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Whitehead contra Einstein
Volume 48A, number 2
PHYSICS LETTERS
WHITEHEAD
3 June 1974
CONTRA EINSTEIN
M. REINHARDT
Astronomische Institute der Universita't Bonn, Germany A. ROSENBLUM
Max-Planck-lnstitut ff~r Physik und Astrophysik, MiJnchen, Germany Received 3 April 1974 None of the classical tests of general relativity can distinguish between Einstein's and Whitehead's theories of gravitation. Claims that geophysical effects exclude Whitehead's theory are shown to be unsubstantiated. Thus Whitehead's theory proves to be one of the so far undefeated rivals of general relativity. A feasible experimentum crueis is delineated.
Presently it is a common belief that the only serious competitor to Einstein's theory of gravitation is the Jordan-Brans-Dicke theory. The purpose o f this note is to show that from the point of view of currently available local observational tests Whitehead's actionat-a-distance theory is also viable, if it is interpreted correctly in terms of modern views of special relativistic versions of field theories of gravitation [ 1 ]. Whitehead's starts by assuming that particles with proper mass move along world lines xU = x u (s), where ds 2 = c 2 d t 2 - d x 2 - d y 2 - d z
2 =~luvdx#dxU,
(1)
is the special relativistiv line element. The adjunct "field" due to all the other particles felt by the particle iis gi.v =
÷
(2)
where = -(mj(2V/c3)/(rJ) 3) with z ~ - xu, / x / being " " the past light " ~uJ - - x~_ m con e o f x * , and I~j = ~!/~0 lu, where ~!~ = dxI# /ds G is the gravitational constant as usual. This means that the "field seen by particle i " is determined by the retarded "fields" of all the other particles. The equation of motion is derived from the variational principle
8fgi~,u(dx~ /dsi) (dx~/dsi)
= O,
(3)
with (dsi) 2 = giuv dx tzidxi"v The equations of motion are formally identical to those of a geodesic in a Riemannian space
d2xU /ds 2i + F~/ (dxX /ds i) (dx'r /ds i) = 0,
(4)
with giuv as a metric tensor and ds i as a line element. Although we have started in Minkowski space the system of particle interactions is described by a Riemannian space• In particular, it was shown by Eddington [2] that the Whitehead line element for the field of a particle at rest can be transformed into the Schwarzschild line element. This implies that the Whitehead theory predicts the same results as general relativity for the classical tests, i.e. gravitational redshift, light deflection, time delay, perihelion shift and gyroscope precission. The fact that Whitehead's equations of motion can be given in the form of eq. (4) means also that Whitehead's theory gives formally the same types o f cosmological solutions as general relativity, since it admits the same group symmetrics. In particular one can reproduce the Robertson-Walker models. Recently it has been claimed [3, 4] that predictions of the Whitehead theory do not agree with certain geophysical effects, namely that there should be an effective anisotropy in the gravitational constant G due to the gravitational attraction of our galaxy. This anisotropy in G should lead to a measurable effect on the tides of the earth which is not observed in high precision gravimeter experiments. Such an effect would not exist in general relativity since it could be removed by a transformation to a local (solar system) reference frame. It has been claimed [3, 4] that one cannot do the same in Whitehead's theory because the 115
Volume 48A, number 2
PHYSICS LETTERS
metric there must be calculated in a global Lorentzian coordinate system or in a space-time of constant curvature. Our purpose is to point out that this is not the case. Only the metric in eq. (4) represents the metric of space-time whereas the Minkowski metric used in formula (2) is only a formal tool used in the computation of the space-time metric in eq. (4). In particular, in the computation of the anisotropy in G, we could ignore the earth for a moment and choose a local frame in which the physical metric g has its Minkowski form, i.e. in which all effects of the galaxy have been removed. We could then use Whitehead's formula to calculate the metric produced by the earth, using this coordinate system. We notice that in the frame as which g has its Minkowski forrl) the Lorentz metric which was used to calculate g and serves no other purpose now has galaxy dependent terms in it. However, the Lorentz metric implicitly used in the calculation of the spacetime metric due to the earth has no connection whatsever with the Lorentz metric which was previously used to calculate the effects of the galaxy. The Lorentz metric is simply used as a calculational tool and since we are now computing the effect of the earth in a local inertial frame we use the Lorentz metric in its Minkowski form. Therefore, as in general relativity, there is no effect due to the galaxy. This point of view is consistent with Eddington's result, who recovered the Schwarzschild sotution from Whitehead's line element by a coordinate transformation in Riemann space. In other words, the anisotropy in G due to the gravitational field of the Galaxy has the same standing in both, Einstein's and Whitehead's
116
3 June 1974
theory, and can be transformed away. We therefore conclude that Whitehead's theory, if properly interpreted, is viable, as far as existing tests of gravitational theories are concerned. Whitehead's theory can possibly be tested by observations of effects concerning gravitational radiation from binary star systems [5]. Whitehead's theory predicts an energy gain due to emitted gravitational radiation for two body systems, i.e. an increase with time of the orbital period, in disagreement with physical intuition, but in agreement with self-consistent first order approximations of the general relativistic two-body problem [ 6 - 8 ] . Higher order calculations using the field equations of general relativity will probably lead to clear-cut differences between Whitehead's and Einstein's theory. This would provide an experimental test via observations of period changes o f and/or gravitational radiation from binary systems [5]. We would like to thank Jiirgen Ehlers for helpful remarks.
References [1 ] J.L. Anderson, Principles of relativity theory, (Academic Press, New York, 1967). [2] A.S. Eddington, Nature 113 (1914) 192. [3] C.M. Will, Astrophys. J. 169 (1971) 141. [41 C.M. Will, Physics Today 25, No. 10 (1972) 23. [5] M. Reinhardt and A. Rosenblum, (1973) to be published. [6] S.F. Smith and P. Havas, Phys. Rev. 138 (1965) B495. [7] P. Havas and A. Rosenblum, (1973) to be published. [8] P. Havas and A. Rosenblum, (1973) to be published.
PHYSICS LETTERS
WHITEHEAD
3 June 1974
CONTRA EINSTEIN
M. REINHARDT
Astronomische Institute der Universita't Bonn, Germany A. ROSENBLUM
Max-Planck-lnstitut ff~r Physik und Astrophysik, MiJnchen, Germany Received 3 April 1974 None of the classical tests of general relativity can distinguish between Einstein's and Whitehead's theories of gravitation. Claims that geophysical effects exclude Whitehead's theory are shown to be unsubstantiated. Thus Whitehead's theory proves to be one of the so far undefeated rivals of general relativity. A feasible experimentum crueis is delineated.
Presently it is a common belief that the only serious competitor to Einstein's theory of gravitation is the Jordan-Brans-Dicke theory. The purpose o f this note is to show that from the point of view of currently available local observational tests Whitehead's actionat-a-distance theory is also viable, if it is interpreted correctly in terms of modern views of special relativistic versions of field theories of gravitation [ 1 ]. Whitehead's starts by assuming that particles with proper mass move along world lines xU = x u (s), where ds 2 = c 2 d t 2 - d x 2 - d y 2 - d z
2 =~luvdx#dxU,
(1)
is the special relativistiv line element. The adjunct "field" due to all the other particles felt by the particle iis gi.v =
÷
(2)
where = -(mj(2V/c3)/(rJ) 3) with z ~ - xu, / x / being " " the past light " ~uJ - - x~_ m con e o f x * , and I~j = ~!/~0 lu, where ~!~ = dxI# /ds G is the gravitational constant as usual. This means that the "field seen by particle i " is determined by the retarded "fields" of all the other particles. The equation of motion is derived from the variational principle
8fgi~,u(dx~ /dsi) (dx~/dsi)
= O,
(3)
with (dsi) 2 = giuv dx tzidxi"v The equations of motion are formally identical to those of a geodesic in a Riemannian space
d2xU /ds 2i + F~/ (dxX /ds i) (dx'r /ds i) = 0,
(4)
with giuv as a metric tensor and ds i as a line element. Although we have started in Minkowski space the system of particle interactions is described by a Riemannian space• In particular, it was shown by Eddington [2] that the Whitehead line element for the field of a particle at rest can be transformed into the Schwarzschild line element. This implies that the Whitehead theory predicts the same results as general relativity for the classical tests, i.e. gravitational redshift, light deflection, time delay, perihelion shift and gyroscope precission. The fact that Whitehead's equations of motion can be given in the form of eq. (4) means also that Whitehead's theory gives formally the same types o f cosmological solutions as general relativity, since it admits the same group symmetrics. In particular one can reproduce the Robertson-Walker models. Recently it has been claimed [3, 4] that predictions of the Whitehead theory do not agree with certain geophysical effects, namely that there should be an effective anisotropy in the gravitational constant G due to the gravitational attraction of our galaxy. This anisotropy in G should lead to a measurable effect on the tides of the earth which is not observed in high precision gravimeter experiments. Such an effect would not exist in general relativity since it could be removed by a transformation to a local (solar system) reference frame. It has been claimed [3, 4] that one cannot do the same in Whitehead's theory because the 115
Volume 48A, number 2
PHYSICS LETTERS
metric there must be calculated in a global Lorentzian coordinate system or in a space-time of constant curvature. Our purpose is to point out that this is not the case. Only the metric in eq. (4) represents the metric of space-time whereas the Minkowski metric used in formula (2) is only a formal tool used in the computation of the space-time metric in eq. (4). In particular, in the computation of the anisotropy in G, we could ignore the earth for a moment and choose a local frame in which the physical metric g has its Minkowski form, i.e. in which all effects of the galaxy have been removed. We could then use Whitehead's formula to calculate the metric produced by the earth, using this coordinate system. We notice that in the frame as which g has its Minkowski forrl) the Lorentz metric which was used to calculate g and serves no other purpose now has galaxy dependent terms in it. However, the Lorentz metric implicitly used in the calculation of the spacetime metric due to the earth has no connection whatsever with the Lorentz metric which was previously used to calculate the effects of the galaxy. The Lorentz metric is simply used as a calculational tool and since we are now computing the effect of the earth in a local inertial frame we use the Lorentz metric in its Minkowski form. Therefore, as in general relativity, there is no effect due to the galaxy. This point of view is consistent with Eddington's result, who recovered the Schwarzschild sotution from Whitehead's line element by a coordinate transformation in Riemann space. In other words, the anisotropy in G due to the gravitational field of the Galaxy has the same standing in both, Einstein's and Whitehead's
116
3 June 1974
theory, and can be transformed away. We therefore conclude that Whitehead's theory, if properly interpreted, is viable, as far as existing tests of gravitational theories are concerned. Whitehead's theory can possibly be tested by observations of effects concerning gravitational radiation from binary star systems [5]. Whitehead's theory predicts an energy gain due to emitted gravitational radiation for two body systems, i.e. an increase with time of the orbital period, in disagreement with physical intuition, but in agreement with self-consistent first order approximations of the general relativistic two-body problem [ 6 - 8 ] . Higher order calculations using the field equations of general relativity will probably lead to clear-cut differences between Whitehead's and Einstein's theory. This would provide an experimental test via observations of period changes o f and/or gravitational radiation from binary systems [5]. We would like to thank Jiirgen Ehlers for helpful remarks.
References [1 ] J.L. Anderson, Principles of relativity theory, (Academic Press, New York, 1967). [2] A.S. Eddington, Nature 113 (1914) 192. [3] C.M. Will, Astrophys. J. 169 (1971) 141. [41 C.M. Will, Physics Today 25, No. 10 (1972) 23. [5] M. Reinhardt and A. Rosenblum, (1973) to be published. [6] S.F. Smith and P. Havas, Phys. Rev. 138 (1965) B495. [7] P. Havas and A. Rosenblum, (1973) to be published. [8] P. Havas and A. Rosenblum, (1973) to be published.