Theory Of Gravity Essay

· = 4G.

This formulation is entirely equivalent to Newton's law of gravitation. Because a test particle's acceleration depends only on the potential generated by matter in the surroundings, the theory respects the weak equivalence principle: the motion of a particle is independent of its internal structure or composition. As the subject of Galileo Galilei's apocryphal experiment at the tower of Pisa, this principle is supported by a series of high precision experiments culminating in those directed by Baron Lorand von Eötvos in Budapest in 1922, Robert Dicke at Princeton in 1964, and Vladimir Braginsky in Moscow in 1972.

Highly successful in everyday applications, newtonian gravitation has also proved accurate in describing motions in the solar system (except for tiny relativistic effects), the internal structure of planets, the sun and other stars, orbits in binary and multiple stellar systems, the structure of molecular clouds, and, in a rough way, the structure of galaxies and clusters of galaxies (but see below).

THE GENERAL THEORY OF RELATIVITY

According to newtonian theory, gravitational effects propagate from place to place instantaneously. With the advent of Einstein's special theory of relativity in 1905, a theory uniting the concepts of space and time into that of four dimensional flat space-time (named Minkowski space-time after the mathematician Hermann Minkowski), a problem became discernible with newtonian theory. According to special relativity, which is the current guideline to the form of all physical theory, the speed of light, c = 3 x 1010 cm s-1, is the top speed allowed to physical particles or forces: There can be no instantaneous propagation. After a decade of search for new concepts to make gravitational theory compatible with the spirit of special relativity, Einstein came up with the theory of general relativity (1915), the prototype of all modern gravitational theories. Its crucial ingredient, involving a colossal intellectual jump, is the concept of gravitation, not as a force, but as a manifestation of the curvature of space-time, an idea first mentioned in rudimentary form by the mathematician Ceorg Bernhard Riemann in 1854. In Einstein's hands gravitation theory was thus transformed from a theory of forces into the first dynamical theory of geometry, the geometry of four dimensional curved space-time.

Why talk of curvature? One of Einstein's first predictions was the gravitational redshift: As any wave, such as light, propagates away from a gravitating mass, all frequencies in it are reduced by an amount proportional to the change in gravitational potential experienced by the wave. This redshift has been measured in the laboratory, in solar observations, and by means of high precision clocks flown in airplanes. However, imagine for a moment that general relativity had not yet been invented, but the redshift has already been measured. According to a simple argument owing to Alfred Schild, wave propagation under stationary circumstances can display a redshift only if the usual geometric relations implicit in Minkowski space-time are violated: The space-time must be curved. The observations of the redshift thus show that space-time must be curved in the vicinity of masses, regardless of the precise form of the gravitational theory.

Einstein provided 10 equations relating the metric (a tensor with 10 independent components describing the geometry of space-time) to the material energy momentum tensor (also composed of 10 components, one of which corresponds to our previous ). These Einstein field equations, in which both of the previously mentioned constants G and c figure as parameters, replace Poisson's equation. Einstein also replaced the newtonian law of motion by the statement that free test particles move along geodesics, the shortest curves in the space-time geometry. The influential gravitation theorist John Archibald Wheeler has encapsulated general relativity in the aphorism ``curvature tells matter how to move, and matter tells space-time how to curve.'' The Eötvos-Dicke-Braginsky experiments demonstrate with high precision that free test particles all travel along the same trajectories in space-time, whereas the gravitational redshift shows (with more modest precision) these universal trajectories to be identical with geodesics.

Despite the great contrast between General Relativity and Newtonian theory, predictions of the former approach the latter for systems in which velocities are small compared to c and gravitational potentials are weak enough that they cannot cause larger velocities. This is why we can discuss with newtonian theory the structure of the earth and planets, stars and stellar clusters, and the gross features of motions in the solar system without fear of error.

Einstein noted two other predictions of General Relativity. First, light beams passing near a gravitating body must suffer a slight deflection proportional to that body's mass. First verified by observations of stellar images during the 1919 total solar eclipse, this effect also causes deflection of quasar radio images by the sun, is the likely cause of the phenomenon of ``double quasars'' with identical redshift and of the recently discovered giant arcs in clusters of galaxies (both probably effects of gravitational lensing), and is part and parcel of the black hole phenomenon. In a closely related effect first noted by Irwin Shapiro, radiation passing near a gravitating body is delayed in its flight in proportion to the body's mass, a time delay verified by means of radar waves deflected by the sun on their way from Earth to Mercury and back.

The second effect is the precession of the periastron of a binary system. According to newtonian gravitation, the orbit of each member of a binary is a coplanar ellipse with orientation fixed in space. General relativity predicts a slow rotation of the ellipse's major axis in the plane of the orbit (precession of the periastron). Originally verified in the motion of Mercury, the precession has of late also been detected in the orbits of binary pulsars.

All three effects mentioned depend on features of General Relativity beyond the weak equivalence principle. Indeed, Einstein built into general relativity the much more encompassing ``strong equivalence principle'': the local forms of all nongravitational physical laws and the numerical values of all dimensionless physical constants arc the same in the presence of a gravitational field as in its absence. In practice this implies that within any region in a gravitational field, sufficiently small that space-time curvature may be ignored, all physical laws, when expressed in terms of the space-time metric, have the same forms as required by special relativity in terms of the metric of Minkowski space-time. Thus in a small region in the neighborhood of a black hole (the source of a strong gravitational field) we would describe electromagnetism and optics with the same Maxwell equations used in earthly laboratories where the gravitational field is weak, and we would employ the laboratory values of the electrical permittivity and magnetic susceptibility of the vacuum.

SCALAR TENSOR THEORIES

The strong equivalence principle effectively forces gravitational theory to be General Relativity. Less well tested than the weak version of the principle mentioned earlier, the strong version requires Newton's constant expressed in atomic units to be the same number everywhere, in strong or weak gravitational fields. Stressing that there is very little experimental evidence bearing on this assertion, Dicke and his student Carl Brans proposed in 1961 a modification of general relativity akin to a theory considered earlier by Pascual Jordan. In the Brans-Dicke theory the reciprocal of the gravitational constant is itself a one-component field, the scalar field , that is generated by matter in accordance with an additional equation. Then as well as matter has a say in determining the metric via a modified version of Einstein's equations. Because it involves both metric and scalar fields, the Brans-Dicke theory is dubbed scalar-tensor. Although not complying with the strong equivalence principle, the theory does respect a milder version of it, the Einstein equivalence principle, which asserts that only nongravitational laws and dimensionless constants have their special relativistic forms and values everywhere. Gravitation theorists call theories obeying the Einstein equivalence principle metric theories.

The Brans-Dicke theory also reduces to Newtonian theory for systems with small velocities and weak potentials: It has a newtonian limit. In fact, Brans-Dicke theory is distinguishable from general relativity only by the value of its single dimensionless parameter which determines the effectiveness of matter in producing . The larger , the closer the Brans-Dicke theory predictions are to general relativity. Both theories predict the same gravitational redshift effect, although they predict slightly different light deflection and periastron precession effects; the differences vanish in the limit of infinite . Measurements of Mercury's perihelion precession, radar flight time delay, and radio wave deflection by the sun indicate that is at least several hundred.

Initially a popular alternative to General Relativity, the Brans-Dicke theory lost favor as it became clear that must be very large-an artificial requirement in some views. Nevertheless, the theory has remained a paradigm for the introduction of scalar fields into gravitational theory, and as such has enjoyed a renaissance in connection with theories of higher dimensional space-time.

However, constancy of is not conceptually required. In the generic scalar-tensor theory studied by Peter Bergmann, Robert Wagoner, and Kenneth Nordtvedt, is itself a general function of (). It remains true that in regions of space-time where () is numerically large, the theory's predictions approach those of general relativity. It is even possible for () to evolve systematically in the favored direction. Thus in the variable mass theory (VMT, see Table 1), a scalar-tensor theory devised to test the necessity for the strong equivalence principle, the expansion of the universe forces evolution of toward a particular value at which () diverges. Thus, late in the history of the universe (and today is late), localized gravitational systems are accurately described by general relativity although the assumed gravitational theory is scalar-tensor.


TheoryMetricOther FieldsFree ElementsStatus
Newtonian (1687)1Nonmetric Potential None Nonrelativistic theory Nordstrom (1913)1,2Minkowski Scalar None Fails to predict observed light detection Einstein's General Relativity (1915)1, 2Dynamic None None Viable Belifante-Swihart (1957)2Nonmetric Tensor K parameter Contradicted by Dicke-Braginsky experiments Brans-Dicke (1961)1-3 Generic Scalar Dynamic Scalar parameter Viable for > 500 Tensor (1970)2Dynamic Scalar 2 free functions Viable Ni (1970)1, 2Minkowski Tensor, Vector, and Scalar One parameter,
3 functions Predicts unobserved preferred-frame effects Will-Nordtvedt (1972)2Dynamic Vector None Viable Rosen (1973)2Fixed Tensor None Contradicted by binary pulsar data Rastall (1976)2Minkowski Tensor, vector None Viable VMT (1977)2Dynamic Scalar 2 parameters Viable for a wide range of the parameters MOND (1983)4Nonmetric Potential Free function Nonrelativistic theory
1 Misner, Thorne, and Wheeler (1973) 2 Will (1981) 3 Dicke (1965) 4 Milgrom (1989)

OTHER THEORIES

More than two score relativistic theories of gravitation have been proposed. Some have no metric; others take the metric as fixed, not dynamic. These have usually fared badly in light of experiment. Among metric theories those involving a vector field or a tensor field additional to the metric can display a preferred frame of reference or spatial anisotropy effects (phenomena that depend on direction in space). Both effects may contradict a variety of modern experiments. Table 1 gives a sample of theories of gravitation, summarizing the main ingredients of each theory and its experimental status.

All relativistic gravitational theories mentioned so far have a newtonian limit, a tacit requirement of candidate relativistic gravitational theories until very recently. Now, if the correct gravitational theory is general relativity or any of its traditional imitations, then newtonian theory should satisfactorily describe galaxies and clusters of galaxies, astrophysical systems involving small velocities and weak potentials. But there is mounting observational evidence that this can be the case only if galaxies and clusters of galaxies are postulated to contain large amounts of dark matter. Thus far this dark matter has not been detected independently of the preceding argument.

Might not this missing mass puzzle signal instead the break-down of the newtonian limit of gravitational theory for very large systems? In this connection several schemes alternative to Newtonian theory have been proposed. A well developed one is the modified newtonian dynamics or MOND (see Table 1), in which the relation between newtonian potential and the resulting acceleration is regarded as departing from newtonian form for gravitational fields with magnitude of below 10-8 cm s-2. In galaxies and clusters of galaxies (with no dark matter assumed) the gravitational fields are weaker than this, and a breakdown of newtonian predictions having nothing to do with dark matter is expected. With its one postulated relation, MOND ties together a number of empirical relations in extragalactic astronomy. A nonrelativistic gravitational theory containing the MOND relation has been set forth, and relativistic generalizations of these ideas are currently under study.

Additional Reading

    Dicke, R. (1965). The Theoretical Significance of Experimental Relativity. Gordon and Breach, New York. Milgrom, M. (1989). Alternatives to dark matter. Comments Astrophysics13 215. Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973). Graviation. W.H. Freeman, San Francisco. Will, C. (1981). Theory and Experiment in Gravitational Physics. Cambridge University Press, Cambridge. Will, C. (1986). Was Einstein Right? Basic Books, New York. See alsoBlack Holes, Stellar, Observational Evidence; Black Holes, Theory; Dark Matter, Cosmological; Gravitational Lenses; Missing Mass, Galactic; Pulsars, Binary; Stars, Neutron, Physical Properties and Models.

A billion years ago, two dancing black holes make a final spin, merge, and – in a matter of seconds – release a cataclysmic amount of energy. Much as a falling pebble spreads waves on the surface of a still lake, the merger initiates gravitational waves in the space-time continuum. Fast-forward to planet Earth and the year 2015. After an immense journey, the gravitational waves from the black-hole merger pass through our solar system. On the morning of 14 September, they oh-so-slightly wiggle the arms of the twin Laser Interferometer Gravitational-Wave Observatory (LIGO) detectors in Louisiana and Washington state. A pattern of light-waves shifts in a distinctive, long-sought way. A computer sounds the alarm. 

Niayesh Afshordi at the University of Waterloo in Canada first heard of LIGO’s seminal detection over lunch in a bistro. It was late 2015 and still weeks to go until the results were officially released. But rumours were buzzing, and a colleague who had seen the unpublished paper spilled the beans. Afshordi, an astrophysicist who also works at the Perimeter Institute in Waterloo, instantly appreciated the importance of the news – both for the physics community at large, and for his own unconventional theory about the construction of the Universe. 

‘I had an existential crisis at some point. I thought all the problems in cosmology had been solved,’ Afshordi recalled. ‘But then I came up with this idea that dark energy is made by black holes.’ Studies of distant stellar explosions and other lines of evidence show that our universe grows at an accelerating pace, but nobody knows the cause. Matter alone cannot have this effect, so cosmologists blame the expansion on a peculiar type of energy, called dark energy. Its origin and nature were, and are, a mystery.  

In 2009, Afshordi, together with his colleagues Chanda Prescod-Weinstein and Michael Balogh, put forward a theory according to which black holes seed a long-range field that mimics dark energy. The field spills out from black holes and spans through the Universe. It’s an intriguing explanation for the origin of dark energy and, by Afshordi’s calculations, the number of black holes estimated to exist should create just about the right amount of field energy to fit the observations. 

But Afshordi’s idea overthrows what physicists believed they knew about black holes. In Albert Einstein’s theory of general relativity, the event horizon of a black hole – the surface beyond which there is no escape – is insubstantial. Nothing special happens upon crossing it, just that there is no turning around later. If Afshordi is right, however, the inside of the black hole past the event horizon no longer exists. Instead, a Planck-length away from where the horizon would have been, quantum gravitational effects become large, and space-time fluctuations go wild. (The Planck length is a minuscule distance: about 10-35 metres, or 10-20 times the diameter of a proton.) It’s a complete break with relativity. 

When he heard of the LIGO results, Afshordi realised that his so-far entirely theoretical idea could be observationally tested. If event horizons are different than expected, the gravitational-wave bursts from merging black holes should be different, too. Events picked up by LIGO should have echoes, a subtle but clear signal that would indicate a departure from standard physics. Such a discovery would be a breakthrough in the long search for a quantum theory of gravity. ‘If they confirm it, I should probably book a ticket to Stockholm,’ Afshordi said, laughing. 

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Quantum gravity is the missing unification of general relativity with the quantum field theories of the standard model of particle physics. If just thrown together, the two theories lead to internal contradictions, and fail to make sense. Black holes are one of the most studied examples for such a contradiction. Use quantum field theory near the horizon and you find that the black hole emits particles, slowly evaporating. Those particles carry away mass but, as Stephen Hawking demonstrated in the 1970s, they cannot carry information about what formed the black hole. And so, if the black hole evaporates entirely, all the information about what fell in has been destroyed. In quantum field theory, however, information is always preserved. Something in the mathematics, therefore, doesn’t fit correctly. 

The culprit, most physicists think, is that the calculation doesn’t take into account the quantum behaviour of space and time because the theory for this – quantum gravity – is still unknown. For decades, physicists thought that the quantum gravitational effects necessary to solve the black-hole conundrum were hidden behind the event horizon. They thought that it is only near the singularity, at the centre of the black hole, that the effects of quantum gravity become relevant. But recently, they have had to rethink.

In 2012, a group of researchers from the University of California, Santa Barbara, found an unexpected consequence of the currently favoured idea that information somehow escapes with the radiation from a black hole. To make the idea work, large deviations from general relativity are required, not only near the singularity but also at the event horizon. Those deviations would create what the researchers dubbed a ‘black hole firewall’, a barrier of high energy just outside the horizon. 

Such a firewall (if it exists) would become noticeable only for an infalling observer, and would not emit observable signatures that could show up in our telescopes. However, the firewalls lent support to Afshordi’s earlier idea that black holes create a field that acts as dark energy. If that was so, then the near-horizon region of black holes should be very different from what general relativity predicts; a firewall that solves the black hole information-loss problem could be one effect of that deviation. Afshordi’s proposal for how to modify general relativity might therefore hold the key to resolving the tension between quantum theory and general relativity. It was an idea that wouldn’t let him go. 

Instead of a splash followed by dissipating ripples, the gravitational waves should come as faint echoes of the original event

When he learned of LIGO’s first detection, Afshordi began to explore whether the gravitational waves emitted from a black-hole merger could reveal intimate details about what happens near the black-hole horizon. At first it seemed too much to hope for. ‘I didn’t really think we could see quantum gravity effects in the gravitational-wave signal because we had already looked in so many places,’ Afshordi said. ‘But I changed my mind about this.’ 

What made Afshordi reconsider was work by Vítor Cardoso and colleagues at the Instituto Superior Técnico in Portugal on gravitational-wave echoes from black holes. Cardoso had laid out on general grounds that a merger of two objects that are compact but do not have an event horizon would produce gravitational waves very similar to those of black holes – similar, but not identical. The key feature indicating the horizon’s absence, Cardoso argued, would be a periodic recurrence in the signal from the merger. Instead of a single peak followed by a ringdown (think a big splash on a pond, and then rapidly dissipating ripples), the gravitational waves should come as a series of fading pulses – fainter echoes of the original event. Afshordi found that the near-horizon modification described by his theory would cause exactly such echoes. Moreover, he could calculate their recurrence time as a function of the final black hole’s mass, allowing a precise prediction. 

Nobody had ever sought such a signal before, and finding it would not be easy. So far there are only two public, well-defined gravitational-wave detections from LIGO. Together with a collaborator, Afshordi analysed the LIGO data for traces of echoes. By comparing the openly available recordings to random noise, they found an echo at the calculated recurrence time. The statistical significance is not high, however. In scientific terms, it has an estimated significance of 2.9 sigma. Such a signal can be caused by pure noise with a chance of about a one-in-200. In physics, an event of such low confidence is interesting but does not amount to a discovery.

The LIGO experiment is really just getting started, however. The most remarkable thing about the first two gravitational wave events is that the facilities were able to record them at all. The technological challenges were tremendous. Each site, both in Louisiana and in Washington state, has an interferometer with two perpendicular arms about 4 kilometres long in which a laser beam bounces back and forth between mirrors; when recombined, the beams interfere with each other. Interference of the laser’s light-waves is sensitive to deformations in the arms’ relative length as little as a thousandth the diameter of a proton. That is the level of sensitivity required to pick up gravitational effects of colliding black holes. 

A gravitational wave passing through the interferometer deforms both arms at different times, thereby skewing the interference pattern. Requiring an event to be recorded at both sites provides protection against false alarms. By design, LIGO detects gravitational waves best at wavelengths of hundreds to thousands of kilometres, the range expected for black-hole mergers. Other gravitational-wave detectors are planned to cover different parts of the spectrum, each tuned to different types of phenomena. 

Gravitational waves are an unavoidable prediction of general relativity. Einstein recognised that space-time is dynamic – it stretches, it curves, and it wiggles in response to gravitational disturbances. When it wiggles, the waves can travel freely into the far distance, carrying away energy and manifesting themselves by a periodic expansion and contraction of space in orthogonal directions. We have long had indirect evidence for gravitational waves. Because they carry away energy, they cause a small but measurable decay in the mutual orbit of binary pulsars. This effect was first observed in the 1970s, and was awarded a Nobel Prize in 1993. But until LIGO’s detection, we had no direct evidence for the existence of gravitational waves.

This is basic research at its finest. What kind of black hole and compact stellar systems are there? Where are they within the galaxies? 

LIGO’s first event – the September 2015 detection that so excited Afshordi – was remarkable, and not only because it happened just a few days after a long-planned instrumental upgrade. It stood out also because the merging black holes were so heavy, with masses estimated at 29 and 36 times the Sun’s mass. ‘A lot of people expected the black hole events to have lower masses,’ said Ofek Birnholtz, a member of the LIGO collaboration’s group on compact binary coalescence and a physicist at the Max Planck Institute for Gravitational Physics in Germany. The strikingly clean signal, together with the collaboration’s openness in sharing the data, has been an inspiration for physicists in other communities who, like Afshordi, are now exploring how to use the new observations for their own work. 

On 26 December 2015, LIGO recorded a second event. The age of gravitational-wave astronomy had officially begun, after many years of slow progress and false starts. ‘Some of my PhD colleagues had left the field of gravitational-wave astronomy,’ Birnholtz said and added, laughing, ‘but are returning because suddenly it’s hot again.’ This is uncharted territory, basic research at its finest. What kind of black hole and compact stellar systems are there? Where are they within the galaxies? What do the gravitational waves reveal about their origins? If a neutron star merges with a black hole, what can be learned about matter in such extreme conditions? Do black holes behave the way that our calculations predict? 

Afshordi’s theory of black holes and dark energy is just one example of the kinds of enquiries that are now possible. A wealth of information is waiting to be explored, openly, around the world.

A few days after Afshordi’s result appears on the open-access server arXiv.org, members of the LIGO collaboration scrutinise the analysis. It takes only a few weeks until they publish a reply, criticise the methodology, and call for different statistical tools. Birnholtz is one of the authors of that criticism.  

‘The claim is surprising,’ said Birnholtz. ‘I have no prior as to whether or not there should be echoes. That’s physics nobody can guess at. But I do have a strong intuition, working with LIGO data, that the amplitude is probably not large enough to claim such a significance at this stage.’ Birnholtz has suggestions for how to improve the analysis, but avoids making statements about the chances of confirming the result. Alex Nielsen, another member of the LIGO collaboration and one of Birnholtz’s co-authors, reiterates the need for caution: ‘As members of the LIGO collaboration, we have to be very careful about what statements we make in public, before we have full collaboration approval. But the data is public and people can do with it what they want.’  

The LIGO collaboration has an open science centre, where data recorded for one hour around the time of confirmed gravitational events is publicly available. ‘People are welcome to use it and contact us for any questions,’ Birnholtz said. ‘If they find anything interesting, they can share it with us, and we can work on it together. This is part of the scientific experience.’ 

The collaboration has several thousand members worldwide, distributed at more than a hundred institutions. They meet twice a year; the most recent meeting was in March in Pasadena, California. Some members of the collaboration are now trying to reproduce Afshordi’s analysis. Birnholtz expects the effort to take several months. ‘The result might be disappointing,’ he warned. ‘Not in that it says there are no echoes, but that we can’t say whether there are echoes.’ Gravitational-wave astronomy is still a field in its infancy, though, and much more data are on the way. The collaboration estimates that by the completion of the third observing run in 2018, LIGO is likely to have made 40 high-quality detections of black-hole mergers. Each will offer another opportunity to test Afshordi’s theory. 

Because they interact so weakly and deposit so little energy as they pass by, gravitational waves are exceedingly difficult to measure. The deformation they cause is tiny, and enormous care is necessary to extract a clean signal. The discovery threshold used by the LIGO collaboration is 5 sigma, corresponding to a chance of less than one in 3 million that the signal was coincidence, which is far above the significance level of Afshordi’s signal. The weak interaction of gravitational waves, however, is also the reason why they are excellent messengers. Unlike particles or light, they are barely affected on their way to us, carrying with them pristine information about where and how they were generated. They allow entirely new precision tests of general relativity in a regime that has never before been explored.  

Cosmologists would also want to look much more closely at the implications of this new explanation for dark energy

If black hole echoes should be confirmed, that would almost certainly indicate a stark deviation from general relativity. Finding echoes would not uniquely confirm Afshordi’s theory that black holes seed dark energy. But some truly novel idea would be needed to explain it. ‘I don’t know of such echoes in any simulation that we have done to date,’ Birnholtz said. ‘If we were to confirm that there was an echo, that would be very interesting. We would have to look into what could produce such an echo.’ 

Afshordi has research plans in case the statistical significance of his signal increases. He wants to improve his model of black-hole mergers, and run a numerical simulation to support the analytical estimate of what the echoes should look like. The next step would then be to better understand the underlying theory of space-time that could give rise to such a behaviour of the black-hole horizon. Cosmologists would also want to look much more closely at the implications of this new explanation for dark energy.

Afshordi is aware just how speculative it is to alter general relativity so drastically. But he’s a rebel with a mission: ‘I want to encourage people to keep an open mind and not to dismiss ideas because they don’t match their preconceived notions.’ With LIGO exposing the workings of the Universe in ways never before studied, a lot of preconceived notions may soon be set aside.

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Sabine Hossenfelder

is a research fellow at the Frankfurt Institute for Advanced Studies, with a special interest in the phenomenology of quantum gravity. Her writing has appeared in ForbesScientific American, and New Scientist, among others.

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