De Sitter effect

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In astrophysics, the term De Sitter effect (named after the Dutch physicist Willem de Sitter) has been applied to two unrelated phenomena.

The De Sitter effect was first described by de Sitter in 1913 and used to support the special theory of relativity against a competing 1908 theory by Walter Ritz that postulated a variable speed of light. de Sitter showed that Ritz's theory predicted that the orbits of binary stars would appear more eccentric than consistent with experiment and with the laws of mechanics.[1]

The second de Sitter effect was introduced in 1916 and arises in general relativity; it describes a particular secular precession of astronomical orbits, equivalent to the rotation of the Laplace-Runge-Lenz vector.[2]

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[edit] History

Willem de Sitter was born in 1872. He studied Mathematics in Groningen and was obligated to take some classes in experimental physics . De Sitter went to the observatory of Jacobus Kapteyn and during his research, he was invited for some work in Kaapstad. During his stay in Kaapstad, De Sitter began to like physics and decided to become an astronomer.
Together with Albert Einstein he developed the De Sitter-universe, a simple model for an expanding universe.
Nowadays, people tend to say that De Sitter was, until his death in 1934, one of the few people who fully understood the General theory of relativity.

[edit] De Sitter precession

The De Sitter precession is very similar to the Thomas precession. To derive the precession, assume the system is in a rotating Schwarzschild metric.


\boldsymbol{ds}^{2} = (1-\frac{2m}{r})dt^{2} - (1 - \frac{2m}{r})^{-1}dr^{2} - r^{2}(d\theta^{2} + \sin^{2}\theta d\phi'^{2})

Using units in which c = 1.
We introduce a rotating coordinate system, with an angular velocity ω. This gives us


d\phi = d\phi' - \omega dt \frac{}{}


If we use this in the Schwarzschild metric, and assume that θ is a constant, we find

\boldsymbol{ds}^{2} = (1-\frac{2m}{r}-r^{2}\beta\omega^{2})(dt-\frac{r^{2}\beta\omega}{1-\frac{2m}{r}-r^{2}\beta\omega^{2}}d\phi)^{2} - (1-\frac{2m}{r})^{-1}dr^{2} - \frac{r^{2}\beta - 2mr\beta}{1-\frac{2m}{r}-r^{2}\beta\omega^{2}}d\phi^{2}

with β = sin(θ)2. Now, the metric is in the canonical form


\boldsymbol{ds}^{2} = e^{\frac{2\Phi}{c^{2}}}\left(c dt - \frac{1}{c^{2}} w_{i} dx^{i}\right)^{2} - k_{ij} dx^{i}dx^{j}

From this canonical form, we can easily determine the rotational rate of a gyroscope

\Omega = \frac{\sqrt{2}}{4}e^{\Phi}[k^{ik}k^{jl}(\omega_{i,j}-\omega_{j,i})(\omega_{k,l} - \omega{l,k})]^{1/2} = -\sqrt{\beta}\omega

We are at rest in our rotating coordinate system, so there is no acceleration, and thus Φ,i = 0. This leads to

\Phi,_{i} = \frac{\frac{2m}{r^{2}} - 2r\beta\omega^{2}}{2(1-\frac{2m}{r}-r^{2}\beta\omega^{2})} = 0

From this, we can distill ω,

\omega^{2} = \frac{m}{r^{3}\beta}


Since α', the precession of the gyroscope relative to the rotating coordinate system is given by α' = ΩΔτ, with

\Delta \tau = (1-\frac{2m}{r}-r^{2}\beta\omega^{2})^{1/2}dt = (1-\frac{3m}{r})^{1/2}dt

the precession in the coordinate system at rest is given by:

\alpha = \alpha' + 2\pi = -2 \pi \sqrt{\beta}\Bigg( (1-\frac{3m}{r})^{1/2} - 1 \Bigg)

With a first order Taylor series we find

\alpha \approx \frac{3\pi G m}{c^{2}r}\sqrt{\beta} = \frac{3\pi G m}{c^{2}r}\sin(\theta)

[edit] References

  1. ^ de Sitter, W (1913). "Unknown". Phokatische Zeitschrift 14: pp. 429, 1267. 
  2. ^ de Sitter, W (1916). "On Einstein's Theory of Gravitation and its Astronomical Consequences". Mon. Not. Roy. Astron. Soc. 77: 155–184. 

[edit] Textbooks