What Determines the Nature of Gravity? A Phenomenological Approach

Transcription

1 Space Sci Rev (2009) 148: DOI /s What Determines the Nature of Gravity? A Phenomenological Approach Claus Lämmerzahl Received: 27 March 2009 / Accepted: 24 June 2009 / Published online: 16 July 2009 Springer Science+Business Media B.V Abstract The gravitational field can only be explored through the motion of test objects. To achieve this one first has to set up the correct equations of motion. Initially these equations are based on Newton s laws. Corresponding experiments that support Newton s laws are described. Furthermore, the basic characteristics of the motion of test objects in gravitational fields are described. This leads to the notion of Einstein s Equivalence Principle which has as consequence a metric theory of gravity. One particular metric theory is General Relativity based on Einstein s field equations with its particular predictions for effects like periastron advance, light deflection, etc. An overview over the experimental confirmation of General Relativity, in particular those presented at this workshop, is given. This workshop summary ends with open problems. We also describe some of the strategies for the experimental search for a quantum gravity theory. Keywords General relativity Special relativity Newton s axioms Experimental relativity Equivalence principle Solar system tests Quantum gravity phenomenology 1 Introduction In this article we present a general frame of how to define and explore the nature of gravity as well as the mathematical formalism and the equations that represent gravitational phenomena. In principle there are two ways to state physical equations: The first way a top down scheme is to postulate the equations. For General Relativity (GR), for example, one may postulate a Lorentzian space time manifold, the Einstein field equations and the geodesic equations for point-like masses and light, as well as the observables of the theory. All observable consequences will follow from these statements. A second way corresponding to a bottom up approach is to base the physical laws on a few basic observations and to build up the theory in a constructive manner. Here we like to proceed along the second way as far as possible. C. Lämmerzahl ( ) ZARM, University of Bremen, Am Fallturm, Bremen, Germany laemmerzahl@zarm.uni-bremen.de

2 502 C. Lämmerzahl The advantages of the second way are: (i) a physical understanding of mathematical schemes, (ii) each mathematical structure is directly related to an observable phenomenon and, thus, is immediately physically interpretable, and (iii) if generalizations are necessary (like those expected from quantum gravity) natural generalizations are offered through this way. In what follows we can only give a very rough, short and incomplete description of the scheme. However, we hope to show all the ingredients and what in principle has to be taken into account in order to reach a certain stage of complete mathematical description of the nature of gravity. At many instances we refer to other contributions to this workshop which expand several short remarks given here. 2 How to Explore Gravity The gravitational field and its properties can only be explored through the observation of the dynamics of test objects: point particles, light rays or quantum fields (Ehlers 2006). Their dynamics is governed by equations of motion. The simplest such equations are Newton s laws. Much more complicated laws are feasible: one may consider dynamical equations with higher order time derivative (Lämmerzahl and Rademaker 2009), for example. After we setting the main structure of equations of motion for test objects we ask for characteristic features of the interaction of these objects with the gravitational field. These are governed by the Einstein Equivalence Principle. From that we conclude that gravity is a metric theory. Each of these theories shows the typical effects like perihelion shift, light deflection, time delay, Lense Thirring and Schiff effects, etc. Only for certain values of these effects gravity is described by the Einstein field equations. These field equations are then extrapolated to the strong field regime and can be confronted with observations of binary systems and black holes. We always consider fundamental equations only. Effective equations for the motion of test objects with largely different features may come out from complicated calculations that take, for example, radiation reaction into account. 3 The Structure of Dynamics The notion of an inertial system, of the inertial law and the law of reciprocal actions (actio = reactio) is assumed in all equations of motion, either non-relativistic or relativistic. Any test or exploration of the gravitational interaction has to account for the structure of the equations of motion which are used to measure gravitational effects. In many instances we have equations of motion which are more general than those related to Newton s axioms. Examples are the equations taking into account radiation reaction, or dynamics with memory. However, these are effective equations of motion. In our bottom up approach we are interested in the fundamental equations of motion only. 3.1 Existence of Inertial Frames A condition for the existence of inertial frames is content of the first of Newton s laws. In an intuitive sense an inertial frame is a local reference frame where all freely falling particles

3 What Determines the Nature of Gravity? A Phenomenological Approach 503 move uniformly along straight lines. Here we already have to put in an intuitive understanding of free motion. Though it is not clear of how to characterize a force-free motion (nongravitational forces) uniquely in an experimental way, Finsler space time (Lämmerzahl and Perlick 2009) provides a model for the non-existence of inertial frames. 1 An indefinite Finslerian geometry is given by the line element ds 2 = F(x,dx) with F(x,λdx) = λ 2 F(x,dx), (1) for all λ R and where F is a function homogeneous of degree two. Then ds 2 = g μν (x, dx)dx μ dx ν with g μν (x, y) = 2 F(x,y) y μ y ν, (2) where g μν (x, dx) is a Finslerian metric which, however, depends on the vector it is acting on. The motion of light rays and point particles is given by the action principle 0 = δ ds 2 and leads to the equation of motion d 2 x μ = { μ } dx ρ dx σ ds 2 ρσ (x, ẋ) ds ds. (3) Since the Finsler Christoffel connection { μ } 1 ρσ (x, ẋ) = 2 gμν (x, ẋ) ( ρ g σν (x, ẋ) + σ g ρν (x, ẋ) ν g ρσ (x, ẋ) ) (4) (here g μν (x, ẋ) is the inverse of g μν (x, ẋ) defined through g μν (x, ẋ)g νρ (x, ẋ) = δρ μ) depends on the velocity ẋ μ, it cannot be transformed away. As a consequence, there is no frame in which all particles move uniformly along straight lines. We always have accelerated particles. This provides a model for that gravity cannot be transformed away and, thus, for the non-existence of inertial systems. This is true for all equations of motion with a non-linear connection of the form (3). Another consequence of a Finslerian metric is the anisotropy of, e.g., light propagation violating Lorentz invariance (see below and Lämmerzahl et al. 2009). Free fall experiments and orbits of planets and satellites yield that the order of magnitude of any hypothetical Finslerian deviation from ordinary Riemannian space times should be smaller than 10 9 m/s 2 (Lämmerzahl and Perlick 2009). 3.2 The Inertial Law The inertial law ṗ = mẍ = F (5) is characterized by its order of differentiation and the linear relation between force and acceleration. We highlight both properties. Any change in these characteristic features of the inertial law dramatically influences the interpretation of, e.g., orbits of satellites, planets or stars. 1 We leave out Berwald Finsler space times where the space time metric depends on the connecting vector while the equation of motion still is the ordinary Riemannian geodesics.

4 504 C. Lämmerzahl Order of Equations of Motion Newtons second law (5) with p = m i v where m i is the inertial mass implies an equation of motion of second order for the position. The relation between momentum and velocity has the structure of a constitutive law and, thus, may be generalized to p = p(m,v, v, v,...). This implies higher-order equations of motion. Higher-order equations of motion may also come from metrical fluctuations with a certain time correlation. Here we are concerned only with fundamental equations: In the contrary effective equations of motion in general contain higher order derivatives from radiation reaction. The most simple model for a higher order dynamics is based on a second order Lagrangian L = L(t, x, ẋ, ẍ). This gives an equation of motion of fourth order. With a small additional term of appropriate sign (see Lämmerzahl and Rademaker 2009 for more details) the solution of this fourth order equation gives the standard solution of the usual second order equation together with a kind of zitterbewegung. Therefore, the usual second order equations of motion seem to be rather robust against small higher-order additions. In order to be consistent we introduce interactions with external fields through a gauge principle. Such higher order gauge principles result in novel gauge fields. As we expect from this approach only a small zitterbewegung, an experimental detection of such a phenomenon is rather difficult. One potentially feasible idea is to look for fundamental noise in electronic devices with characteristics differing from the standard Nyquist or 1/f noise. Corresponding proposals will be worked out Linearity Newton s inertial law (5) is a definition of the force F. Measuring the path of a test object and knowing its characteristic parameters determines the force. Theories modifying this relation by introducing a function f(a) on the left hand side, mf (a) = F (x), as MOND does, (Milgrom 2002), are equivalent to a theory of modified gravity provided the function f possesses an inverse. Then ma = mf 1 (F (x)/m), and this is a mere redefinition of the force equivalent to a modification of the gravitational influence. This modified Newtonian dynamics or modified gravity is rather successful in modeling galactic rotations curves. The function f (a) is mainly determined by a characteristic acceleration scale a 0 of the order ms 2. Though the inertial law defines the force, there is one aspect which may be subject to experimental proof: If the force acting on a body is given by a gravitating mass, F = m U with U = G ρ(x )/ x x dv (G is Newton s graviational constant and ρ the mass density), then one may ask the question whether the acceleration decreases linearly with decreasing gravitating mass which can be measured through its weight. If the gravitating mass M is spherically symmetric, U = GM/r, then the question is whether ẍ αẍ for M αm, in particular in the case of small M. This is an operationally well defined question which is worth to be explored experimentally. A recent laboratory experiment performed tests of the linearity between force and acceleration in the extremely weak force regime, (Gundlach et al. 2007). No deviation from Newton s inertial law has been found for accelerations down to ms 2. This experiment, however, does not test MOND. Within MOND it is required that the full acceleration has to be smaller than approx ms 2 while in the above experiment only two components of the acceleration were small while the acceleration due to the Earth attraction was still present. Therefore such tests of MOND have to be performed in space (for the constraints of tests on Earth, see Ignatiev 2007). An earlier test (Abramovici and Vager 1986) wentdown

5 What Determines the Nature of Gravity? A Phenomenological Approach 505 to accelerations of ms 2. In both cases the applied force was non gravitational. It might be speculated whether the MOND ansatz applies to all forces or to the gravitational force only. It has also been speculated whether the MOND ansatz can describe the Pioneer anomaly (Milgrom 2002; Anderson et al. 2002) but this has not been convincingly confirmed. In any case, it is a very remarkable coincidence that the Pioneer acceleration, the MOND characteristic acceleration a 0 as well as the cosmological acceleration are all of the same order of magnitude, a Pioneer a 0 ch, whereh is the Hubble constant. 3.3 Law of Reciprocal Action A key model for the violation of the law of reciprocal action is a difference in active and passive gravitational masses. The notion of active and passive masses and their possible nonequality has first been introduced and discussed by (Bondi 1957). The active mass m a is the source of the gravitational field (here we restrict to the Newtonian case with the gravitational potential U) U = 4πm a δ(x), whereas the passive mass m p reacts to it m i ẍ = m p U(x). (6) Here, m i is the inertial mass and x the position of the particle. The equations of motion for a gravitationally bound two body system then are x 2 x 1 m 1i ẍ 1 = Gm 1p m 2a x 2 x 1, m x 1 x 2 2iẍ 3 2 = Gm 2p m 1a x 1 x 2, (7) 3 where 1, 2 refer to the two particles and G is the gravitational constant. For the equation of motion of the center of mass X, wefind Ẍ = G m 1pm 2p M i C 21 x x 3 with C 21 = m 2a m 2p m 1a m 1p (8) where M i = m 1i + m 2i and x is the relative coordinate. Thus, if C 21 0 then active and passive masses are different and the center of mass shows a self-acceleration along the direction of x. This is a violation of Newton s actio equals reactio. A limit has been derived by Lunar Laser Ranging (LLR): no self-acceleration of the moon has been observed yielding a limit of C Al Fe (Bartlett and van Buren 1986). The dynamics of the relative coordinate ẍ = G m ( ) 1pm 2p m 1a m 2a x m 1 + m 2 (9) m 1i m 2i m 1p m 2p x 3 has been probed in the laboratory by (Kreuzer 1968) with the result C Similar considerations have been made for active and passive charges or for magnetic moments (Lämmerzahl et al. 2007a). 4 The Structure of Gravity After having set up the fundamental equations of motion for test objects one can start to explore the structure of interactions. The gravitational interaction is first characterized by a number of universality principles put together in the Einstein Equivalence Principle (EEP).

6 506 C. Lämmerzahl It consists of (i) the Universality of Free Fall (UFF), (ii) the Universality of the Gravitational Redshift (UGR), and (iii) Local Lorentz Invariance (LLI), see Will (1993). These principles further constrain the structure of the equations of motion of test objects: The EEP leaves only freedom for a symmetric second rank tensor field to couple to the equation of motion of test objects. As a consequence one arrives at a metric theory of gravity. Any metric theory of gravity shows the standard Solar system effects like perihelion shift, light deflection, gravitational time delay, and the Lense Thirring and Schiff effect. Only for GR given by the Einstein field equations these effects attain certain values. There is no constructive way to derive these Einstein equations. However, the Parametrized Post Newtonian formalism (PPN), Will (1993), with approximate 10 undetermined parameters (the number of parameters depends on the chosen version) provides a very powerful method of parameterizing deviations from GR. For GR these parameters attain certain values. The PPN formalism gives a theoretical frame within which by means of a finite number of observations and experiments it is possible to single out GR from other theories of gravity. In the following two sections we describe the experiments and observations first leading to a metric theory of gravity and second singling out GR from all other metric theories (see also the contribution of C. Will). 5 The Foundations of Metric Gravity 5.1 Universality of Free Fall The UFF states that all neutral point like particles move in a gravitational field in the same way: The path of these bodies is independent of the composition of the body. The corresponding tests are described in terms of the acceleration of these particles in the reference frame of the gravitating body: the Eötvös factor compares the normalized accelerations of two bodies η = a 2 a (a 2+a 1 ) in the same gravitational field. In the frame of Newtons theory this can be expressed as η = μ 2 μ (μ 2+μ 1 ),whereμ = m g/m i is the ratio of the (passive) gravitational and inertial mass. There are two principal schemes to perform tests of UFF. The first scheme uses the free fall of bodies. In this case the full gravitational attraction towards the Earth can be exploited. However, these experiments suffer from the fact that the time-of-flight is limited to roughly 1 s and that a repetition needs new adjustment. The other scheme uses a restricted motion confined to one dimension only, namely a pendulum or a torsion balance. The big advantage is the periodicity of the motion which by far outweights the disadvantage that only a fraction of the gravitational attraction is used. In fact, the best test today of the UFF uses a torsion pendulum and confirms it to the order of Altogether we then have the amazing equality m i = m g = m a = m p. New proposed tests in space, the approved mission MICROSCOPE, (Touboul2001 and the contribution of P. Touboul), and the proposal STEP, (Lockerbie et al. 2001), will combine the advantages of free fall and periodicity (see also the contributions by J. Mester and T. Sumner). There are hints from quantum gravity inspired scenarios that the UFF might be violated below the level, Damour et al. (2002a, 2002b) and the contribution of T. Damour. Also from cosmology with a dynamical vacuum energy (quintessence) one can derive a violation of UFF at the level (Wetterich 2003). The validity of the UFF has also been used for setting bounds on the time variability of various constants such as the fine structure constant and the electron-to-proton mass ratio (Dent 2006).

7 What Determines the Nature of Gravity? A Phenomenological Approach 507 According to GR, spinning particles couple to the space time curvature (Hehl 1971; Audretsch 1981) and, thus, violate the UFF. However, the effect is far beyond any experimental detectability. Therefore testing the UFF for spinning matter amounts to a search for an anomalous coupling of spin to gravity. Motivations for anomalous spin couplings came from the search for the axion, a candidate for the dark matter in the universe which also can resolve the strong PC puzzle in Quantum Chromodynamics (Moody and Wilczek 1984). In these models spin may couple to the gradient of the gravitational potential or to gravitational fields generated by the spin of the gravitating body. The first case can easily be tested by weighting polarized bodies what showed that for polarized matter the UFF is valid up to the order of 10 8 (Hsieh et al. 1989). Also charged particles do couple to the space time curvature (DeWitt and Brehme 1960) but this effect is again too small to be detectable. Further, it is possible to introduce a chargedependent violation of the UFF by assuming a charge-dependent anomalous inertial and/or gravitational mass. It is also possible to choose the model such that for a neutral atom UFF is fulfilled exactly while it is violated for isolated charges (Dittus et al. 2004). It has been suggested to carry out a corresponding experiment in space (Dittus et al. 2004). 5.2 Universality of Gravitational Redshift A test of the universal influence of the gravitational field on clocks based on different physical principles requires clock comparison during their common transport through different gravitational potentials. There is a large variety of clocks which can be compared: (i) light clocks (optical resonators), (ii) various atomic clocks, (iii) various molecular clocks, (iv) gravitational clocks based on the revolution of planets or binary systems, (v) the rotation of the Earth, (vi) pulsar clocks based on the spin of stars, and (vii) clocks based on particle decay. On a phenomenological level the comparison of two collocated clocks is given by ( ν clock 1 (x 1 ) ν clock 2 (x 1 ) = 1 (α clock 2 α clock 1 ) U(x ) 1) U(x 0 ) νclock 1 (x 0 ) (10) c 2 ν clock 2 (x 0 ) where α clock i are clock-dependent parameters. If this frequency ratio does not depend on the gravitational potential then the gravitational redshift is universal. This is a null-test of α clock 2 α clock 1. It is obviously preferable to use large differences in the gravitational potential which clearly shows the need for space experiments. In experiments today the variation of the gravitational field is induced by the motion of the Earth around the Sun. The best test up to date has been performed by comparing the frequency ratio of the 282 nm 199 Hg + optical clock transition to the ground state hyperfine splitting in 133 Cs over 6 years. The result is α Hg α Cs (Ashby et al. 2007; Fortier et al. 2007). Other tests compare Cs clocks with the hydrogen maser, Cs or electronic transitions in I 2 with optical resonators. We are looking forward to ultrastable clocks on the ISS and on satellites in Earth orbit or even in deep space as proposed by SPACETIME (Maleki 2001), OPTIS (Lämmerzahl et al. 2004) and SAGAS (Wolf et al. 2008), which should considerably improve the scientific results (see also the contribution by S. Reynaud). So far there are no tests using anti clocks, that is, clocks made of anti-matter. However, since the production of anti-hydrogen is a working technique today, there are attempts to perform high-precision spectroscopy of anti-hydrogen. These measurements first should test special relativistic CPT invariance but, as a long-term goal, they could also be used to test the Universality of the Gravitational Redshift for a clock based on anti-hydrogen.

8 508 C. Lämmerzahl In many scenarios it is assumed that constants vary with (cosmological) time. Since different atomic or molecular states depend differently on these constants the question of th constancy of constants is related to the UGR, cf. the contributions by J.-P. Uzan, N. Kolachevsky, P. Petitjean, and E. Fischbach. 5.3 Local Lorentz Invariance Lorentz invariance, the symmetry of Special Relativity (SR) which also holds locally in GR, is based on the constancy of the speed of light and the relativity principle. For a recent review, see Amelino-Camelia et al. (2005) The Constancy of the Speed of Light The constancy of the speed of light has many aspects: 1. The speed of light does not depend on the velocity of the source. Using the model c = c + κv,wherev is the velocity of the source and κ some parameter one gets from astrophysical observations κ (Brecher 1977). 2. The speed of light does not depend on the frequency and polarization. The best results come from astrophysics. From radiation at frequencies Hz and Hz of Gamma Ray Burst GRB one obtains c/c (Schaefer 1999). Analysis of the polarization of light from distant galaxies yielded an estimate c/c (Kostelecky and Mewes 2002). 3. The speed of light is universal. This means that the velocity of all other massless particles as well as the limiting maximum velocity of all massive particles coincides with c. The maximum speed of electrons, neutrinos and muons has been shown in various laboratory experiments to coincide with the velocity of light at a level (c particle c)/c 10 6 (Brown et al. 1973; Guiragossian et al. 1975; Alspector et al. 1976; Kalbfleisch et al. 1979). Astrophysical observations of radiation from the supernova SN1987A yield for the comparison of photons and neutrinos an estimate which is two orders of magnitude better (Stodolsky 1988; Longo 1987). 4. The speed of light does not depend on the velocity of the laboratory. This can be tested in Kennedy Thorndike experiments which is a clock clock comparison experiment where the laboratory moves with varying speed (e.g. a laboratory on the surface of the Earth moves with a velocity consisting of the rotation around its own axis and its revolution around the Sun). The two clocks can be either two light clocks (different resonators or a Michelson-type interferometer with different arm lengths) or a light clock and an atomic clock. The best comparison yields c/c (Müller et al. 2007). 5. The speed of light depends not on the direction of propagation. This has been confirmed by modern Michelson Morley experiments using optical resonators to a relative accuracy of c/c (Müller et al. 2007). 6. A bit more involved is the combination of a finite velocity of signal propagation with quantum systems and quantum measurements involving entanglement ( spooky action at a distance ). Though quantum systems may be entangled over long distances and a measurement of one part of the system has some influence on the properties of the other part of the quantum system it is not possible to communicate with velocities larger than the velocity of light. This altogether means that the velocity of light is a universal structure and, thus, can be interpreted as part of a space time geometry.

9 What Determines the Nature of Gravity? A Phenomenological Approach The Relativity Principle The relativity principle states that the outcome of all experiments when performed identically within a laboratory without reference to the external word, is independent of the orientation and the velocity of the laboratory. For the photon sector this can be tested with Michelson Morley and Kennedy Thorndike type experiments already discussed above. Regarding the matter sector the corresponding tests are Hughes Drever type experiments. In general, these are nuclear or electronic spectroscopy experiments. Such effects can be modeled by an anomalous inertial mass tensor (Haugan 1979) of the corresponding particle. For nuclei one then gets estimates of the order δm/m (Chupp et al. 1989). Also an anomalous coupling of the spin to some given cosmological vector or tensor fields destroys the Lorentz invariance. All anomalous spin couplings are absent to the order of GeV, see Walsworth (2006) for a review. Also higher-order derivatives in the Dirac and Maxwell equations in general lead to anisotropy effects (Lorek and Lämmerzahl 2008). A further aspect of anisotropy is that there might be some anisotropies in the Coulomb or Newtonian potential (Kostelecky and Mewes 2002; Kostelecky 2004). Anisotropies in the Coulomb potential may affect the length of, e.g., optical cavities which may influence the frequency of light in the cavity. However, it has been shown that the influence of the anisotropies of the Coulomb potential are smaller than the corresponding anisotropies in the velocity of light (Müller et al. 2003). Anisotropies in the Newtonian potential of the Earth has recently been looked for by means of atomic interferometry; these measurements constrain the anisotropies to the 10 8 level (Müller et al. 2008). Future spectroscopy of anti-hydrogen may yield further information about the validity of the PCT symmetry. 5.4 The Consequence The consequence of the validity of the EEP is that gravity is described by a Riemannian metric g μν, a symmetric second rank tensor defined on a differentiable manifold being the collection of all possible physical events. The purpose of this metric is twofold: First, it governs the rate of clocks, that is, s = ds, ds = g μν dx μ dx ν (11) is the time shown by clocks where the integration is along the worldline of these clocks. Second, the metric gives the equation of motion for massive point particles as well as for light rays, 0 = d2 x μ ds 2 + { μ } dx ρ dx σ ρσ ds ds where { μ ρσ }= 1 2 gμν ( ρ g νσ + σ g νρ ν g ρσ ) is the Christoffel symbol. Here x = x(s) is the worldline of the particle parametrized by its proper time. It can be shown that the metric also describes the propagation of, e.g, the spin vector, D v S = 0, where S is a particle spin. (12) 6 Motivating Einstein s Field Equations There is no derivation of Einstein s field equations from a few key observations. However, a PPN formalism (Will 1993), makes it possible to parametrize in terms of ten or more

10 510 C. Lämmerzahl parameters deviations of the metric from a metric following from Einstein s field equations R μν 1 2 Rg μν = κt μν (13) where R μν and R are the Ricci tensor and scalar, respectively, T is the energy-momentum tensor of the matter creating the gravitational field, and κ the relativistic coupling constant. In the case of the validity of Einstein s field equations these parameters take specific values. As a consequence the precise measurement of the effects described in the next section will also give a justification of the validity of Einstein s field equations. While all this is going in the weak field and low-velocity regime, one extrapolates the field equation (13) to the strong field and large velocity regime. This extrapolation then can be examined by observations of binary systems and effects near black holes where higher order terms are needed for their correct description (Blanchet 2006). One largely discussed generic deviation from GR is a modification of the Newtonian 1/r potential. Such deviations described by V(r)= M r (1 + αer/λ ) are parametrized by the strength α and range λ. Various experiments yield estimates for α for a given range λ. The high precision of LLR and ephemerides give very tight restrictions on α for interplanetary ranges. Higher-dimensional models predict deviations from the 1/r potential at short distances which motivated big experimental efforts in that direction, see e.g. the contribution by R. Newman. 7 Proving Consequences of General Relativity Gravity can be explored only through its action on test particles (or test fields). Accordingly the gravitational interaction has been studied through the motion of stars, planets, satellites and of light. There are only very few experiments which demonstrate the effects of gravity on quantum fields. There are two classes of tests: Weak gravity effects, mostly observed within the Solar system, and strong gravity effects present in binary systems and near black holes. 7.1 Solar System Effects For the calculation of the effects to be described one needs a solution of Einstein s field equations or an approximate solution in the frame of the PPN formalism The Gravitational Redshift In a stationary gravitational field the gravitational redshift between two positions with radial coordinates r 1 and r 2 is given by ν 2 g tt (r 1 ) = ν 1 g tt (r 2 ) 1 GM ( 1 1 ), (14) c 2 r 1 r 2 where r 1 and r 2 are the radial positions of the two observers. The right hand side of the equation comes out if we assume the validity of Einstein theory of gravity. This effect has best been observed in a space experiment where the time of a hydrogen maser in a rocket has been compared with the time of an identical hydrogen maser on ground yielding a conformation of GR at the level of 1 part in 10 4 (Vessot et al. 1980).

11 What Determines the Nature of Gravity? A Phenomenological Approach Light Deflection The deflection of light was the first prediction of Einstein s GR; it has been confirmed by observation four years after the theory has been completed. In the frame of the PPN formalism we obtain ϕ = 1 2 (1 + γ)m b, (15) where M is the mass of the Sun and b the impact parameter. Today s observations use Very Long Baseline Interferometry (VLBI), and this has led to γ (Shapiro et al. 2004) Perihelion/Periastron Shift Within the PPN formalism we obtain the perihelion shift δϕ = 1 6πM (2 + 2γ β) 3 a 2 (1 e 2 ), (16) where a is the semimajor axis and e the eccentricity of the orbit. Today this post-newtonian perihelion shift has been determined as per century with an error of the order 10 4 (Pitjeva 2005). Recently a huge periastron shift of a candidate binary black hole in the quasar OJ287 has been observed where one black hole is small compared to the other (Valtonen et al. 2008). The observed perihelion shift is approximately 39 per revolution, which takes 12 years Gravitational Time Delay In the vicinity of masses, electromagnetic signals move slower than in empty space, when compared in a coordinate system attached to spatial infinity. This is the gravitational time delay. There are two ways to confirm this effect: (i) direct observation, that is, by comparing the time of flight of light signals in two situations for fixed sender and receiver, and (ii) by observing the change in the frequency induced by this gravitational time delay. Direct measurement The gravitational time delay for signals which pass through the vicinity of a body of mass M is given by δt = 2(1 + γ) GM c 3 ln 4x Satx Earth b 2, (17) where x Sat and x Earth are the distances of the satellite and the Earth, respectively, from the gravitating mass. If the gravitating body is the Sun and if we take b to be the radius of the Sun then the effect would be of the order 10 4 s which is clearly measurable. This has been measured using Mars ranging data of the Viking Mars mission giving γ (Reasenberg et al. 1979). Measurement of frequency change Though the time delay is comparatively small, the induced modification of the received frequency can indeed be measured with higher precision. The reason is that clocks are very precise and, thus, can resolve frequencies also very precisely.

12 512 C. Lämmerzahl The corresponding change in the frequency is y(t) = ν ν 0 ν 0 = 2(1 + γ) GM c 3 1 b(t) db(t), (18) dt where ν 0 is the emitted frequency. It is the time dependence of the impact parameter which is responsible for the effect. This effect has been measured by the Cassini mission. One important issue in the actual measurement was that three different wavelengths for the signals have been used. This made it possible to eliminate dispersion effects near the Sun and to verify with this time delay GR with an accuracy of γ (Bertotti et al. 2003) Lense Thirring Effect For the Einstein field equation as well as within the PPN formalism a rotating gravitating body gives metric components J i dt dx i,wherej is the angular momentum of the rotating body. On the level of the equations of motion this results in a Lorentz type gravitational force acting on bodies called gravitomagnetism, see also the contribution of G. Schäfer. The influence of this field on the trajectory of satellites results in a motion of the nodes, which has been measured by observing the LAGEOS satellites via laser ranging. Together with new data of the Earth s gravitational field obtained from the CHAMP and GRACE satellites the confirmation recently reached the 10% level (Ciufolini 2004, see also the contribution of I. Ciufolini for the LAGEOS results and the contribution of L. Stella for the Lense Thirring effect in astrophysics). In the meantime the LARES mission has been approved. This is another satellite of the same tye as LAGEOS which orbit will have a different inclination than LAGEOS. This makes is possible to eliminate multipole moments of the Earth from the joint LAGEOS and LARES data. The launch is scheduled for early This gravitomagnetic field also influences the proper time and, thus, the rate of clocks. It can be shown that the difference of the proper time of two counterpropagating clocks is s + s = 4πJ/M. It should be remarked that this quantity does not depend on G and r. This effect for clocks in satellites orbiting the Earth can be as large as 10 7 s per revolution (Mashhoon et al. 2001) Schiff Effect The gravitational field of a rotating gravitating body also influences the rotation of gyroscopes. This effect is right now under consideration by the data analysis group of the GP-B mission flown in Data analysis is expected to be completed early Though the mission met all requirements and, thus, was a big technological success it turned out after the mission that contrary to all expectations and requirements the gyroscopes lost more energy than calculated. This requires the determination of further constants characterizing this spinning down effect which effects the overall accuracy of the measurement of the Schiff effect which was expected to be of the order of 0.5%. Nevertheless, recent reports of the GP-B data analysis group indicate that finally the error may go down to 1% (see the contribution of F. Everitt and the GP-B team). For updates of the data analysis one may contact GP-B s website. 2 It should be noted that though both effects within GR are related to the gravitomagnetic field of a rotating gravitational source, the Lense Thirring effect and the Schiff effect are 2 See

13 What Determines the Nature of Gravity? A Phenomenological Approach 513 conceptually different and measure different quantities and, thus, should be regarded as independent tests of GR. In a generalized theory of gravity spinning objects may couple to different gravitational fields (like torsion) than the trajectory of orbiting satellites. Furthermore, the Lense Thirring effect is a global effect related to the whole orbit while the Schiff effect observes the Fermi-propagation, a characterization of a torque-free dynamics, of the spin of the gyroscope The Strong Equivalence Principle The gravitational field of a body contains energy which adds to the rest mass of the gravitating body. The strong equivalence principle now states that EEP is valid also for self gravitating systems, that is, that UFF is valid for the gravitational energy, too. This has been confirmed by LLR with an accuracy of 10 3 (Will 1993), where the validity of UFF had to be assumed. The latter has been tested separately for artificial bodies of a composition similar to that of the Earth and the Moon yielding a confirmation with an accuracy of (Baeßler et al. 1999). 7.2 Strong Gravity Effects While most of the observations and tests of gravity are being performed in weak fields: Solar system tests, galaxies, galaxy clusters, recently it became possible to observe phenomena in strong gravitational fields: in binary systems and in the vicinity of black holes. The observation of stars in the vicinity of black holes (Schoedel et al. 2007) may in one or two decades give new improved measurements of the perihelion shift or of the Lense Thirring effect. Binary systems present an even better laboratory for observing strong field effects. See, e.g., the binary black hole candidate observed by (Valtonen et al. 2008). The inspiral of binary systems which has been observed with very high precision can be completely explained by the loss of energy through the radiation of gravitational waves as calculated within GR (Blanchet 2006). The various data from such systems can be used to constrain hypothetical deviations from GR. As an example, it can be used for a test of the strong equivalence principle (Damour and Schäfer 1991) and of preferred frame effects and conservation laws Bell and Damour (1996) in the strong field regime. Recently, double pulsars have been detected and studied. These binary systems offer the new possibility to analyze spin effects and, thus, open up a new domain of exploration of gravity in the strong field regime (Kramer et al. 2006a, 2006b). Accordingly, the dynamics of spinning binary objects has been intensively analyzed recently (Faye et al. 2006; Blanchet et al. 2006; Steinhoff et al. 2008). A consequence of strong gravity is the emission of gravitational waves. At present ground experiments are reaching their projected sensitivity and collect data. The space mission LISA is sensitive to a lower frequency range more adapted to the long inspiral period of binary systems and is a cornerstone mission of ESA/NASA. LISA is presently prepared through the technology testing LISA Pathfinder mission (see the contribution of S. Vitale). 8 Open Problems Unexplained Observations There are several observations which have not yet found a convincing explanation. In most cases there is no doubt concerning the data. The main problem is the interpretation of the observations and measurements.

14 514 C. Lämmerzahl 8.1 Dark Matter Dark matter is needed to describe the motion of galaxy clusters, as has been first speculated by F. Zwicky (1933), and for stars in galaxies, and has been also confirmed with gravitational lensing, see e.g. Sumner (2002). Also structure formation needs this dark matter. However, until now there is no single observational hint at particles which could make up this dark matter. As a consequence, there are attempts to describe the same effects by a modification of the gravitational field equations, e.g., of Yukawa form (Sanders 1984), or nonlocal gravity (Hehl and Mashhoon 2008), or by a modification of the dynamics of particles, like the MOND ansatz (Milgrom 2002; Sanders and McGough 2002), recently formulated in a relativistic frame (Bekenstein 2004). Due to the lack of direct detection of Dark Matter particles, all those attempts are on the same footing. There are suggestions that at least a considerable part of the observations which usually are explained by dark matter can be related to a stronger gravitational field which come out while taking the full Einstein equations into account (Cooperstock and Tieu 2005; Balasin and Grumiller 2006). 8.2 Dark Energy Observations of type Ia supernovae, (Riess et al. 1998; Perlmutter et al. 1999), WMAP measurements of the cosmic microwave background (Spergel et al. 2007), the galaxy power spectrum and the Lyman alpha forest data lines (van de Bruck and Priester 1998; Overduin and Priester 2001; Tegmark et al. 2004), indicate an accelerating expansion of the universe and that 75% of the total energy density consist of a dark energy component with negative pressure (Peebles and Ratra 2003). Buchert and Ehlers (1997) have shown first in a Newtonian framework that within a spatial averaging of matter and the gravitational field, rotation and shear of matter can influence the properties of the averaged gravitational field which are described in effective Friedman equations. This also holds in the relativistic case (Buchert 2008). Therefore it is an open question whether dark energy is just a result of a correct averaging procedure. An influence of the averaging has been found in existing data (Li and Schwarz 2007; Li et al. 2008). These topics are illuminated in more detail in the contributions by Zakharov, Lasenby, Caldwell, and Goobar. 8.3 Pioneer Anomaly The Pioneer anomaly, an unexplained anomalous acceleration of the Pioneer 10 and 11 spacecraft of a Pioneer = (8.74 ± 1.33) m/s 2 toward the Sun, is discussed in Anderson et al. (2002) and the contribution of S. Turyshev. This acceleration seemed to turn on after the last flyby at Jupiter and Saturn and stayed constant within a 3% range. Until now no convincing explanation has been found. An anisotropy of the thermal radiation might explain the acceleration. However, while the power provided by the plutonium decay decreases exponentially, the acceleration stays constant. Nevertheless, further work on a good thermal modeling of the spacecraft is going on at ZARM (Rievers et al. 2008). Moreover, an analysis of the early tracking data is on the way. Improvements of ephemerides also helps to rule out various suggested explanations and theories (Standish 2008).

15 What Determines the Nature of Gravity? A Phenomenological Approach Flyby Anomaly It has been observed at various occasions that satellites having been subjected to an Earth swing-by possess a significant unexplained velocity increase by a few mm/s. This unexpected and still unexplained velocity increase is called the flyby anomaly. For a summary of recent investigations of this phenomenon, see Lämmerzahl et al. (2007b). Anderson et al. (2008) have proposed the heuristic formula v = v ωr c (cos δ 2 in cos δ out ) (19) which describes all flybys. Here R and ω are the radius and the angular velocity, respectively, of the Earth, and δ in and δ out are the inclinations of the incoming and outgoing trajectory. However, the recent observation of a Rosetta flyby could not verify this empirical formula. 3 Until now no explanation has been found but, currently, it is expected that it is a mismodeling of either (i) the thermal influence of the Earth and the Sun s radiation on the satellite, (ii) of reference systems (this is supported by the fact that all the flybys can be modeled by (19), which contains geometrical terms only), (iii) of the flyby since this takes place at an accelerated body, or (vi) of the satellite s body being described by a point mass. There was an ISSI workshop on this topic in March 2009 (cf. footnote 3). 8.5 Increase of Astronomical Unit From the analysis of radiometric measurements of distances between the Earth and the major planets including observations from Martian orbiters and landers from 1961 to 2003 a secular increase of the Astronomical Unit of approximately 10 m per century has been reported (Krasinsky and Brumberg 2004) (see also the article Standish 2005 and the discussion therein). This increase cannot be explained by a time-dependent gravitational constant G because the Ġ/G that would be needed is larger than the restrictions obtained from LLR. Such an increase might be mimicked, e.g., by a long-term increase of the density of the Sun plasma. 8.6 Quadrupole and Octupole Anomaly Recently an anomalous behavior of the low-l contributions to the cosmic microwave background has been reported. It has been shown that (i) there exists an alignment between the quadrupole and octupole with >99.87% C.L. (de Oliveira-Costa et al. 2005), and (ii) that the quadrupole and octupole are aligned to Solar system ecliptic to >99% C.L. (Schwarz et al. 2004). No correlation with the galactic plane has been found. The reason for this is totally unclear. One may speculate that an unknown gravitational field within the Solar system slightly redirects the incoming cosmic microwave radiation (in the similar way as a motion with a certain velocity with respect to the rest frame of the cosmological background redirects the cosmic background radiation and leads to modifications of the dipole and quadrupole parts). Such a redirection should be more pronounced for low-l components of the radiation. It should be possible to calculate the gravitational field needed for such a redirection and then to compare that with the observational data of the Solar system and the other observed anomalies. 3 Team meeting Investigation of the flyby anomaly, ISSI, Bern, March 2 6, 2009; teams/investflyby/.

16 516 C. Lämmerzahl 9 The Search for Signals of Quantum Gravity There are many experiments proving that matter has to be quantized and, in fact, all experiments in the quantum domain are in full agreement with quantum theory with all its somehow strange postulates and consequences. Consistency of the theory also requires that the fields to which quantized matter field couple has to be quantized, too. Therefore, also the gravitational interaction has to be quantized. In particular, there is no meaning of the Einstein equation if the right hand side consists of quantized matter while the left hand side is purely classical. Also the semiclassical Einstein equation with an expectation value on the right hand side has been shown to lead to unwanted effects like faster than light propagation. However, though gravity is an interaction between particles it also deforms the underlying geometry. This double-role of gravity seems to prevent all quantization schemes from being successful in the gravitational domain. The incompatibility of quantum mechanics and GR also shows up in the role of time which plays a different role in quantum mechanics and in GR. Furthermore, it is expected that a quantum theory of gravity would solve the problem of the singularities appearing within GR. As a last issue, it is the wish that such a new theory also would lead to a true unification of all interactions and, thus, to a better understanding of the physical world. Any theory is characterized by their own set of constants. It is believed that the Planck energy E Pl ev sets the scale of quantum gravity effects. As a consequence, all expected effects scale with this energy or the corresponding Planck length, Planck time, etc. In string theory other scales influence the modifications (as is explained in the contributions by T. Damour and by B. Schutz). The implications of deviations from the standard model of cosmology is the subject of the article by S. Sarkar. 9.1 Theoretical Approaches The low energy limit of string theory, a quasiclassical limit of loop quantum gravity as well as results from noncommutative geometry suggest that many of the standard laws of physics will suffer modifications. At a basic level these modifications show up in the equations of the standard model and in Einstein s field equations. These modifications then result in violation of Lorentz invariance different limiting velocities of different particles modified dispersion relation leading to birefringence in vacuum modified dispersion relation leading to frequency-dependent velocity of light in vacuum orientation and velocity dependence of effects time and position-dependence of constants (varying α, G, etc.) modified Newton potential at short and large distances. In recent years there have been increased activities to search for these possible effects. However, until now nothing has been found. 9.2 Experimental Approaches The experimental search for signals of a new theory requires to measure effects which have never been measured before. A strategy to find new things is (i) to explore new parameter regions in extreme situations, (ii) to use more precise devices, (iii) to use high-precision methods for new tests, or (iv) to test or measure exotic things.