Gravitation

      A singularity in the integration invalidated Albert Einstein's paper, "Erklärung Perihelbewegung des Merkur aus". Einstein made an unnecessary approximation on the geodesic equation to calculate the angle of precession for the orbit of Mercury.

Unknown to Einstein, there is singularity in the integration which produces an infinite number for the angular difference between the aphelion and the perihelion.

      There is singularity in the geodesic at the aphelion and the perihelion. The angular difference between the perihelion and the aphelion is infinity.

Karl Schwarzschild's metric, derived from Albert Einstein's theory of gravitation, is not compatible with the orbit of any planet in the solar system. The space-time geometry is not a representation of gravitation.

      The orbit of Mercury according to NASA's data is different from any geodesic in Schwarzschild manifold.  There are two geodesic constants for Schwarzschild's metric. Both constants can be calculated from NASA's data for Mercury. The calculation shows that there are two singularities in the geodesic at both the aphelion and the perihelion.

    Schwarzschild's metric, derived from Albert Einstein's theory of gravitation, is not compatible with the orbit of Mercury. The space-time geometry is not a representation of gravitation.

      The  contracted Bianchi identities in the Riemannian geometry are not equivalent to the covariant derivatives of energy tensor in Albert Einstein's theory of general relativity.

  Both Kerr metric and Schwarzschild metric show that the energy tensor for vacuum is not always zero.

  The metric contradicts the theory of general relativity by proving that energy momentum tensor is not covariant.

The energy tensors of the metrics from four vacuum solutions of general relativity are calculated for each metric. The calculation shows that most of the energy tensors and the contra tensors are not zero for vacuum. 

The four metrics are Schwarzschild metric, Reissner–Nordström metric, Kerr metric and Newman metric.

Another error in Albert Einstein's paper, "Erklärung Perihelbewegung des Merkur aus", invalidated his confirmation of the precession of perihelion for Mercury. Einstein included an extra term in an equation to calculate the angle of precession.

Unknown to Einstein, the equation contradicts Schwarzschild's equations. Einstein was using the wrong equation.

An error in Albert Einstein's paper, "Erklärung Perihelbewegung des Merkur aus", renders his confirmation of the precession of perihelion for Mercury invalid. Einstein made an assumption on an equation to calculate the angleof precession.

 Unknown to Einstein, the assumption caused two roots of this equation to become imaginary numbers.

   Space-time geometry started with Henri Poincare's Lorentz transformation. It was adapted by Hermann Minkowski and adopted by Albert Einstein for gravitation. Marcel Grossmann incorporated Ricci tensor into the space-time geometry. David Hilbert used Ricci scalar as Lagrangian to show that the time coordinate is not suitable for the elapsed time. Hermann Weyl also avoided time coordinate for the elapsed time. 

  Eric Su proved that Newton's gravity can be represented by Schwarschild's metric only if the proper time is used for the elapsed time.

  In Kerr's manifold, the null geodesic is not deflected by the gravity. The black hole in Kerr's manifold is not equivalent to the black hole in the universe. The manifold fails to represent the gravitation and the universe. Time is not in the space-time geometry.

  Hermann Weyl derived a metric for a static mass in an isotropic manifold. His metric is not equivalent to Karl Schwarzschild's metric which is the only valid solution to Albert Einstein's field equation for Hermann Minkowski's space-time geometry.

  The main error in Weyl's metric is his equation of motion with three positive roots while there are only two positive roots in Schwarzschild's metric. 

 Another related error is that the perihelion and aphelion can not co-exist in Schwarzschild's metric but there is geodesic for both in Weyl's metric.

In Kerr manifold, all null geodesics except the radial geodesic are confined within a sphere with radius proportional to the mass at the origin. There is no gravitational deflection on light from a remote source. 

The radial geodesic extends from the infinity toward the origin with a constant radial velocity. There is no gravity effect on the radial velocity  of the radial geodesic. There is no event horizon. The radial speed is negative and toward the origin. The null geodesic is circular if it forms a closed path.

The black hole in Kerr manifold is very different from the black hole in the universe.

    In Schwarzschild manifold, the radial geodesic extends between the origin and the infinity in a straight path.

    Light can travel toward and away from the origin. There is no event horizon. The speed of light increases with the mass at the origin. The null geodesic is circular if the geodesic is a closed path. The origin is excluded from all open geodesics. The open geodesic takes the shape of either a parabola or a spiral. Light from the remote star can not fall into the origin.

The singularity at the origin is mathematics, not physics. There is no black hole in Schwarzschild manifold.

The geodesic in Schwarzschild manifold is different from the orbit of Mercury according to the data from NASA. Schwarzschild metric, based on Albert Einstein's theory of gravitation, is not fully compatible with Newton's gravity.

Consequently, the space-time geometry from Minkowski is not a good representation of gravitation.

Kerr metric is a generalized version of Schwarzschild metric with the addition of angular momentum to the mass at the origin. The angular momentum alters the value of a geodesic constant that is associated with the angular velocity. The constant is unique to the geodesic. However, the data from NASA shows that the value of this constant depends on the location of the aphelion and the perihelion for the planet of Mercury. 

Kerr metric is not a realistic representation of the orbit of Mercury unless the unique constant is unique for the whole geodesic. The geodesic in Kerr manifold is very different from the orbit of Mercury because the geodesic is exclusively for the massless object.

For Kerr metric to become Minkowski metric at the infinite radial distance, the radial distance for the perihelion of the null geodesic has to be greater than the angular momentum per unit mass of the mass at the origin. Therefore, there is no null geodesic within this minimum distance.

For light to be represented by the null geodesic, there is no star light passing by the surface of the Sun. According to Kerr metric, there is no gravitational deflection on star light near the surface of the Sun.

The Schwarzschild metric, based on Albert Einstein's theory of gravitation, is not compatible with Newton's gravity nor the data from NASA.

The acceleration in the radial geodesic consists of extra term in addition to gravity. The geodesic is drastically different from the orbit of Mercury.  The trajectory is not an ellipse but a segment of arc. The 

angular difference between the radial directions of the aphelion andthe perihelion is meaningless.

Karl Schwarzschild derived a metric for an isolated system of a single mass point at the origin. The geodesic specified by this isotropic metric exhibits a gravitational deflection caused by the single mass point.

The deflection is the variation in the direction of the tangent velocity of the geodesic connecting a remote star to the earth. For null geodesic, the angle of deflection from the Sun to the earth is approximately 1.0797 arc second.

The commonality among the geodesics from 4 related metrics (Minkowski, Schwarzschild, Su, Kerr) includes:

1) No elliptic geodesic for all metrics. 

2) Limited circular geodesic for all metrics except Minkowski metric. 

3) Radial geodesic for all metrics except Kerr metric. 

4) Limited spiral geodesic for all metric except Minkowski metric. 

The metrics are more suitable for the spiral galaxy than the solar system.

David Hilbert derived a different metric for the same manifold described by Schwarzschild's metric. The geodesic in Hilbert's manifold is a curve in Newton's manifold with radial acceleration similar to Newton's gravity. 

The similarity exists only if time is excluded from Hilbert's metric of Schwarzschild manifold.