In theoretical physics, Whitehead's theory of gravitation was introduced by the mathematician and philosopher Alfred North Whitehead in 1922.[1] While never broadly accepted, at one time it was a scientifically plausible alternative to general relativity. However, after further experimental and theoretical consideration, the theory is now generally regarded as obsolete.

## Principal features

Whitehead developed his theory of gravitation by considering how the world line of a particle is affected by those of nearby particles. He arrived at an expression for what he called the "potential impetus" of one particle due to another, which modified Newton's law of universal gravitation by including a time delay for the propagation of gravitational influences. Whitehead's formula for the potential impetus involves the Minkowski metric, which is used to determine which events are causally related and to calculate how gravitational influences are delayed by distance. The potential impetus calculated by means of the Minkowski metric is then used to compute a physical spacetime metric $\displaystyle{ g_{\mu\nu} }$, and the motion of a test particle is given by a geodesic with respect to the metric $\displaystyle{ g_{\mu\nu} }$.[2][3] Unlike the Einstein field equations, Whitehead's theory is linear, in that the superposition of two solutions is again a solution. This implies that Einstein's and Whitehead's theories will generally make different predictions when more than two massive bodies are involved.[4]

Following the notation of Chiang and Hamity[5] , introduce a Minkowski spacetime with metric tensor $\displaystyle{ \eta_{ab}=\mathrm{diag}(1, -1, -1, -1) }$, where the indices $\displaystyle{ a, b }$ run from 0 through 3, and let the masses of a set of gravitating particles be $\displaystyle{ m_a }$.

The Minkowski arc length of particle $\displaystyle{ A }$ is denoted by $\displaystyle{ \tau_A }$. Consider an event $\displaystyle{ p }$ with co-ordinates $\displaystyle{ \chi^a }$. A retarded event $\displaystyle{ p_A }$ with co-ordinates $\displaystyle{ \chi_A^a }$ on the world-line of particle $\displaystyle{ A }$ is defined by the relations $\displaystyle{ (y_A^a = \chi^a - \chi_A^a, y_A^a y_{Aa} = 0, y_A^0 \gt 0) }$. The unit tangent vector at $\displaystyle{ p_A }$ is $\displaystyle{ \lambda_A^a = (dx_A^a/d\tau_A)p_A }$. We also need the invariants $\displaystyle{ w_A = y_A^a \lambda_{Aa} }$. Then, a gravitational tensor potential is defined by
$\displaystyle{ g_{ab} = \eta_{ab} - h_{ab}, }$
where
$\displaystyle{ h_{ab} = 2\sum_A \frac{m_A}{w_A^3} y_{Aa} y_{Ab}. }$

It is the metric $\displaystyle{ g }$ that appears in the geodesic equation.

## Experimental tests

Fowler argued that different tidal predictions can be obtained by a more realistic model of the galaxy.[6][2] Reinhardt and Rosenblum claimed that the disproof of Whitehead's theory by tidal effects was "unsubstantiated".[7] Chiang and Hamity argued that Reinhardt and Rosenblum's approach "does not provide a unique space-time geometry for a general gravitation system", and they confirmed Will's calculations by a different method.[5] In 1989, a modification of Whitehead's theory was proposed that eliminated the unobserved sidereal tide effects. However, the modified theory did not allow the existence of black holes.[8]

Subrahmanyan Chandrasekhar wrote, "Whitehead's philosophical acumen has not served him well in his criticisms of Einstein."[9]

## Philosophical disputes

Clifford M. Will argued that Whitehead's theory features a prior geometry.[10] Under Will's presentation (which was inspired by John Lighton Synge's interpretation of the theory[11][12]), Whitehead's theory has the curious feature that electromagnetic waves propagate along null geodesics of the physical spacetime (as defined by the metric determined from geometrical measurements and timing experiments), while gravitational waves propagate along null geodesics of a flat background represented by the metric tensor of Minkowski spacetime. The gravitational potential can be expressed entirely in terms of waves retarded along the background metric, like the Liénard–Wiechert potential in electromagnetic theory.

A cosmological constant can be introduced by changing the background metric to a de Sitter or anti-de Sitter metric. This was first suggested by G. Temple in 1923.[13] Temple's suggestions on how to do this were criticized by C. B. Rayner in 1955.[14][15]

Will's work was disputed by Dean R. Fowler, who argued that Will's presentation of Whitehead's theory contradicts Whitehead's philosophy of nature. For Whitehead, the geometric structure of nature grows out of the relations among what he termed "actual occasions". Fowler claimed that a philosophically consistent interpretation of Whitehead's theory makes it an alternate, mathematically equivalent, presentation of general relativity.[6] In turn, Jonathan Bain argued that Fowler's criticism of Will was in error.[2]

## References

1. Whitehead, A. N. (2011-06-16) (in en). The Principle of Relativity: With Applications to Physical Science. Cambridge University Press. ISBN 978-1-107-60052-2.
2. Bain, Jonathan (1998). "Whitehead's Theory of Gravity". Stud. Hist. Phil. Mod. Phys. 29 (4): 547–574. doi:10.1016/s1355-2198(98)00022-7. Bibcode1998SHPMP..29..547B.
3. Synge, J. L. (1952-03-06). "Orbits and rays in the gravitational field of a finite sphere according to the theory of A. N. Whitehead" (in en). Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 211 (1106): 303–319. doi:10.1098/rspa.1952.0044. ISSN 0080-4630. Bibcode1952RSPSA.211..303S.
4. Cite error: Invalid <ref> tag; no text was provided for refs named :0
5. Chiang, C. C.; Hamity, V. H. (August 1975). "On the local newtonian gravitational constant in Whitehead's theory" (in en). Lettere al Nuovo Cimento. Series 2 13 (12): 471–475. doi:10.1007/BF02745961. ISSN 1827-613X.
6. Fowler, Dean (Winter 1974). "Disconfirmation of Whitehead's Relativity Theory -- A Critical Reply". Process Studies 4 (4): 288–290. doi:10.5840/process19744432.
7. Reinhardt, M.; Rosenblum, A. (1974). "Whitehead contra Einstein". Physics Letters A (Elsevier BV) 48 (2): 115–116. doi:10.1016/0375-9601(74)90425-3. ISSN 0375-9601. Bibcode1974PhLA...48..115R.
8. Hyman, Andrew (1989). "A New Interpretation of Whitehead's Theory". Il Nuovo Cimento 387 (4): 387–398. doi:10.1007/bf02725671. Bibcode1989NCimB.104..387H.
9. Chandrasekhar, S. (March 1979). "Einstein and general relativity: Historical perspectives" (in en). American Journal of Physics 47 (3): 212–217. doi:10.1119/1.11666. ISSN 0002-9505.
10. Will, Clifford (1972). "Einstein on the Firing Line". Physics Today 25 (10): 23–29. doi:10.1063/1.3071044. Bibcode1972PhT....25j..23W.
11. Synge, John (1951). Relativity Theory of A. N. Whitehead. Baltimore: University of Maryland.
12. Tanaka, Yutaka (1987). "Einstein and Whitehead-The Comparison between Einstein's and Whitehead's Theories of Relativity". Historia Scientiarum 32.
13. Temple, G. (1924). "Central Orbit in Relativistic Dynamics Treated by the Hamilton-Jacobi Method". Philosophical Magazine. 6 48 (284): 277–292. doi:10.1080/14786442408634491.
14. Rayner, C. (1954). "The Application of the Whitehead Theory of Relativity to Non-static Spherically Symmetrical Systems". Proceedings of the Royal Society of London 222 (1151): 509–526. doi:10.1098/rspa.1954.0092. Bibcode1954RSPSA.222..509R.
15. Rayner, C. (1955). "The Effects of Rotation in the Central Body on its Planetary Orbits after the Whitehead Theory of Gravitation". Proceedings of the Royal Society of London 232 (1188): 135–148. doi:10.1098/rspa.1955.0206. Bibcode1955RSPSA.232..135R.