Relativistic Perihelion Research Articles

The Weak Field Approximation of General Relativity and the Problem of Precession of the Perihelion for Mercury

In this paper we represent a different approach to the calculation of the perihelion shift than the one presented in common text books. We do not rely on the Schwarzschild metric and the Hamilton–Jacobi technique to obtain our results. Instead we use a weak field approximation, with the advantage that we are not obliged to work with a definite static metric and can consider time dependent effects. Our results support the conclusion of Křížek regarding the significant influence of celestial parameters on the indeterminacy of the perihelion shift of Mercury’s orbit. This shift is thought to be one of the fundamental tests of the validity of the general theory of relativity. In the current astrophysical community, it is generally accepted that the additional relativistic perihelion shift of Mercury is the difference between its observed perihelion shift and the one predicted by Newtonian mechanics, and that this difference equals 43″ per century. However, as it results from the subtraction of two inexact numbers of almost equal magnitude, it is subject to cancellation errors. As such, the above accepted value is highly uncertain and may not correspond to reality.

Relativistic perihelion shift of Mercury revisited

AbstractIn 1915, Albert Einstein derived a formula for an additional relativistic perihelion shift of Mercury. We give several critical comments of how this formula is treated, since the perihelion shift of Mercury is an ill‐conditioned problem. We also present various sources of nonnegligible errors, which should be taken into account.

The detection of relativistic corrections in cosmological N-body simulations

Cosmological N-body simulations are done on massively parallel computers. This necessitates the use of simple time integrators, and, additionally, of mesh-grid approximations of the potentials. Recently, Adamek et al. (2015); Barrera-Hinojosa et al. (2019) have developed general relativistic N-body simulations to capture relativistic effects mainly for cosmological purposes. We therefore ask whether, with the available technology, relativistic effects like perihelion advance can be detected numerically to a relevant precision. We first study the spurious perihelion shift in the Kepler problem, as a function of the integration method used, and then as a function of an additional interpolation of forces on a 2-dimensional lattice. This is done for several choices of eccentricities and semi-major axes. Using these results, we can predict which precisions and lattice constants allow for a detection of the relativistic perihelion advance in N-body simulation. We find that there are only small windows of parameters---such as eccentricity, distance from the central object and the Schwarzschild radius---for which the corrections can be detected in the numerics.

A simple exact series representation for relativistic perihelion advance

The small discrepancy between the observed orbit of Mercury and the orbit predicted by Newtonian gravity was a key test of Einstein's theory, and a dramatic verification of the correctness of General Relativity. This `anomalous precession' of the perihelion is here calculated using a particularly simple method and resulting in a new and elegant series for the precession which converges very quickly.

Computation of the General Relativistic Perihelion Precession and of Light Deflection via the Laplace-Adomian Decomposition Method

We study the equations of motion of the massive and massless particles in the Schwarzschild geometry of general relativity by using the Laplace-Adomian Decomposition Method, which proved to be extremely successful in obtaining series solutions to a wide range of strongly nonlinear differential and integral equations. After introducing a general formalism for the derivation of the equations of motion in arbitrary spherically symmetric static geometries and of the general mathematical formalism of the Laplace-Adomian Decomposition Method, we obtain the series solution of the geodesics equation in the Schwarzschild geometry. The truncated series solution, containing only five terms, can reproduce the exact numerical solution with a high precision. In the first order of approximation we reobtain the standard expression for the perihelion precession. We study in detail the bending angle of light by compact objects in several orders of approximation. The extension of this approach to more general geometries than the Schwarzschild one is also briefly discussed.

Comparing Relativistic Theories Against Observed Perihelion Shifts of Icarus and Mercury

This paper compares the post-Newtonian approximation (PNA) to general relativity (GR) for the relativistic perihelion shift calculations. Nelson’s PNA predicts 5/6 of GR’s perihelion shift. Using the original Universal Time (UT), Shapiro’s accurate, highly elliptical orbit for Icarus corroborates PNA while GR exceeds the error boundary. The Icarus result was λ = 0.75 ± 0.08 where λ=1 for GR and λ=0 for Newtonian theory. Studies of Mercury’s perihelion shift used timescales equivalent to lunar Ephemeris Time (ET) with the present Système International (SI) second, the basic time unit for all atomic timescales like International Atomic Time (TAI). Atomic timescales run faster than UT, because the SI second is 2.468E-8 s shorter than the original UT second. This is confirmed by the two observational reports using the original calibration data of 1955-1958, by the Improved Lunar Ephemeris used in the original calibration, by the linear divergence of TAI versus UT during 1958-1998, and by the 2.1 ms mean excess between a UT day and TAI day during 1958-1998. Time dilation was not included in the lunar theory, which is confirmed by timekeeping authorities. So, the undilated lunar ET second is shorter than Earth’s proper UT second. An ET timescale creates an additional, artificial perihelion shift for Mercury of 6.433”/cy. Other renowned relativists used a 1973 update for Earth’s general precession that now excludes the GR prediction while including the PNA prediction if the artificial Mercury shift is included in the calculations. Apparently, Nelson’s PNA is more accurate than GR.

English

The history of experimentation relevant to general relativity covers the time post-1928. Classes of investigation are the weak equivalence principle (equivalence of inertial and gravitational mass and gravitational redshift), orbital precession of a body in gravitational fields (the relativistic perihelion advance of the planets, the relativistic periastron advance of binary pulsars, geodetic precession and Lense-Thirring effect), light propagation in gravitational fields (gravitational optical light deflection, gravitational radio deflection due to the Sun, gravitational lensing, time dilation and atomic clocks) and strong gravity implications (Nordtved effect and potential gravitational waves). The results of experiments are analysed to conclude to what extent they support general relativity. A number of questions are then answered: (a) how much evidence exists to support general relativity, (b) is it a reasonable way of thinking and (c) what is the niche it may occupy?

Higher-order corrections to the relativistic perihelion advance and the mass of binary pulsars

We study the general relativistic orbital equation and using a straightforward perturbation method and a mathematical device first introduced by d'Alembert, we work out approximate expressions of a bound planetary orbit in the form of trigonometrical polynomials and the first three terms of the power series development of the perihelion advance. The results are applied to a more precise determination of the total mass of the double pulsar J0737-3039.

Relativistic perihelion precession of orbits of Venus and the Earth

Abstract Among all the theories proposed to explain the “anomalous” perihelion precession of Mercury’s orbit first announced in 1859 by Le Verrier, the general theory of relativity proposed by Einstein in November 1915 alone could calculate Mercury’s “anomalous” precession with the precision demanded by observational accuracy. Since Mercury’s precession was a directly derived result of the full general theory, it was viewed by Einstein as the most critical test of general relativity from amongst the three tests he proposed. With the advent of the space age, the level of observational accuracy has improved further and it is now possible to detect this precession for other planetary orbits of the solar system — viz., Venus and the Earth. This conclusively proved that the phenomenon of “anomalous” perihelion precession of planetary orbits is a relativistic effect. Our previous papers presented the mathematical model and the computed value of the relativistic perihelion precession of Mercury’s orbit using an alternate relativistic gravitational model, which is a remodeled form of Einstein’s relativity theories, and which retained only experimentally proven principles. In addition this model has the benefit of data from almost a century of relativity experimentation, including those that have become possible with the advent of the space age. Using this model, we present in this paper the computed values of the relativistic precession of Venus and the Earth, which compare well with the predictions of general relativity and are also in agreement with the observed values within the range of uncertainty.

LISA capture sources: Approximate waveforms, signal-to-noise ratios, and parameter estimation accuracy

Captures of stellar-mass compact objects (COs) by massive $(\ensuremath{\sim}{10}^{6}{M}_{\ensuremath{\bigodot}})$ black holes (MBHs) are potentially an important source for LISA, the proposed space-based gravitational-wave (GW) detector. The orbits of the inspiraling COs are highly complicated; they can remain rather eccentric up until the final plunge, and display extreme versions of relativistic perihelion precession and Lense-Thirring precession of the orbital plane. The amplitudes of the strongest GW signals are expected to be roughly an order of magnitude smaller than LISA's instrumental noise, but in principle (i.e., with sufficient computing power) the GW signals can be disentangled from the noise by matched filtering. The associated template waveforms are not yet in hand, but theorists will very likely be able to provide them before LISA launches. Here we introduce a family of approximate (post-Newtonian) capture waveforms, given in (nearly) analytic form, for use in advancing LISA studies until more accurate versions are available. Our model waveforms include most of the key qualitative features of true waveforms, and cover the full space of capture-event parameters (including orbital eccentricity and the MBH's spin). Here we use our approximate waveforms to (i) estimate the relative contributions of different harmonics (of the orbital frequency) to the total signal-to-noise ratio, and (ii) estimate the accuracy with which LISA will be able to extract the physical parameters of the capture event from the measured waveform. For a typical source (a ${10M}_{\ensuremath{\bigodot}}$ CO captured by a ${10}^{6}{M}_{\ensuremath{\bigodot}}$ MBH at a signal-to-noise ratio of 30), we find that LISA can determine the MBH and CO masses to within a fractional error of $\ensuremath{\sim}{10}^{\ensuremath{-}4},$ measure ${S/M}^{2}$ (where S and M are the MBH's mass and spin) to within $\ensuremath{\sim}{10}^{\ensuremath{-}4},$ and determine the location to the source on the sky to within $\ensuremath{\sim}{10}^{\ensuremath{-}3}$ stradians.

Relativistic perihelion advance as a centrifugal effect

Relativistic perihelion advance as a centrifugal effect

Where is the energy stored in a gravitational field?

The meaning and location of energy density in gravitational fields is considered in analogy with the energy density in electrostatic fields. The fact that energy density of gravitational fields acts as a source of gravitation makes the localization, in principle, an observable possibility. A nonlinear Newtonian theory of gravity is explored, and the sensitivity of the predicted perihelion precession to the location of energy density is discussed. Problems associated with identification of gravitational energy density in general relativity are pointed out, together with the similar effects generated by gravitational stresses. Comparison with the nonlinear Newtonian theory allows one to determine the various effects which contribute to the general relativistic perihelion precession.

Testing general relativity: Progress, problems, and prospects

Recent results from radar and radio gravity experiments are: (1) Retardation of radar signals: 1.02 +−0.05 (planetary radar), 1.00 +− 0.04 (spacecraft); (2) Deflection of radio waves: 0.99 +− 0.12 (short baseline interferometry), +0.15 1.04 (short baseline interferometry), −0.10. 0.90 +− 0.05 (short baseline interferometry); (3) Relativistic perihelion advance of Mercury: 0.99 +− 0.03 (planetary radar), 1.00 +− 0.01 (radar plus optical); (4) Time variation of the gravitational constant: (G/G) < 4 ×10−10/yr (planetary radar). The first three results are expressed as fractions of the corresponding effects predicted by general relativity. In terms of the Eddington-Robertson parameters, the first two are approximately equal to (1+γ)/2 and the third to (2 + 2γ-β)/3. The third result depends on the assumption that the solar gravitational quadrupole moment vanishes; its effects cannot be separated usefully from those of general relativity with the present data set. The uncertainties are as given by the individual experimenters; in some cases they represent formal standard errors, in others an allowance for possible systematic errors is included.

Relativistic Perihelion Shift Refined after Icarus Approach

The asteroid Icarus passed within 6 million km of the earth on 14 June and was studied as never before in the 19 years since its discovery. Its highly eccentric orbit makes it suitable for studying the relativistic precession of its perihelion, and its small size and close approach to the earth allow accurate determination of its position.

Icarus: Further Confirmation of the Relativistic Perihelion Precession.

Icarus: Further Confirmation of the Relativistic Perihelion Precession.

Separation of the Relativistic Perihelion Precession from the Precession due to the Gravitational Quadrupole Moment

The gravitational quadrupole moment is considered insofar as its effect on the perihelion precession of Mercury and pericenter precession of satellites is concerned. It is found possible to separate the quadrupole precession from the precession due to the general theory of relativity. As a corollary, it is found that an exchange of angular momentum between planet and sun takes place due to the quadrupole moment. This exchange may help to explain the phenomenon of sunspots and solar flares.