[Paper Review] Einstein's Discovery of Gravitational Waves 1916-1918

Galina Weinstein|arXiv (Cornell University)|Feb 12, 2016

Relativity and Gravitational Theory4 references3 citations

TL;DR

This paper analyzes Einstein's 1916–1918 development of gravitational waves, revealing a critical mathematical error in his derivation that led to three types of waves—longitudinal, transverse, and a novel type—due to his restrictive unimodular coordinate condition. After colleagues demonstrated that only the third wave type carried energy in those coordinates, Einstein abandoned the condition, ultimately leading to the correct discovery of plane transverse gravitational waves.

ABSTRACT

In his 1916 ground-breaking general relativity paper Einstein had imposed a restrictive coordinate condition, his field equations were valid for coordinate systems which are unimodular. Later, Einstein published a paper on gravitational waves. The solution presented in this paper did not satisfy the above restrictive condition. In his gravitational waves paper, Einstein concluded that gravitational fields propagate at the speed of light. The solution is the Minkowski flat metric plus a small disturbance propagating in a flat spacetime. Einstein calculated the small deviation from Minkowski metric in a manner analogous to that of retarded potentials in electrodynamics. However, in obtaining the above derivation, Einstein made a mathematical error. This error caused him to obtain three different types of waves compatible with his approximate field equations: longitudinal waves, transverse waves and a new type of wave. Einstein added an Addendum in which he suggested that in systems in unimodular coordinates only waves of the third type occur and these waves transport energy. He became obsessed with his unimodular coordinates. Einstein's colleagues demonstrated to him that in the coordinate system in unimodular coordinates the gravitational wave of a mass point carry no energy, but Einstein tried to persuade them that he had actually not made a mistake in his gravitational waves paper. Einstein, however, eventually accepted his colleagues results and dropped the restrictive unimodular coordinates. This finally led him to discover plane transverse gravitational waves.

Motivation & Objective

• To clarify the historical development of Einstein’s gravitational wave theory between 1916 and 1918.
• To identify the mathematical error in Einstein’s derivation of gravitational waves and its consequences.
• To analyze Einstein’s reliance on unimodular coordinates and its impact on wave energy transport.
• To trace the resolution of the conflict between Einstein’s results and his colleagues’ counterarguments.
• To explain how the rejection of unimodular coordinates led to the correct formulation of transverse gravitational waves.

Proposed method

• Analyzes Einstein’s 1916 general relativity paper, focusing on the imposition of unimodular coordinates and its restrictive effect on field equations.
• Examines the 1918 gravitational wave paper, where Einstein derived solutions using a perturbative approach analogous to retarded potentials in electrodynamics.
• Identifies the mathematical error in the derivation that produced three distinct wave types instead of the correct transverse waves.
• Evaluates the role of the Addendum in which Einstein claimed only the third wave type transported energy in unimodular coordinates.
• Reviews the arguments by Einstein’s colleagues demonstrating that in unimodular coordinates, gravitational waves from a point mass carry no energy.
• Traces the shift in Einstein’s thinking after these criticisms, culminating in his abandonment of unimodular coordinates and discovery of transverse waves.

Experimental results

Research questions

• RQ1 What mathematical error did Einstein commit in his 1918 gravitational wave derivation?
• RQ2 Why did Einstein insist that only the third type of wave carried energy in unimodular coordinates?
• RQ3 How did Einstein’s colleagues challenge his conclusion about energy transport in unimodular coordinates?
• RQ4 What role did the restrictive unimodular coordinate condition play in distorting the nature of gravitational waves?
• RQ5 How did the rejection of unimodular coordinates lead to Einstein’s correct formulation of transverse gravitational waves?

Key findings

• Einstein’s derivation of gravitational waves in 1918 contained a mathematical error that produced three wave types: longitudinal, transverse, and a novel type.
• In unimodulr coordinates, Einstein claimed only the third wave type transported energy, but his colleagues proved this was incorrect for a point mass source.
• Einstein’s colleagues demonstrated that gravitational waves in unimodular coordinates carried no energy, contradicting Einstein’s assertion.
• After prolonged resistance, Einstein accepted his colleagues’ analysis and abandoned the unimodular coordinate condition.
• The removal of the unimodular constraint allowed Einstein to correctly derive plane transverse gravitational waves.
• This correction marked the final discovery of the physically correct form of gravitational waves as transverse, propagating at the speed of light.