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Operational Foundation of Einstein's General Theory of Relativity
Abstract
The measurement of time, space, and space-time intervals, and the related geometries in any gravitational field are analyzed carefully. It is pointed out that synchronization is arbitrary and very complicated operationally; and a method for doing away with it is given. Then it is shown how space geodesics can be used to construct several useful coordinate systems in an arbitrary field, and the associated coordinate conditions are derived. These coordinate systems are then used to find a unique physically meaningful solution to the field equations in the cases of zero, linear, rotation, and spherically symmetric fields.
... Whereas in special relativity two clocks B and C, synchronized with a clock A, are synchronized with each other (transitivity of synchronization), the same property does not hold in general relativity. Basri [5] argues that, since Lorentz contraction and time dilation depend on the procedure of synchronization of clocks, such effects have received, in the scientific literature in matter of relativistic theory, an excessive emphasis, while the clock effect, based on the comparison of proper durations (independent of the procedure of synchronization), measured by real clocks along two nonequivalent world lines between two fixed extreme events, plays, within the Einstein theory, a very important theoretical and empirical role. ...
... While according to Newton's theory real clocks provide a relative, approximate 5 and external measure of the absolute duration, therefore of the variable t, according to Einstein's theory clocks measure the length τ of the world line, hence not t, which appears to be a simple mathematical label without physical meaning. In general relativity the evolution of bodies and phenomena is not a function of an independent and preferential variable as, instead, it happens with Newtonian time, that plays the role of the independent parameter to which every evolution is referred. ...
... As claimed by Basri[5], clocks whose operation is linked to different forms of interaction do not necessarily behave in an equivalent manner: the equivalence, in the current state of experimental findings, is practically gained for clocks whose operation is related to strong nuclear or electromagnetic interactions, but it is problematic about the behaviour of radioactive (weak nuclear force) or gravitational clocks. Possible abnormalities of behaviour, explicitly recognized also by Finzi[31], have never been subjected to a careful investigation in relation to a critical analysis or an operational redefinition of physical time. ...
This paper puts forward a broad critical analysis of the concept of physical time. Clock effect is conceived as a consequence of the variation of the gravitational or pseudo gravitational potential, and it is remarked that only some real clocks measure durations in agreement with the predictions of general relativity. A probable disagreement is expected between radioactive and atomic clocks, in the light of Rovelli's thermal time hypothesis. According to the recent contributions by Rugh and Zinkernagel, the relationship between time and clocks is investigated in order to found on a physical basis the concept of cosmic time. In the conclusive section is argued the impossibility of reducing thermal time to relativistic time.
... The main lines of this demonstration were the following. Einstein's fundamental tensor of components , is the solution of the system of equations (44) of Einstein (1916) obtained from pure mathematical reasoning not related a priori only to gravitation. This is the reason why, in my opinion, the title of his famous paper: "The foundation of the general theory of relativity", do not comport the word 'gravitation'. ...
... We use the Einstein summation, the time derivative ⁄ of y is denoted simply . A particle of mass m and electric charge q will be denoted ( , ). denotes Einstein's tensor which is a solution of the system of equations (44) of Einstein (1916). In the present paper, we do not take into account dark matter since we consider only individual particles. ...
... One recalls that since Einstein's Eqs. (44) of Einstein (1916) were obtained by pure mathematical reasoning, one can suppose that these equations are more general than defining only the gravitational field. Indeed, as we show now, they define also the exact electromagnetic equations that complete the Maxwell equations shown to be an approximation, by taking into account the general relativity. ...
In previous publications, we showed that Maxwell’s equations are an approximation to those of General Relativity when V<<c, where V is the velocity of the particle submitted to the electromagnetic field. This was demonstrated by showing that the Lienard-Wiechert potential four-vector A_u created by an electric charge is the equivalent of the gravitational four-vector G_u created by a massive neutral point when V<<c. In the present paper, we generalize these results for V non-restricted to be small. To this purpose, we show first that the exact Lagrange-Einstein function of an electric charge q submitted to the field due an immobile charge q_0 is of the same form as that of a particle of mass m submitted to the field created by an immobile particle of mass m_0. Maxwell’s electrostatics is then generalized as a case of the Einstein’s general relativity. In particular, it appears that an immobile q_0 creates also an electromagnetic horizon that behaves like a Schwarzschild horizon. Then, there exist ether gravitational waves constituted by gravitons in the same way as the electromagnetic waves are constituted by photons. Now, since A_u and G_u, are equivalent, and as we show, G_u produces the approximation, for V<<c, of g_u4 created by m_0 mobile, where the g_uv are the components of Einstein’s fundamental tensor, it follows that A_u+u_u produces the approximation, for V<<c, of Bet_u4 , where the Bet_uv created by m_0 and by q_0, generalize the g_uv.
... After the discovery of that various developing, modification and magnifications have done. As a strict definition " the relativity is the relative interaction of anything with respect to others " [1], [2]. Such as the electron moves inside the atom within a proper circumstantial obit, it has always a time, velocity and energy instantaneously [3], [4]. ...
... In this relevant area the Lorentz's equation collaborate with speed of light and the speed of the object is very useful and salient. Which is given by [1], [2] ...
... The given expression shows the variation of time which is correlated with height and space. This expression is a moderated form of Lorentz's equation where the space and the height is installed instead of the velocity of object v and speed of light c respectively [1][2][3][4][5]. In fact for the change of height and space of any object, it is keen difficult to predict the time which changes simultaneously with these two parameters. ...
This article contributes a mathematical visualization on the relativity among space-time-height accordingly with the A. Einstein’s special relativity theory. The variation of time with the increment and decrement of space and height is truly focused and elaborated inside. The Lorentz expression about the relativity interprets the relativity of mass correspondingly with the energy of matter. This proposed theory visualize the relativity among three dimensions at a time i.e. space, time and height. Space, time and height are taken as three individual dimensions and all 3D plots are sketched collaborating with these dimensions. To comprehend the theory several explanations are established and several expressions are derived and plotted manually. All the expressions conclude with a great satisfaction that the relativity of time also exists in the collaboration with space and height both instantaneously.
Proposed theory imparts a biggest role to provide a great practical comparativeness on the relativity, among all the visible or invisible substances. This special relativity justifies, the existence of relativity in a different dimension, by the precise mathematical model.
In 1905, Albert Einstein invented and published in his revolutionize paper about the general relativity. This was the first initiation of the modern physics, which is still ongoing towards the future. The relativity comprises an outstanding relationship among all the particles even after the quantum and nanoparticles. The space & time is relatively dependent upon themselves, this statement is the fundamental concept of relativity. One causes the effective change occurrence to another.
In this paper one new concept is proposed. This theory appends one extra dimension in the relativity theory. This proposed theory consists of the space-time-height, these three dimensions concurrently. The time dilation due to the variance of these three dimensions are clearly exhibited by a set of mathematical expressions and by the help of graphs. The correlation between the time and the energy is also proposed mathematically. The incentive of this work is to briefly visualize the existing relationship among all these three dimensions in the right way.
In this paper the speed of object is related to the speed of light mathematically, which exposes out the major connections among the space time height. Variance of time associated with height and space of the object is theoretically visualized and comprehended in such a way that the appearance of the relativity got congenial to the readers to understand perfectly.
... Due to the steadily increasing number of gravitational wave (GW) observation from coalescing binaries [2] new techniques to tackle the strong field regime have also been developed [3]. Besides these efforts and achievements there are some fundamental issues that are left aside in these developments [4]. For instance, although the emission of GW is a relativistic effect, its detection could be understood as non-relativistic, depending on the way we use light to measure distances. ...
... Comparing with LIGO (VIRGO), with finesse F LG ≈ 2 × 10 2 (F VG ≈ 50) the quality factor for the mirrors reduces up to Q ≈ 10 8 which translates in F SR ≈ 22 × 10 3 . It seems reasonable to reach ≈ 15 × 10 3 bounces for 4 By diffraction effects we mean those due to irregularities on the mirror surfaces and which do not lead in a loss of overall power, but rather in a redistribution of the laser intensity. 5 Work in progress. ...
We propose a novel approach to detect gravitational waves based on a semi-rigid detector. The approach relies upon the time delay that the light takes to travel from a fixed mirror at the end of a rigid bar to a nearby free mirror. We show that the dimensions of the experimental device can be shortened in comparison to the conventional ones based on two free mirrors.
... The deflection of a ray of light incident on a massive body such as a star is well known [3] , with the total angle of deflection given by θ E = 4 G M / R 2 , where G is Newton's gravitation constant, M is the mass of the star and R is the impact parameter, the closest distance of approach of the incident light ray to the center of the star. Our aim is to obtain this deflection amount to first order in v/c in the rocket ship analogy. ...
... By Kepler's third law, G M = 4 π 2 a 3 / T 2 where T is the period of the elliptical orbit. Substituting this into the right hand side of (54) yields Einstein's result [3] , ...
The nature of the principle of equivalence is explored. The light ray travel path in an accelerated reference frame, a rocket ship, is described and the rocket ship model is used to derive the deflection of light by a massive body. This model is applied to the deflection of light by a central gravitational field. Also, a novel approach is used to determine the precession of the perihelion of a planet.
... In particular, substantial efforts have been invested in understanding the so-called cosmological constant problem [1][2][3], more precisely why the vacuum energy does not produce a huge value for the cosmological constant, many orders of magnitude above the observed value. Within this context a gravitational theory, nearly as old as general relativity (GR) itself [4], the so-called unimodular gravity (UG) [5], has once again been analyzed [6] as a potential way to approach the problem. ...
... Originally, the idea of unimodular gravity was conceived when Einstein considered the unimodular condition [4], √ −g = 1, as a convenient way to partially fix a coordinate system in GR. The definition of unimodular gravity is usually based on the invariance under a restricted group of diffeomorphisms that leave the determinant of the metric invariant, so that the determinant of the metric can be set equal to a fixed scalar density ǫ 0 , √ −g = ǫ 0 . ...
We study the quantization of two versions of unimodular gravity, namely fully diffeomorphism-invariant unimodular gravity and unimodular gravity with fixed metric determinant, utilizing standard path integral approach. We derive the BRST symmetry of effective actions corresponding to several relevant gauge conditions. We observe that for some gauge conditions, the restricted gauge structure may complicate the formulation and effective actions, in particular, if the chosen gauge conditions involve the canonical momentum conjugate to the induced metric on the spatial hypersurface. The BRST symmetry is extended further to the finite field-dependent BRST transformation, in order to establish the mapping between different gauge conditions in each of the two versions of unimodular gravity.
... Moreover, future observations can potentially have important consequences also for cosmology. The discovery comes nearly one century after gravitational waves were first predicted by Einstein[3], in 1916, one year after he wrote down the gravitational field equations[4]. Gravitational waves are one of the most remarkable predictions of Einstein's theory of General Relativity, and the one for which it proved hardest to find direct experimental confirmation. The reason for this is the very weak coupling of systems of ordinary masses and length scales to gravity, as expressed by the smallness of the gravitational constant. ...
... In these lectures notes we have presented a field theoretical approach to the construction of General Relativity, starting from free tensor fields of spin s = 2, and leading after summing infinitely many interaction terms to the action (3.50) originally presented by Einstein[4]. In this approach the geometric interpretation of gravity arises from the field theory, instead of being the starting point of the construction of the theory. ...
The aim of these notes is to give an accessible and self-contained introduction to the theory of gravitational waves as the theory of a relativistic symmetric tensor field in a Minkowski background spacetime. This is the approach of a particle physicist: the graviton is identified with a particular irreducible representation of the Poincar\'e group, corresponding to vanishing mass and spin two. It is shown how to construct an action functional giving the linear dynamics of gravitons, and how General Relativity can be obtained from it. The Hamiltonian formulation of the linear theory is examined in detail. We study the emission of gravitational waves and apply the results to the simplest case of a binary Newtonian system.
... However, considering time quantum degree of freedom as a dynamical degree of freedom which is connected to a physical system has more physical significance. This representation of clock to define time can be described as an operational definition of time which is appropriate for proper time [22,23]. In whole of the paper we assign time quantum degree of freedom as dynamical degree of the freedom which is relevant to some physical system. ...
We apply quantum description of time, based on Page and Wootters' mechanism, for near horizon region of black hole. It is indicated that this method of quantization of time which is a modified form of PaW approach and has been proposed in [21] can not be applied in this particular quantum region of black hole, fully. The time evolving of the system and flow of time, measurement problem, recording the history of near-horizon zone and also a consistent definition of the arrow of time are controversial in PaW approach in this particular region even by considering the previous modifications which exist in this literature. The clock ambiguity also is investigated and a unique definition of clock is defined in near horizon region of black hole.
... Due to the steadily increasing number of Gravitational Wave (GW) observation from coalescing binaries [2] new techniques to tackle the strong field regimen are also been developed [3]. Besides these efforts and achievements there are some fundamental issues that are left aside in these developments [4]. Although GW is a relativistic effect, its detection could be understood as non-relativistic, depending on the way we measure distances, essentially whether or not we use light. ...
We propose a novel approach to detect gravitational waves. It relies on simple ideas and allows us to redesign the interferometer setup shortening the Fabry-Perot cavity. As a consequence the number of bounces could be increased and the signal enhanced.
... equivalence principle postulates the exact equality between inertial and gravitational mass of any object, regardless of composition. This fundamental equality paved the way to a metric theory of gravity and is a vital pillar of general relativity [1,2]. It implies the universality of the gravitational interaction: all systems and forms of energy are affected equally by gravity in a sufficiently small region of space. ...
The equivalence principle in combination with the special relativistic equivalence between mass and energy, , is one of the cornerstones of general relativity. However, for composite systems a long-standing result in general relativity asserts that the gravitational mass is not simply equal to the total energy. This seeming anomaly is supported by all explicit, general relativistic derivations of the dynamics of bound systems, and is only avoided after time-averaging. Here we rectify this misconception and derive from first principles the correct gravitational mass of a generic bound system in curved space-time. Our results clarify a lasting conundrum in general relativity and show how the weak and strong equivalence principles naturally manifest themselves for composite systems. The results are crucial for describing new effects due to the quantization of the interaction between gravity and composite systems.
... modern theoretical physics. Indeed, for Einstein's geometrical general relativity theory (GRT) [3] gravity is curvature of the "non-material" space-time, and in this sense, space-time presents the Newtonian "immaterial" agent of gravity. While for Feynman's relativistic quantum field theory (FGT) [4] gravity is a consequence of exchange by "material" gravitons -the quanta of gravitation field in Minkowski space-time. ...
Contemporary discussions [1] of theoretical problems generated by classical (non-quantum) Black Hole event horizon and by gravitational wave energy transfer and localization, have demonstrated that gravity quantization is unavoidable forthcoming paradigm shift in gravity physics. The key question is to choose a way for gravity quantization - one approach is the space-time quantization, while another approach is quantization of material gravitation field in Minkowski space-time, as for other fundamental physical interactions.
... modern theoretical physics. Indeed, for Einstein's geometrical general relativity theory (GRT) [3] gravity is curvature of the "non-material" space-time, and in this sense, space-time presents the Newtonian "immaterial" agent of gravity. While for Feynman's relativistic quantum field theory (FGT) [4] gravity is a consequence of exchange by "material" gravitons -the quanta of gravitation field in Minkowski space-time. ...
Detection of gravitational waves by LIGO-Virgo observatories disclosed the agent of gravity (gravitons) and together with information paradox of black hole event horizon points to paradigm shift from non-quantum geometry to quantum gravity physics. Historical, philosophical, mathematical, physical and astronomical arguments are presented for demonstration that in the beginning of the 21st century we are on the verge of the forthcoming change in understanding of gravity nature, which will bring new ideas in performing gravitational experiments and interpretation of astrophysical phenomena.
... The twentieth century saw the building of cosmology theoretical frame : Einstein general relativity [Ein52], its solution the Friedman's equation in 1922 [Fri99], the observational discovery of redshift by V. Slipher [Sli13], the proposal of the Big Bang model by Lemaître [Lem33] in 1933 and the establishing by Edwin Hubble of the Hubble's law ( [Hub26], [Hub36]). Subsequently is modern cosmology based on the current standard Hot Big Bang Model which states that the universe expanded from an initially hot and dense state, and that it is ever-expanding today. ...
The presence of mass inhomogeneities along the line of sight of propagation of light fromdistant objects can induce deflection in the flight path of the photon. This phenomenon is called as gravitational lensing. Lensing can have both distortion (shear) and isotropic magnification effects on the source. We studied the effect of lensing magnification on supernova (SN) Ia in this thesis. Presence of lensing would introduce a source of contamination to the brightness distribution of the source (SN Ia in our case). Thus it also enables one to compute the lensing effect indirectly from the Hubble diagram (i.e. from the residual to the Hubble diagram). In this thesis we computed the correlation between these two effects : the Hubble residual and the computed lensing magnification for the SN by the line of sight foreground dark matter haloes. A detection of positive correlation between these two would signify the positivity of lensing signal detection. The data sample is the spectroscopic SNe Ia sample from the five years full SNLS data and the Hubble residuals are those of the preliminary cosmology analysis performed on SNLS5 data. We obtain a signal of \rho = 0.177 at 2.51 \sigma. This result is consistent with the previous SNLS three years data lensing analysis results.
... General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or space-time [5]. In particular, the curvature of space-time is directly related to the energy and momentum of whatever matter and radiation are present. ...
This paper was written as a response to the contradiction between the collapse of a black hole to a gravitational singularity and the Pauli exclusion principle that could form a quark star at the centre. An investigation of the Kerr and Scwarzschild models inside the event horizon will show that a stable radius (stellar core) is possible near the centre.
It has been found that quarks would be liberated from the neutron by extreme gravitational energy of large stars and generate an ultra-relativistic fermion gas of quarks. The maximum density for a fully degenerate non-strange quark star will be 1.1 x 10 25 kg.m-3. Thus, a quark star could reside behind an event horizon and behave like a black hole.
The non-strange quark matter of the core is found to be stable for a range of masses from 3Msun to 21 million Msun. Empirical confirmation of the quark star is claimed in the event horizon of GW150914, the gravitational wave detection for binary 'black holes'.
... The Einstein's original field equation was of the form (Einstein, 1916): ...
Mapana Journal of Sciences
2018, Vol. 17, No. 1, 25–32. //
One of the unresolved problems in cosmology is that the measured mass density of the universe has revealed a value that was about 30% of the critical density. Since the universe is very nearly spatially flat, as is indicated by measurements of the cosmic microwave background, about 70% of the energy density of the universe was left unaccounted for. Another observation seems to be connected to this mystery. Generally one would expect the rate of expansion to slow down once the universe started expanding. The measurements of Type Ia supernovae have revealed that the expansion of the universe is actually accelerating. This accelerated expansion is attributed to the so-called dark energy (DE).Here we give a brief overview on the observational basis for DE hypothesis and how cosmological constant, initially proposed by Einstein to obtain a static universe, can play the role of dark energy.
... It is worth emphasizing that the curved spacetime is an excellent mathematical model for the photon-embodied vacuum. 55 For example, the quanta of light that propagate from the universal vacuum into the local gravitational field of a body will increase in energy density, i.e., blue-shift to maintain thermodynamic balance in the denser surroundings. ...
A change in momentum will inevitably perturb the all-embracing vacuum, whose reaction we understand as inertia. Since the vacuum’s physical properties relate to light, we propose that the vacuum embodies photons, but in pairs without net electromagnetic fields. In this physical form the free space houses energy in balance with the energy of matter in the whole Universe. Likewise, we reason that a local gravitational potential is the vacuum in a local balance with energy that is bound to a body. Since a body couples to the same vacuum universally and locally, we understand that inertial and gravitational masses are identical. By the same token, we infer that gravity and electromagnetism share the similar functional form because both are carried by the vacuum photons as paired and unpaired.
... • parts of this work have been published as: [1] and [2] Signed:.. [3] the theory of General Relativity (GR). The theory came as a radical generalisation of Newton's law of gravity and describes gravity in terms of a geometrical curvature of spacetime. ...
This project makes progress towards a first calculation of the second-order gravitational self-force in extreme-mass-ratio binaries. This is an important component in the modeling of these key astrophysical sources of gravitational waves. Computing the secondorder self-force requires the second-order metric perturbation, which can be calculated by solving the Einstein field equations through second order in the mass ratio. Here we have developed, for the first time, a practical scheme for solving the second-order equations. The main ingredient is a certain “puncture” field, which describes the local metric perturbation near the small member of the binary, and for which we obtain a useful covariant-form expression. We apply this method to the case of a quasicircular binary of nonrotating black holes. As a first test we numerically solve the first-order field equations and compute the first-order self-force, finding good agreement with previous results obtained using a different method. The calculation of the second-order metric perturbation brings about two additional technical difficulties: the need for a certain regularization at infinity and on the event horizon of the large black hole, and the strong divergence of the second-order source of the field equations near the small object. We show how these issues can be resolved, first in a simple scalar-field toy model, and then in the second-order gravitational problem. We finally apply our method in full in order to numerically solve the second-order perturbation equations in the quasicircular case, focusing on the monopole piece of the perturbation as a first example.
... The latter can be seen as acting either actively, by displacing the fields on the spacetime manifold, or passively, by relabeling the points of the spacetime manifold. The so-called 'hole argument' shows that this symmetry implies that spacetime points have no physical meaning per se, i.e. in absence of fields, and events can only be localized with respect to each other, rather than with respect to the underlying manifold [10,11]. Similarly to gauge theories, therefore, also in general relativity physical boundaries are defined by the presence of 'something'. ...
We analyze the edge mode structure of Euclidean three dimensional gravity from within the quantum theory as embodied by a Ponzano-Regge-Turaev-Viro discrete state sum with Gibbons-=-Hawking-York boundary conditions. This structure is encoded in a pair of dual statistical models of the vertex and face kind, which for specific choices of boundary conditions turn out to be integrable. The duality is just the manifestation of a pervasive dual structure which manifests at different levels of the classical and quantum theories. Emphasis will be put on the geometrical interpretation of the edge modes which leads in particular to the identification of the quantum analogue of Carlip's would-be normal diffeomorphisms. We also provide a reinterpretation of our construction in terms of a non-Abelian 2+1 topological phase with electric boundary conditions.
... After accomplishing his general relativity theory in 1916, Einstein asserted in 1933 [17]: "Physics constitutes a logical system of thought which is in a state of evolution, whose basis (principles) cannot be distilled, as it were, from experience by an inductive method, but can only be arrived at by free invention. " He reaffirmed in 1934 [18]: "Pure thought can grasp reality" and "Nature is the realization of the simplest conceivable mathematical ideas. " He asserted in 1940 [19]: "For the time being we have to admit that we do not possess any general theoretical basis for physics which can be regarded as its logical foundation. ...
The road from fuzzy sets to logically definable causality is surveyed. It is shown that YinYang bipolar fuzzy set theory as an extension to fuzzy sets provides a logical basis for the definability of causality. In turn, bipolar causality leads to a ubiquitous analytical paradigm of quantum cognition and quantum intelligence. It is pointed out that these would have been impossible without Lotfi’s recognition of bipolar fuzzy sets. It is concluded that Lotfi’s scientific spirit transcends the boundaries of his achievement in science, and on the road to definable causality and quantum intelligence he will be remembered as a Giant whose feet are on the ground but whose head is above the clouds.
... However, when it is ignored, the galaxy rotation is attributed to unobservable dark matter [31,39,40]. On the solar system's scale, a small but distinct fraction of the advancing Mercury perihelion, albeit accounted for by general relativity, is referred to as anomalous [41,42]. On the planetary scale an additional velocity gain of a spacecraft during its flyby of Earth is perceived as anomalous [43] when the energy density gradient due to Earth's rotation is omitted from analysis. ...
Evolution of the Universe is described by the principle of least action. The general law is motivated because it accounts for ubiquitous characteristics. These manifest themselves on the cosmic scale, for example, as power laws in distributions of galaxies and voids. When free energy is consumed in the least-time, the Universe will invariably evolve toward isotropy and homogeneity on the largest scale without superluminal expansion. Notably, the least time evolution results in history, i.e., unique trajectories. This path-dependence of nascent evolution manifests itself as correlated multipoles in angular decomposition of cosmic microwave background radiation. These characteristics of the least-time universal evolution distinguish from those resulting from the standard cosmological model.
... Satisfaction of the Einstein field equations is a necessary but insufficient condition for determination of Einstein's gravitational field. Einstein's field equations "in the absence of matter" [29] are, ...
The LIGO Scientific Collaboration and Virgo Collaboration have announced [1] that on the 14th of September 2015, at 09:50:45 UTC, they detected a transient Einstein gravitational wave, designated GW150914, produced by two merging black holes forming a single black hole. Not so long ago similar media excitement surrounded the announcement by the BICEP2 Team of detection of primordial gravitational waves imprinted in B-mode polarisations of a Cosmic Microwave Background, which proved to be naught. The two black holes that merged are reported to have been at a distance of some 1.3 billion light years from Earth, of �29 solar masses and �36 solar masses respectively, the newly formed black hole at �62 solar masses, radiating away �3 solar masses as Einstein gravitational waves. The insurmountable problem for the credibility of the LIGO-Virgo Collaboration claims is the falsity of the theoretical assumptions upon which they are based.
... Three years from now, in 2016, we will celebrate the century of the publication of Einstein's General Relativity (Einstein, 1916), event that can be considered as the starting point of cosmology. Scientists got really excited by this new concept of a non-static cosmos and it gave rise to many theories and predictions as well as observational campaigns. ...
p>This work focuses on the use of deep, wide field surveys to extract new cosmo- logical information. Precise photometry plays a large role in this quest, through the determination of photometric redshifts and the propagation of photometric errors into the cosmological results. This is a unifying theme which effectively ties both parts of the thesis together.
After a general review of cosmology and the measurements that support the ΛCDM model, and a description of the Large Synoptic Survey Telescope (LSST), the first part of this work deals with the influence of the variation of main atmospheric con- stituents on ground-based photometry, focusing particularly on the LSST site at Cerro Pachón, Chile. We process all recent available data on ozone, water vapor and aerosols to construct a long-term atmospheric simulation and estimate quanti- tatively how the spatial and temporal gradients of these constituents would affect LSST calibration process.
The second part of this work starts with a theoretical description of gravitational lensing, concentrating on the weak lensing aspect. After discussing the advantages and difficulties of cosmic shear measurements, we explore the use of cosmic magni- fication, together with the redshift tomography enabled by the LSST, to constrain cosmological models. We find that cosmic magnification covariance is beset by in- trinsic clustering but nevertheless represents a useful probe of galaxy bias and dark energy that complements cosmic shear, and which can increase the robustness of cosmological constraints from lensing surveys. </p
... The proof of the energy-momentum conservation law, T µν ;ν = 0, is new. This law was explicitly rejected in the past [6][7][8], which gave rise to a host of ambiguous claims in the literature. Therefore, the proof given here is a crucial step forward. ...
It is first shown that there is no exchange of energy-momentum during planetary motion. Then, starting with the field equations, it is shown that the gravitational field does not exchange energy-momentum with any form of matter. Conclusion: In general relativity, there is no gravitational energy, momentum, stress, force or power.
... For A ∈ C I 1 ×···×I N ×K 1 ×···×K N and B ∈ C K 1 ×···×K N ×J 1 ×···×J M , the Einstein product [12] of tensors A and B is defined by the operation * N via ...
The notion of the Moore–Penrose inverse of tensors with the Einstein product was introduced, very recently. In this paper, we further elaborate on this theory by producing a few characterizations of different generalized inverses of tensors. A new method to compute the Moore–Penrose inverse of tensors is proposed. Reverse order laws for several generalized inverses of tensors are also presented. In addition to these, we discuss general solutions of multilinear systems of tensors using such theory.
... Then the advances in atomic physics and in space navigation led to a revolution in metrology with the elimination of these old idealized notions and their replacements with realistic standards. For instance one can compare the 1965 point of view of Ref. (Basri S.A., 1965) with the 1997 one of Ref. (Guinot B., 1997). ...
... In the 20 century, two great physical theories, quantum theory and relativity, led two standard models respectively, the micro-particle standard model characterized by SU C (3)×SU L (2)×U Y (1) gauge field based on quantum theory and the macro-cosmos standard model characterized by big-bang cosmology based on general relativity [1], which construct theoretical core of contemporary physics [2]. However, the two standard models are so different that they cannot be unified within a harmony theory, which caused a theoretical poser called as quantum gravity. ...
A mathematical multi-vector consisted of a complex scalar, a complex vector and a bivector, which constructs a physical linked-measure, yielding a linked-field. When the linked-measure is applied as the world measure, its strong symmetric links generate electromagnetic field and its strong micro-inner links do strong field, while its weak micro-inner symmetric links synthesize electroweak field. With adding outer space-time metric, the linked-field leads to gravitational field with a new understanding of dark matter and dark energy. In the linked-field, the micro-particle standard model and the macro-cosmos standard model are unified, where double dynamic sources drive the universe, in which one initialized big-bang and another pushed rotation. Dark matter and dark energy are structural effects of the whole universe, caused respectively by rotation and big-bang. A rotated vacuum explosive experiment is suggested to verify the hypothesis.
... In the 20 century, two great physical theories, quantum theory and relativity, led two standard models respectively, the micro-particle standard model characterized by SU C (3)×SU L (2)×U Y (1) gauge field based on quantum theory and the macro-cosmos standard model characterized by big-bang cosmology based on general relativity [1], which construct theoretical core of contemporary physics [2]. However, the two standard models are so different that they cannot be unified within a harmony theory, which caused a theoretical poser called as quantum gravity. ...
A mathematical multi-vector consisted of a complex scalar, a complex vector and a bivector, which constructs a physical linked-measure, yielding a linked-field. When the linked-measure is applied as the world measure, its strong symmetric links generate electromagnetic field and its strong micro-inner links do strong field, while its weak micro-inner symmetric links synthesize electroweak field. With adding outer space-time metric, the linked-field leads to gravitational field with a new understanding of dark matter and dark energy. In the linked-field, the micro-particle standard model and the macro-cosmos standard model are unified, where double dynamic sources drive the universe, in which one initialized big-bang and another pushed rotation. Dark matter and dark energy are structural effects of the whole universe, caused respectively by rotation and big-bang. A rotated vacuum explosive experiment is suggested to verify the hypothesis.
... Einstein's gravitational field is not matter. According to Einstein [1], " We make the distinction hereafter between 'gravitational field' and 'matter' in this way, that we denote everything but the gravitational field as 'matter'. " ...
Cosmologists always claim that their black holes, mathematical fabrications entire as they are, have a finite mass. This mass, they say, is concentrated at their ‘singularity’, where volume is zero, density is infinite, and their spacetime infinitely curved. Their singularity they say is not a mathematical artifice, not a limiting fiction, but a real physical entity which absorbs all other matter that encounters it. In this way cosmologists have even asserted that their black holes can become obese. Their black hole was first conjured from their solution to what Einstein called The Field Equations of Gravitation in the Absence of Matter. Notwithstanding the absence of matter, Einstein claimed that a material source is still present, because his gravitational field is spacetime curvature induced by the presence of a material source. Without matter there is no gravitational field. And what is matter? According to Einstein it is everything except his gravitational field. Now there is only one other form of Einstein’s field equations: The Field Equations of Gravitation in the Presence of Matter. Thus, in both cases, Einstein and his followers claim that a material source is present. However, in a mathematical theory, matter cannot be both present and absent by the very same mathematical constraint. Cosmologists routinely call their contradictions ‘paradoxes’ that defy ‘common sense’. Nonetheless, common sense does in fact know that a contradiction is a contradiction; no less than a rose by any other name is still a rose. Not only does the black hole defy common sense, it defies physics and mathematics.
The conventionality of simultaneity thesis as established by Reichenbach and Grünbaum is related to the partial freedom in the definition of simultaneity in an inertial reference frame. An apparently altogether different issue is that of the conventionality of spatial geometry, or more generally the conventionality of chronogeometry when taking also into account the conventionality of the uniformity of time. Here we will consider Einstein’s version of the conventionality of (chrono)geometry, according to which we might adopt a different spatial geometry and a particular definition of equality of successive time intervals. The choice of a particular chronogeometry would not imply any change in a theory, since its “physical part” can be changed in a way that, regarding experimental results, the theory is the same. Here, we will make the case that the conventionality of simultaneity is closely related to Einstein’s conventionality of chronogeometry, as another conventional element leading to it.
Universe history in -gravity is studied from "beginning" up to the present epoch. It is assumed that initially the curvature scalar R was sufficiently large to induce the proper duration of inflation. Gravitational particle production by the oscillating R(t) led to a graceful exit from inflation, but the cosmological evolution in the early universe was drastically different from the standard one till the universe age reached the value of the order of the inverse decay rate of the oscillating curvature R(t). This deviation from the standard cosmology might have a noticeable impact on the formation of primordial black holes and baryogenesis. At later time, after exponential decay of the curvature oscillations, cosmology may return to normality.
One of the greatest scientists Albert Einstein laid the foundation of Special and General Relativity. The creation of General Relativity is considered one of the greatest discoveries in the 20th century. General Relativity is the representation of the physical universe in the form of a four-dimensional space-time manifold. The General Relativity, the seminal work of Einstein, is the basis of modern cosmology. Einstein used a cosmological constant
to represent the universe static. He has prepared the model of the universe on the basis that the universe is static, isotropic and homogeneous. In this study the steady state model of the Einstein universe is discussed in some details. The purpose of this article is to highlight the aspects of the Einstein’s universe with easier mathematical analysis.
The covariant, canonical and path integral approach trilemma.
LIGO的GW150914信号有可能是地面机车启动信号经过缩小处理后的结果，也有可能是实验生成的信号经过缩小处理后的结果，而决不可能是螺旋双星合并的引力波信号。这里介绍GW15094信号的实验生成方法。
One of the greatest scientists Albert Einstein laid the foundation of Special and General Relativity. The creation of General Relativity is considered one of the greatest discoveries in the 20th century. General Relativity is the representation of the physical universe in the form of a four-dimensional space-time manifold. The General Relativity, the seminal work of Einstein, is the basis of modern cosmology. Einstein used a cosmological constant to represent the universe static. He has prepared the model of the universe on the basis that the universe is static, isotropic and homogeneous. In this study the steady state model of the Einstein universe is discussed in some details. The purpose of this article is to highlight the aspects of the Einstein’s universe with easier mathematical analysis.
We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (M) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (M) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.
The understanding of nature has been radically changed by the advent of quantum mechanics and general relativity.
Considering clocks as physical systems in quantum mechanics allows for operationally well-defined scenarios where proper time can display quantum properties—i.e. a single clock can run different proper-times in superposition or two clocks can be entangled in their proper times, as discussed in the preceding chapters. This chapter discusses the regime where quantum theory has to be applied also to the massive systems–which according to General Relativity define casual relations between events.
An energy condition, in the context of a wide class of spacetime theories (including general relativity), is, crudely speaking, a relation one demands the stress-energy tensor of matter satisfy in order to try to capture the idea that “energy should be positive”. The remarkable fact I will discuss in this paper is that such simple, general, almost trivial seeming propositions have profound and far-reaching import for our understanding of the structure of relativistic spacetimes. It is therefore especially surprising when one also learns that we have no clear understanding of the nature of these conditions, what theoretical status they have with respect to fundamental physics, what epistemic status they may have, when we should and should not expect them to be satisfied, and even in many cases how they and their consequences should be interpreted physically. Or so I shall argue, by a detailed analysis of the technical and conceptual character of all the standard conditions used in physics today, including examination of their consequences and the circumstances in which they are believed to be violated.
IoT exhibits characteristics of the presence of diverse physical sensing and actuating devices, complex wireless communication and networking technologies, as well as large-scale heterogeneous data generated in the physical and cyber worlds. The exponentially increasing volume of data places an unprecedent burden on the network infrastructure of IoT systems, where there are two key challenges: how to represent the heterogeneous IoT data as a concise and unified model, and extract the essential core data that are smaller for transmission but consist of the most valuable information; and how to globally and flexibly control the network devices, and dynamically reallocate the bandwidth to improve the communication link utilization ratio. To address the mentioned challenges, this article first transforms structured, semi-structured, and unstructured IoT data to a unified tensor model, and employs the HO-SVD approach for extraction of the high-quality core data. Then this article applies SDN technology to IoT for device management, and develops a transition tensor model for routing path recommendation. Finally, a smart home case study is investigated, which reveals that the proposed tensor-based software defined model is feasible and promising. It is strongly suggested that further study on combination of IoT with SDN technology and tensor algebra should be performed.
This article is conceived as an intellectual patchwork of visions and notions, which are considered focal to the development of a posthuman cosmo-ontology. It wishes to experiment in its form, as well as in its contents, in line with the post-dualistic attitude of the posthuman: the " how " is the " what ". It is inspired by Nietzsche's Thus Spoke Zarathustra (1883-85), as well as by lived practice and quantum physics. It focalizes on the significance of the eternal recurrence in relation to the reiteration of patterns of existence and to the hypothetical memory of spacetime. It is a work in progress, aiming to spark original connections and a different set of hermeneutical possibilities. Like pieces of a philosophical puzzle, each subchapter makes full sense after reading the complete article. It is advised to approach this text with scientific joy and existential curiosity.
Quantum entanglement has been visualized for the first time, in view of the spin density distribution and electronic structure for manganese in manganese(II)hexakisimidazole nitrate. Using polarized neutron diffraction and density functional theory modelling we have found for the complex, which crystallizes in the R spacegroup, a = b = 12.4898(3) Å, c = 14.5526(4) Å, α = γ = 90°, β = 120°, Z = 3, that spatially antisymmetric and spatially symmetric shaped regions of negative spin density, in the spin density map for manganese, are a result of quantum entanglement of the high spin d5 configuration due to dative imidazole- manganese π- donation and σ-bonding interactions respectively. We have found leakage of the entangled states for manganese observed as regions of positive spin density with spherical (3.758(2) μB) and non-spherical (1.242(3) μB) contributions. Our results, which are supportive of Einstein's theory of general relativity, provide evidence for the existence of a black hole spin density distribution at the origin of an electronic structure and also address the paradoxical views of entanglement and quantum mechanics. We have also found the complex, which is an insulator, to be suitable for spintronic studies.
It is proposed that the forces between manifest particles are mediated by chains of virtual pair production processes in a plenum of unmanifest vacuum fermions. The world lines of both virtual and real bosons are identified with such sequences, each step coupled to the next by the requirement for macroscopic conservation of energy and momentum. If the short-lived pairs are identified with positronium in the case of electromagnetic force, the coupling strength is controlled by the fine structure constant. Application of first-order perturbation (or scattering) theory yields an inverse-square law incorporating quantized space and time intervals that reflect the density of vacuum fermions. The occurrence of these intervals eliminates infinite self-energies and allows for a number of broken symmetries that yield force laws with an apparent noninverse character (aside from the noninverse square character that arises from relativistic transformations). An important feature of the model is that positive energy mass and charge depress the surrounding vacuum density. This means that propagating pair formation processes must be accompanied by compression-expansion waves of vacuum density, yielding a wave aspect of radiation. The depression of vacuum density produced by mass may be identified with space-curvature, and it is shown that the model incorporates Mach’s principle and the principle of equivalence. Strictly speaking, vacuum particles cannot have mass since they are not absorbers; however, they can inherit mass by acting as transient absorbers. These masses and the structure of the vacuum density induced by the distribution of positive energy mass and charge must satisfy self-consistent equilibrium relations. The time evolution of any system in which these relations are disturbed must exhibit fundamental irreversibility due to the fact that the classical specification of force depends on the vacuum density structure, hence on the wave function of the vacuum. This underlying irreversibility is negligible in simple physical systems, but would be unmasked by the sensitive dynamics of biological systems and measurement instrumentation.
The growing volume of network traffic and gradual deployment of SDN devices initiate a new era in which one distinguished feature is the application of big data technology to SDNs for construction of flexible, scalable, and self-managing networks. The primary purpose of this article is to develop a novel tensor-based model for efficient provisioning of QoS in software defined networks. First, a forwarding tensor model is proposed to formalize the networking functions in the data plane; then a controlling tensor model is presented for routing path recommendation in the control plane. Finally, the article introduces a transition tensor model for network traffic prediction and QoS provisioning. The three models can automatically monitor the network state, recommend routing paths and predict network traffic, respectively. A case study to recommend routing paths is investigated in the article.
Maxwell and Einstein presumed the existence of an ether. In a previous publication, we showed that the Maxwell electrodynamics ensues from an elasticity theory where the elastic medium is the ether denoted CE of which the field of the displacement of its points is denoted xi. These results stimulated us to investigate the physical common nature of the fields and of the particles that eliminate the contradiction and the inconsistency cited by Einstein. To this purpose, we extended this elastic interpretation to the case where the particles can be massive and electrically charged in a gravitational field and an electromagnetic (EM) field. This extension is achieved by showing that the Lagrange Einstein function L-G of such a particle in such fields, not only yields the particle four-motion equation, but also is such that phi, defined by (h) over bard phi/dt = L-G, is the phase of a wave associated to these particles. To the particles are therefore associated waves of phase phi, denoted xi that generalize the EM waves. We show that like the EM waves, the waves xi must be propagated in CE, since when m tends toward zero, a xi wave becomes an EM wave. It follows that a xi wave is also a field of the displacements of the points of CE. The xi waves are of phase velocity V-p and are the solution of an equation that generalizes the EM wave equation, where in particular as we show, the particle trajectory equation generalizes the light ray equation, the fields, and ds are modification of the elastic rotation restoring coefficient of CE which is itself proportional to V-p(2). Then we show that a specific sum of xi waves forms a globule that moves like the particle and contains all its parameters and reciprocally that a xi wave is a sum of such globules, i.e., of particles, we call this property "wave-particle reciprocity property." This property and the elastic nature of particles and of fields that are in accordance with Maxwell's and Einstein's ideas suppress the inconsistency cited by Einstein, and provide a physical interpretation of the quantum mechanics, the duality EM waves-photons that explains the particle photon behavior that causes, e.g., the photo-electric effect. (C) 2013 Physics Essays Publication.
Measuring the solar diameter at all position angles gives the complete figure
of the Sun. Their asphericities have implications in classical physics and
general relativity, and the behavior of the optical systems used in the direct
measurements is to be known accurately. A solar filter is a plane-parallel
glass with given absorption, and here we study the departures from the
parallelism of the faces of a crystal slab 5 mm thick, because of static
deformations. These deformations are rescaled to the filter's dimensions.
Related to the Solar Disk Sextant experiment and to the Reflecting Heliometer
of Rio de Janeiro a simplified model of the influences of the inclination
between the external and the internal surfaces of a glass solar filter, is
discussed.
I extend my thanks to Professor Gerardus ‘t Hooft, Nobel Laureate in Physics, for making more widely known my work on black hole theory, big bang cosmology, and Einstein’s General Theory of Relativity, by means of his personal website, and for providing me thereby with the opportunity to address the subject matter - supported by extensive references to primary sources for further information - in relation to his many comments, by means of this dedicated paper. The extensive mathematical appendices herein are not prerequisite to understanding the text.
We give a consistent quantum description of time, based on Page and Wootters'
conditional probabilities mechanism, that overcomes the criticisms that were
raised against similar previous proposals. In particular we show how the model
allows to reproduce the correct statistics of sequential measurements performed
on a system at different times.
In the present thesis we have found constraints on the fundamental Planck scale within extra dimensional models, by analyzing the torsion induced four-fermion interaction and the LHC data. Additionally, such an interaction induces corrections to the one-loop observables within the Standard Model (SM) of particle physics, which was used to obtain more stringent constraints on the fundamental Planck scale through the SM precision measurements. Further, it was argued that the Yukawa sector of the SM may arise from the condensation of a fourth family of heavy quarks, through the four-fermion interaction in gravity with torsion. This approach gives rise to a composite Higgs boson. Finally, we considered two different scenarios of torsion-descended axions and demonstrated that they are equivalent from the effective theory point of view. We studied the solution to the strong CP problem using such kind of torsion-descended axions and examined some of their cosmological implications.
Reynolds' paper sought to explain the change in character of flow through a pipe from laminar to turbulent that his earlier experiments had shown to occur when the dimensionless group that today bears his name exceeded approximately 2000. This he did by decomposing the velocity into mean and fluctuating components and noting how the average kinetic energy generation and dissipation rates changed with Reynolds number. The paper was only grudgingly accepted by two very distinguished referees and initially raised little external interest. As years went by, however, the averaged form of the equations of motion, known as the Reynolds equations (which were an intermediate stage in Reynolds' analysis) became the acknowledged starting point for computing turbulent flows. Moreover, some 50 years after his paper, a refinement of his strategy for predicting transition was also successfully taken up. For some engineering problems, the continual rapid growth of computing resources has meant that more detailed approaches for computing turbulent flow phenomena can nowadays be employed. However, this growth of computing power likewise makes possible a Reynolds-averaging strategy for complex flow systems in industry or the environment which formerly had to adopt less comprehensive analyses. Thus, Reynolds' approach may well remain in use throughout the present century. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.
The notion of observable in general relativity differs profoundly from the corresponding concept in special relativity and in Newtonian physics. This is because in general relativity fixation of a frame of reference and gauge transformations are intertwined in a manner not encountered in any other area of physics. In other field theories an observable is any (local) component of a field that is gauge-invariant, such as a component of the electric field strength at a stated instant in time and at a given space point. Of course, functions, or functionals of such observables are themselves observables.
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Using an ultracentrifuge rotor, the shift of the 14.4-keV Mössbauer absorption line of Fe57 in a rotating system was measured as a function of the angular velocity omega. An Fe57 absorber was placed at a radius of 9.3 cm from the axis of the rotor. A Co57 source was mounted on a piezoelectric transducer at the center of the rotor. By applying a triangularly varying voltage to the transducer, the source could be moved relative to the absorber. This arrangement makes possible the observation of the entire resonance line at various values of omega. The measured transverse Doppler shift agrees within an experimental error of 1.1% with the predictions of the theory of relativity. Possible sources of systematic errors are discussed.
The construction of a complete set of quantities in general relativity, whose functional form is invariant under coordinate transformations, is indicated. The set obtained is highly redundant. The Cauchy problem for obtaining an independent complete set of such quantities ("observables") is therefore discussed. It is also pointed out that the observables obtained may alternatively be viewed as the metric tensor in a special "gauge" (i.e., with a special coordinate condition). This latter viewpoint may facilitate the quantization of general relativity.
A new approach is presented for defining and obtaining rigid frames of reference. The results are shown to be equivalent to those of Rosen. The advantage of the present approach is that exact solutions can be obtained in certain simple cases, as well as approximate solutions in general.
The effect of gravitational potential on the apparent frequency of ; electro-magnetic radiation was measured by using the sharply defined energy of ; recoil-free 14.4kev gamma rays emitted by Coâµâ· and absorbed in Feâµâ·. ; The mean gravitational shift in frequency was measured to be (-17.6 plus or ; minus 0.6) x l0¹âµ. (C.J.G.);
The results of an investigation into the relation between the time ; measurements performed in an inertial frame and in an accelerated frame are ; presented. A wholly kinematic and not a gravitational scheme is used in this ; study of accelerated frames in empty space-time. (R.E.U.);
It is argued that Einstein's Theory of General Relativity as it stands incorporates Mach's Principle. The boundary conditions for Machian solutions are stated in a coordinate system in which the cosmological background is described by a conformally flat metric. The metric tensor gμν is then written as a product of the scalar density ϕ2 and a tensor density γμν with unit determinant. In the coordinate system that has been so chosen ϕ describes the cosmological structure, while γμν refers to gravitational phenomena. This becomes clear when Einstein's fundamental equations are rewritten in terms of ϕ and γμν. Then κϕ−1 is seen to play the role of the gravitational constant instead of κ in the weak field approximation. The quantity κϕ−1 can be expressed in terms of the radius and the total mass of the universe and the sign of the forces between inhomogeneities of the metric is determined by the requirements of Mach's principle. The forces which derive from ϕ are found to be repulsive for the cosmological background, leading to the expansion of the universe, while attractive gravitational forces arise from the deviations of γμν from the Minkowski metric. Various statements associated with Mach's Principle are discussed in the light of this reformulation of Einstein's Theory.
It is shown that metric measurements, which are essential to the theory of general relativity, are those based only on strong
and electromagnetic interactions. Strong and electromagnetic interactions must not vary throughout the four-dimensional world,
if general relativity is to remain a meaningful theory. The implications of these remarks on cosmology are discussed.
Si fa vedere che le misure di intervalli spaziali e temporali, che sono essenziali per la teoria della relatività generale,
sono quelle basate esclusivamente sulle interazioni forte e elettromagnetica. La consistenza logica della relatività generale
richiede che le interazioni forte e elettromagnetica non variino nello spazio-tempo. Le consequenze di queste osservazioni
in cosmologia sono prese in esame.
Torsion balance measurements of the difference in ratios of gravitational to inertial mass for different materials have been carried out, confirming to higher precision the null results obtained 60 years ago by Eötvös and assumed by Einstein as the Principle of Equivalence upon which the General Theory of Relativity is founded. If the parameter η(A, B) is defined as where M and m represent the passive gravitational and inertial masses respectively of materials A and B, then the results from the most sensitive torsion balance used enable us to conclude with 95% confidence that |η(Au, Al)| < 3 × 10−11. Stated more exactly, the various measurements of η, obtained from the gravitational acceleration toward the sun, gave a substantially Gaussian distribution with mean value η(Au, Al) = (1.3 ± 1.0) × 10−11. The probable error quoted for the mean is based upon the observed scatter in results from individual data runs, assuming a Gaussian distribution. The importance of the Eötvös experiment to contemporary gravitational theories is discussed, and the earlier measurements of Eötvös and J. Renner are examined critically. The torsion balance and associated equipment used in the present experiment are described in detail, along with the considerations involved in their design. Methods of data analysis are also discussed extensively and tables of individual results are presented.
Various statements identified with the principle of equivalence are not acceptable because they are either not generally valid or are simply the definition of an inertial coordinate system. The accepted statement refers, roughly speaking, to the equivalence of inertial mass with passive and active gravitational mass. The clarification of this problem is greatly eased by a detailed study of the static homogeneous gravitational field which precedes the discussion of the equivalence principle. The role of electromagnetic phenomena, and, in particular, the presence of a charge in such a field is analyzed in detail.