affects the instruments in such
a way that the actual geometry is an arbitrary geometry G, while the ob-
served deviation from G is due to universal deformation of the measuring
instruments.”
G0 + F = G (3)
Hence only the combination G0+F is testable. As per Reichenbach’s prin-
ciple one prefers the theory wherein we put F=0. If we accept Reichenbach
principle of putting the Universal Force of gravity to zero, then the arbitrari-
ness in the choice of the measuring procedure is avoided and the question of
the geometrical structure of the physical space has a unique answer deter-
mined by physical measurement. It is this principle which Carnap praises
highly [5, p 171], ” Whenever there is a system of physics in which a certain
universal effect is asserted by a law that specifies under what conditions in
what amount the effect occurs, then the theory should be transformed so that
the amount of effect would be reduced to zero. This is what Einstein did
in regard to contraction and expansion of bodies in gravitational field.” The
left hand side of Einstein’s equation (below) gives the relevant non-Euclideon
geometry
Gµν = 8πG〈φ|Tµν|φ〉 (4)
In the case of gravity, and in as much as Einsteins’s Theory of Relativ-
ity has been well tested experimentally, we treat the above concept as well
placed empirically. But from this single success Reichenbach generalizes this
as a fundamental principle for all cases where Universal forces may arise. As
Carnap states [5, p 171], ” Whenever universal effects are found in physics,
Reichenbach maintained that it is always possible to eliminate them by suit-
able transformation of theory; such a transformation should be made because
of the overall simplicity that would result. This is a useful general principle,
deserving more attention than it has received. It applies not only to relativ-
ity theory, but also to situations that may arise in the future in which other
universal effects may be observed. Without the adoption of this rule there
is no way to give unique answer to the question - what is the structure of
space?”.
As such Reichenbach goes ahead and tries to apply this principle of elimi-
nation of Universal Forces to another universal effect that he finds and which
arises from considerations of topology ( as an additional consideration over
and above that of geometry ) of space-time of the universe.
The Theorem θ is limited to talking about the geometry of space-time
only. It does not take account of specific topological issues that may arise.
To take account of topology of the space-time we shall have to extend the
said theorem appropriately.
What would one experience if space had different topological properties.
To make the point home Reichenbach considers a torus-space [3, p 63]. This
is quite detailed and extensive. However for the purpose of simplifying the
and shortening the discussion here we shall talk of a two dimensional being
who lives on the surface of a sphere. His measurements tell him so. But in
spite of this he insists that he lives on a plane. He may actually do so as per
our discussion above if he confines himself to metrical relations only. With
an appropriate Universal Force he can he can justify living on a plane. But
the surface of a sphere is topologically different from that of a plane. On a
sphere if he starts at a point X and goes on a world tour he may come back
to the same point X. But this is impossible on a plane. And hence to account
for coming back to the ”same point” he has to maintain that on the plane
he actually has come back to a different point Y - which though is identical
to X in all other respects. One option for him is to accept that he is actually
living on a sphere. However if he still wants to maintain his position that he
is living on a plane then he has to explain as to how point Y is physically
identical to point X in spite of the fact that X and Y are different and distinct
points of space. Indeed he can do so by visualizing a fictitious force as an
effect of some kind of ”pre-established harmony” [3, p 65] by proposing that
everything that occurs at X also occurs at the point Y. As it would affect all
matter in the same manner this corresponds to a Universal Force/Effect as
per Reichenbach’s definition.
This interdependence of corresponding points which is essential in this
”pre-established” harmony cannot be interpreted as ordinary causality, as
it does not require ordinary time to transmit it and also does not spread
continuously through intervening space. Hence there is no mysterious causal
connection between the points X and point Y. Thus this necessarily entails
proposing a ”causal anomaly” [3, p 65]. In short connecting different topolo-
gies through a fictitious Universal Effect of ”pre-established harmony” neces-
sarly calls for introduction of ”causal anomalies”. Call this new hypothesize
Universal Force as A and the Theorem θ be extended to read
G0 + F + A = G (5)
where on the right had side we have given a different capital G which
reduces to G of the original Theorem θ when A is set equal to zero.
Now as per Reichenbach’s law of preferring that physical reality wherein
all Universal Forces are put to zero, he advocates of putting A to zero. He
pointed out that this has the advantage of retaining physical ”causality ”
in our science, This he takes as a success of his methodology. As per Re-
ichenbach [3, p 65] ” The principle of causality is one of its (physics) sacred
laws, which it will not abandon lightly; pre-established harmony, however is
incompatible with this law”.
However, as the said ’causal anomaly” is of topological origin we cannot
be sure in what manner it will manifest itself physically. In addition will not
the Universal Force/Effect of ”pre-established harmony” compensate for it in
some manner? So what one is saying is that it is possible that Reichenbach
was wrong in putting all Universal Forces to zero. It was OK to put F to
zero which justified the geometrical interpretation of gravity. But in the case
of this new topological Universal Force we really do not know enough and
let us not be governed by any theoretical prejudice and let the Nature decide
as to what is happening. So to say, let us look at modern cosmology to see
if it is throwing up any new Universal Forces which may be identified with
our ”pre-established harmony” here.
To understand this let us look at the Einstein’s Equation given above.
Harvey and Schucking [11] correcting for Einstein’s error in understanding
the role of the cosmological term λ have derived the most general equation
of motion to be
Gµν + λgµν = 8πG〈φ|Tµν|φ〉 (6)
They showed that [11] the Cosmological Constant λ above provides a
new repulsive force proportional to mass m, repelling every particle of mass
m with a force
F = mc2
x (7)
Recent data [1] on λ is what leads to the crisis of Dark Energy.
Quite clearly this repulsive force is a new Universal Force as per our
definition and hence conforms to the ”pre-established harmony” aspect of
the ”causal anomaly”. Thus we see that indeed as per the recent data on
accelerating universe we have stumbled upon this new Universal Force which
is of topological origin. Hence the source of dark energy is due to ”causal
anomaly” arising from the unique topological structure of our universe. This
solves the mystery of the origin of Dark Energy.
So we would like to emphasize that it is the accelerating universe ( and
hence the Dark Energy ) which is forcing us to accept the incorporation of
this ”causal anomaly” of topological origin. Implications of this new concept
in physics have now to be explored.
Note that as per Theorem θ when one puts F to be zero then one obtains
the proper non-Euclidean Geometry of Einstein’s equation. But now we
know that full structure is the sum of this non-Euclidean geometry plus A
, the new Universal Force ( as per the modified theorem above ) and this is
what the accelerating universe is forcing us to accept. This is what we called
capital G above. We feel that the DASI data on Ω0 being close to one and
thus showing that the Universe is flat [1] is consistent with capital G being
equal to G+ A. In principle just as per the original Theorem θ one may
add a Universal Force F to Einstein’s non-Euclidean geometry to obtain a
physically relevant Euclidean geometry, so in the same manner given a non-
Euclidean geometry of Einstein on can add an appropriate Universal Force
A to provide a flat universe. And this is exactly what capital G is telling us.
Thus the observed flatness of the universe may be treated as a success of the
new idea proposed here.
One would like to ask as to in what other manner incorporation of this
new ”causal anomaly” may help us in understanding Nature better? Will
it provide new perspectives as answers to quantum mechanical puzzles of
quantum jumps, non-locality etc. These are open questions to be tackled in
future.
REFERENCES
1. M S Turner, ”Making sense of the new cosmology”, Int J Mod Phys,
A17S1 (2002) 180-196
2. Hans Reichenbach (1891-1953) can properly be called a philosopher-
scientist. As a leading philosopher of science he was founder of the Berlin
Circle and a proponent of logical positivism. Among his teachers were David
Hilbert, Max Planck, Max Born and Albert Einstein. He wrote extensively
on the theory of probability, theory of relativity and quantum mechanics.
His philosophical writings have a definite scientific touch in them, very much
akin to that of Descartes, Leibniz and Huygens.
3. H Reichenbach, ”The philosophy of space and time”, Dover, New York
(1957) (Original German edition in 1928)
4. C Callender and N Huggett, ” Physics meets philosophy at the Planck
scale”, Cambridge University Press, UK (2001)
5. R Carnap, ”An introduction to the philosophy of science”, Basic Books,
New York (1966)
6. E Nagel, ”The structure of science”, Routledge and Kegan Paul, Lon-
don (1961)
7. D Dieks, ”Gravitation as a Universal Force”, Synthese, 73 (1987)
381-397
8. B Ellis, ”Universal and Differential Forces”, Brit J Phil Sc, 14 (1963)
177-194
9. A Gruenbaum, ”Philosophical problems of space and time”, Dordrecht,
Holland; D Reidel (1973) or Alfred A Knopf, New York (1963)
10. R Torretti, ”Relativity and geometry”, Pergamon Press (1983)
11. A Harvey and E Schucking, ”Einstein’s mistake and the cosmological
constant”, Am J Phys, 68 (2000) 723-727
