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ABSTRACT
In the present investigation, the possible effects of the expansion of the Universe on systems bonded either by gravitational or electromagnetic forces, are reconsidered. It will be shown that the acceleration (positive or negative) of the expanding background, is the determinant factor affecting planetary orbits and atomic sizes. In the presently accepted cosmology (ΛCDM) all bonded systems are expanding at a decreasing rate that tends to be zero as the universe enters in a de Sitter phase. It is worth mentioning that the estimated expansion rates are rather small and they can be neglected for all practical purposes.
Key words: cosmic expansion; cosmology; cosmological models; gravitation
1-INTRODUCTION
The nature of the cosmological expansion is a recurrent debate in cosmology. The most direct evidence for the expansion of the universe appeared with the first spectroscopic observations of extragalactic "nebulae" by Slipher (1915), which revealed that most of these objects have redshifted spectral features. Subsequent observations by Hubble (1929) put in evidence a relation between the distances of these "nebulae" and the observed redshift in their spectra. An interpretation of these observations was provided by Lemaître (1931), who showed that an expanding relativistic universe model could account for the observed radial velocity of galaxies and, in particular, explaining the fact that the redshift increases with the distance. Nevertheless, since the discovery of the redshift phenomenon, alternative explanations to the cosmic expansion interpretation have been proposed. Zwicky (1929) hypothesized that photons lose energy by interacting with the intergalactic medium, the so-called "tired light" theory and its major difficulty is the mechanism behind the loss of energy. Arp (1988) studied different groups of galaxies and remarked on some of them having the presence of objects with discrepant radial velocities. These observations led him and Burbidge to argue that some redshifts might not be cosmological in origin (Burbidge 2001, Burbidge et al. 2006). With the discovery of type Ia supernovae (cosmological "candles") at moderate redshifts, it became clear that most of the alternative theories of the cosmic expansion are unable to explain the time stretching observed in the light curves of these events (Goldhaber et al. 2001).
In the present investigation, the observed redshift of features present in the spectra of distant galaxies is considered as an evidence in favor of the cosmological expansion although, as we shall see later, it cannot be explained simply in terms of a Doppler effect.
A "classical" representation of the expansion of the Universe, frequently used in most scholar texts, was given by Misner et al. (1973) in their classical book "Gravitation". The universe is modeled by the surface of a rubber balloon covered with coins, which represent galaxies. As the balloon inflates, each coin may be considered to be the center of the expansion. The surface of the balloon represents the uniform stretching of space or the background. However, if space itself is stretching, does this means that everything in it is stretching? The aforementioned authors consider that "the coins themselves do not expand", meaning that the expansion would occur only at the scale of galaxies and clusters of galaxies but not at smaller scales. In other words, bodies hold together either by electromagnetic or gravitational forces, would not expand! However, if space is stretching how could these "bonded" systems not be affected? An early tentative of answering these questions was proposed by Dicke and Peebles (1964), who stated that if everything expands in the same proportion, then the expansion could not be detected since the "rules" also expand in the same proportion.
The concept of space in the general relativity theory (GRT) differs from that used in Newtonian physics or in the special theory of relativity. In the former, space is defined as a relation between particles while in the latter, space is not influenced by matter and exists independently of it. In Newtonian physics and special relativity, space has a mathematical character whereas in GRT it is more physical, since matter affects its geometry and dynamics, exemplified by the propagation of gravitational waves, the drag of a reference frame near a rotating body and the cosmic expansion itself. In GRT the geometry of the universe is described by the Friedmann-Robertson-Walker (FRW) metric which, in principle, does not specify any particular scale at which the expansion begins to occur.
Possible cosmic expansion effects on the solar system scale were studied more than 80 years ago by McVittie (1933), who did not reached any definite conclusion. He studied expansion effects on planetary orbits and, according to his calculations, the average orbital radius should remain constant. However, he used a particular coordinate system in which the physical interpretation of the results is not trivial. Jarnefelt (1940, 1942) reanalyzed this problem and confirmed that the planetary orbits do not participate in the expansion of the universe. A similar conclusion was also reached by Einstein and Straus (1945), who adopted a completely different approach. They showed that the Schwarzschild solution can be embedded in a dust-expanding universe and that test particles, representing planets, are unaffected by the expansion. This investigation has been criticized since the match between both metrics requires specific boundary conditions that are not fulfilled in the case of the solar system. After these negative results, Gautreau (1984) reinvestigated the problem of imbedding a Schwarzschild mass into a zero-curvature universe, and he concluded that orbits do expand. According to him, the cosmological fluid crosses a surface of constant R, changing the amount of matter inside such a surface and affecting the planetary orbits, since test bodies experience a gravitational force that decreases with time.Anderson (1995) studied the motion of a pair of gravitating particles in an Einstein-de Sitter universe and he concluded that the expansion does affect the motion, excepting the case of pure circular orbits, which do not expand. This conclusion was confirmed by Cooperstock et al. (1998), who showed that the expansion affects all scales but, as one should expect, the magnitude of the effect is quite small. More recently, Sereno and Jetzer (2007) called the attention to a very important point: in reality planetary orbits are expanding but the effects on the orbital properties depend mainly on the acceleration (deceleration) of the expansion rather than on the expansion itself.
In larger scales, i.e., in scales of clusters or superclusters of galaxies, expansion effects in these bonded structures were investigated by Noerdlinger and Petrosian (1971). They concluded that these structures expand at a rate that depends on the ratio of their average density to the background density.
In the atomic scale, the situation is also controversial. Bonnor (1996) obtained an exact solution of the Einstein-Maxwell equations for a particular case involving charged dust in equilibrium thanks to the balance between electrical and gravitational forces. According to Bonnor (1996) a body in this situation participates in the expansion. Later, Bonnor (1999) showed that in a de Sitter universe, the atomic radius remains strictly constant during the expansion, while in an Einstein-de Sitter universe the atom expands, but the amplitude of the effect is rather small. The effect of the expansion on atomic scales was reconsidered by Price (2005) who concluded that "either the atom expands or...completely ignores the expansion", a rather astonishing statement!
In the present paper, the question of what is expanding in a expanding universe is revisited and, in particular, we address the issue of how this expansion should be interpreted. The main conclusion is that all bodies suffer the consequences of the expansion of the universe but with different strengths. At atomic and solar system scales, effects are presently undetectable and, in practice, can be neglected. It is also shown that, in agreement with Sereno and Jetzer (2007), the acceleration (positive or negative) of the expansion is the dominant factor affecting the behavior of systems bonded gravitationally or electromagnetically. This paper is organized as follows: in Section 2 the interpretation of the receding motion of galaxies as a consequence of the expansion of space is revisited; in Section 3, expansion effects on planetary orbits are analyzed for different cosmological models; in Section 4, the consequences of the expansion for the size of atoms are discussed and, lastly, in Section 5 the main conclusions are given.
...
5-CONCLUSIONS
In the "thought experiment" of a tethered galaxy, the trajectory followed by the test galaxy is considered by some authors as an argument against the interpretation of the Hubble flow as an expansion of space. We have shown that the solution of the radial geodesic equation depends on the initial conditions imposed to the peculiar velocity of the test galaxy. If the test body is dropped into the background with a zero peculiar velocity, then it immediately joins the Hubble flow. On the one hand, if its peculiar velocity is non-zero, the trajectory depends on the initial direction of the motion. In the case of the tethered galaxy, the initial peculiar velocity is negative (opposite to the Hubble flow) and the galaxy, once free from the tether, moves in the direction of the observer, crosses the origin and joins asymptotically the Hubble flow in the opposite side of the sky. On the other hand, if the initial direction is positive, the test galaxy moves away and joins asymptotically the Hubble flow far from its initial position. The expansion of the background produces the well-known decay of the peculiar velocity that is proportional to the inverse of the scale factor. Thus, there is no contradiction with the interpretation of the Hubble flow as an expansion of space.
Planetary orbits and size of atoms are affected by the expansion in the same way, since expansion or contraction rates depend on the acceleration rate of the expansion itself. In our approach, expansion effects on planetary orbits were computed by using a semi-Newtonian formalism, leading to a modification of the Poisson equation, which includes the contribution of the cosmic fluid pressure in the source term. Atoms are quantum systems and the cosmic expansion acts like a perturbation that modifies the energy levels, in particular the ground state and the mean distance of electrons with respect to the atomic nucleus.
In agreement with previous investigations, in a de Sitter cosmology planetary orbits and the size of atoms remain unchanged. However, for the Einstein-de Sitter cosmology, either with a "dust" or a "radiation" equation of state, planetary orbits and atomic sizes increase with time but at a decreasing rate. On the contrary, in quintessence models in which the conditions ε + 3P <0 and ε + P > 0 are satisfied, planetary orbits and atomic sizes shrink. This is not the case if the conditionε + P > 0 is violated (phantom cosmology) since an opposite behavior is expected. In phantom models, planetary orbits and atomic sizes increase and all bonded systems are disrupted near the future singularity present in these cosmologies. For the present accepted model, the ΛCDM cosmology, orbits and atomic sizes are expanding at a decreasing rate, which tends to zero as the expansion enters in a de Sitter phase.
In conclusion, in an expanding universe everything expands (or contracts) but at different rates, according to the interactions that hold the different bodies together. In general the expansion (contraction) rates are quite small and can be neglected for all practical purposes.
http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652015000501915
In the present investigation, the possible effects of the expansion of the Universe on systems bonded either by gravitational or electromagnetic forces, are reconsidered. It will be shown that the acceleration (positive or negative) of the expanding background, is the determinant factor affecting planetary orbits and atomic sizes. In the presently accepted cosmology (ΛCDM) all bonded systems are expanding at a decreasing rate that tends to be zero as the universe enters in a de Sitter phase. It is worth mentioning that the estimated expansion rates are rather small and they can be neglected for all practical purposes.
Key words: cosmic expansion; cosmology; cosmological models; gravitation
1-INTRODUCTION
The nature of the cosmological expansion is a recurrent debate in cosmology. The most direct evidence for the expansion of the universe appeared with the first spectroscopic observations of extragalactic "nebulae" by Slipher (1915), which revealed that most of these objects have redshifted spectral features. Subsequent observations by Hubble (1929) put in evidence a relation between the distances of these "nebulae" and the observed redshift in their spectra. An interpretation of these observations was provided by Lemaître (1931), who showed that an expanding relativistic universe model could account for the observed radial velocity of galaxies and, in particular, explaining the fact that the redshift increases with the distance. Nevertheless, since the discovery of the redshift phenomenon, alternative explanations to the cosmic expansion interpretation have been proposed. Zwicky (1929) hypothesized that photons lose energy by interacting with the intergalactic medium, the so-called "tired light" theory and its major difficulty is the mechanism behind the loss of energy. Arp (1988) studied different groups of galaxies and remarked on some of them having the presence of objects with discrepant radial velocities. These observations led him and Burbidge to argue that some redshifts might not be cosmological in origin (Burbidge 2001, Burbidge et al. 2006). With the discovery of type Ia supernovae (cosmological "candles") at moderate redshifts, it became clear that most of the alternative theories of the cosmic expansion are unable to explain the time stretching observed in the light curves of these events (Goldhaber et al. 2001).
In the present investigation, the observed redshift of features present in the spectra of distant galaxies is considered as an evidence in favor of the cosmological expansion although, as we shall see later, it cannot be explained simply in terms of a Doppler effect.
A "classical" representation of the expansion of the Universe, frequently used in most scholar texts, was given by Misner et al. (1973) in their classical book "Gravitation". The universe is modeled by the surface of a rubber balloon covered with coins, which represent galaxies. As the balloon inflates, each coin may be considered to be the center of the expansion. The surface of the balloon represents the uniform stretching of space or the background. However, if space itself is stretching, does this means that everything in it is stretching? The aforementioned authors consider that "the coins themselves do not expand", meaning that the expansion would occur only at the scale of galaxies and clusters of galaxies but not at smaller scales. In other words, bodies hold together either by electromagnetic or gravitational forces, would not expand! However, if space is stretching how could these "bonded" systems not be affected? An early tentative of answering these questions was proposed by Dicke and Peebles (1964), who stated that if everything expands in the same proportion, then the expansion could not be detected since the "rules" also expand in the same proportion.
The concept of space in the general relativity theory (GRT) differs from that used in Newtonian physics or in the special theory of relativity. In the former, space is defined as a relation between particles while in the latter, space is not influenced by matter and exists independently of it. In Newtonian physics and special relativity, space has a mathematical character whereas in GRT it is more physical, since matter affects its geometry and dynamics, exemplified by the propagation of gravitational waves, the drag of a reference frame near a rotating body and the cosmic expansion itself. In GRT the geometry of the universe is described by the Friedmann-Robertson-Walker (FRW) metric which, in principle, does not specify any particular scale at which the expansion begins to occur.
Possible cosmic expansion effects on the solar system scale were studied more than 80 years ago by McVittie (1933), who did not reached any definite conclusion. He studied expansion effects on planetary orbits and, according to his calculations, the average orbital radius should remain constant. However, he used a particular coordinate system in which the physical interpretation of the results is not trivial. Jarnefelt (1940, 1942) reanalyzed this problem and confirmed that the planetary orbits do not participate in the expansion of the universe. A similar conclusion was also reached by Einstein and Straus (1945), who adopted a completely different approach. They showed that the Schwarzschild solution can be embedded in a dust-expanding universe and that test particles, representing planets, are unaffected by the expansion. This investigation has been criticized since the match between both metrics requires specific boundary conditions that are not fulfilled in the case of the solar system. After these negative results, Gautreau (1984) reinvestigated the problem of imbedding a Schwarzschild mass into a zero-curvature universe, and he concluded that orbits do expand. According to him, the cosmological fluid crosses a surface of constant R, changing the amount of matter inside such a surface and affecting the planetary orbits, since test bodies experience a gravitational force that decreases with time.Anderson (1995) studied the motion of a pair of gravitating particles in an Einstein-de Sitter universe and he concluded that the expansion does affect the motion, excepting the case of pure circular orbits, which do not expand. This conclusion was confirmed by Cooperstock et al. (1998), who showed that the expansion affects all scales but, as one should expect, the magnitude of the effect is quite small. More recently, Sereno and Jetzer (2007) called the attention to a very important point: in reality planetary orbits are expanding but the effects on the orbital properties depend mainly on the acceleration (deceleration) of the expansion rather than on the expansion itself.
In larger scales, i.e., in scales of clusters or superclusters of galaxies, expansion effects in these bonded structures were investigated by Noerdlinger and Petrosian (1971). They concluded that these structures expand at a rate that depends on the ratio of their average density to the background density.
In the atomic scale, the situation is also controversial. Bonnor (1996) obtained an exact solution of the Einstein-Maxwell equations for a particular case involving charged dust in equilibrium thanks to the balance between electrical and gravitational forces. According to Bonnor (1996) a body in this situation participates in the expansion. Later, Bonnor (1999) showed that in a de Sitter universe, the atomic radius remains strictly constant during the expansion, while in an Einstein-de Sitter universe the atom expands, but the amplitude of the effect is rather small. The effect of the expansion on atomic scales was reconsidered by Price (2005) who concluded that "either the atom expands or...completely ignores the expansion", a rather astonishing statement!
In the present paper, the question of what is expanding in a expanding universe is revisited and, in particular, we address the issue of how this expansion should be interpreted. The main conclusion is that all bodies suffer the consequences of the expansion of the universe but with different strengths. At atomic and solar system scales, effects are presently undetectable and, in practice, can be neglected. It is also shown that, in agreement with Sereno and Jetzer (2007), the acceleration (positive or negative) of the expansion is the dominant factor affecting the behavior of systems bonded gravitationally or electromagnetically. This paper is organized as follows: in Section 2 the interpretation of the receding motion of galaxies as a consequence of the expansion of space is revisited; in Section 3, expansion effects on planetary orbits are analyzed for different cosmological models; in Section 4, the consequences of the expansion for the size of atoms are discussed and, lastly, in Section 5 the main conclusions are given.
...
5-CONCLUSIONS
In the "thought experiment" of a tethered galaxy, the trajectory followed by the test galaxy is considered by some authors as an argument against the interpretation of the Hubble flow as an expansion of space. We have shown that the solution of the radial geodesic equation depends on the initial conditions imposed to the peculiar velocity of the test galaxy. If the test body is dropped into the background with a zero peculiar velocity, then it immediately joins the Hubble flow. On the one hand, if its peculiar velocity is non-zero, the trajectory depends on the initial direction of the motion. In the case of the tethered galaxy, the initial peculiar velocity is negative (opposite to the Hubble flow) and the galaxy, once free from the tether, moves in the direction of the observer, crosses the origin and joins asymptotically the Hubble flow in the opposite side of the sky. On the other hand, if the initial direction is positive, the test galaxy moves away and joins asymptotically the Hubble flow far from its initial position. The expansion of the background produces the well-known decay of the peculiar velocity that is proportional to the inverse of the scale factor. Thus, there is no contradiction with the interpretation of the Hubble flow as an expansion of space.
Planetary orbits and size of atoms are affected by the expansion in the same way, since expansion or contraction rates depend on the acceleration rate of the expansion itself. In our approach, expansion effects on planetary orbits were computed by using a semi-Newtonian formalism, leading to a modification of the Poisson equation, which includes the contribution of the cosmic fluid pressure in the source term. Atoms are quantum systems and the cosmic expansion acts like a perturbation that modifies the energy levels, in particular the ground state and the mean distance of electrons with respect to the atomic nucleus.
In agreement with previous investigations, in a de Sitter cosmology planetary orbits and the size of atoms remain unchanged. However, for the Einstein-de Sitter cosmology, either with a "dust" or a "radiation" equation of state, planetary orbits and atomic sizes increase with time but at a decreasing rate. On the contrary, in quintessence models in which the conditions ε + 3P <0 and ε + P > 0 are satisfied, planetary orbits and atomic sizes shrink. This is not the case if the conditionε + P > 0 is violated (phantom cosmology) since an opposite behavior is expected. In phantom models, planetary orbits and atomic sizes increase and all bonded systems are disrupted near the future singularity present in these cosmologies. For the present accepted model, the ΛCDM cosmology, orbits and atomic sizes are expanding at a decreasing rate, which tends to zero as the expansion enters in a de Sitter phase.
In conclusion, in an expanding universe everything expands (or contracts) but at different rates, according to the interactions that hold the different bodies together. In general the expansion (contraction) rates are quite small and can be neglected for all practical purposes.
http://www.scielo.br/scielo.php?script=sci_arttext&pid=S0001-37652015000501915
