Shapiro delay

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The Shapiro time delay effect, or gravitational time delay effect, is one of the four classic solar system tests of general relativity. Radar signals passing near a massive object take slightly longer to travel to a target and longer to return (as measured by the observer) than it would if the mass of the object were not present.

Contents

[edit] History

The time delay effect was first noticed in 1964, by Irwin I. Shapiro. Shapiro proposed an observational test of his prediction: bounce radar beams off the surface of Venus and Mercury, and measure the round trip travel time. When the Earth, Sun, and Venus are most favorably aligned, Shapiro showed that the expected time delay, due to the presence of the Sun, of a radar signal traveling from the Earth to Venus and back, would be about 200 microseconds, well within the limitations of 1960s era technology.

The first test, using the MIT Haystack radar antenna, was successful, matching the predicted amount of time delay. The experiments have been repeated many times since, with increasing accuracy.

[edit] Calculating time delay

The speed of light in meters per given interval of "proper time" is a constant, however the travel time of any electromagnetic wave, or signal, moving at 299,792,458 meters per "second" is affected by the gravitational time dilation in regions of spacetime through which it travels. This is because the coordinate time and proper time diverge as the gravitational field strength increases.

[edit] Time delay due to light travelling around a single mass

For a signal going around a massive object, the time delay can be computed as the following:

\Delta t=-\frac{2GM}{c^3}\ln(1-\mathbf{R}\cdot\mathbf{x})

Here \mathbf{R} is the unit vector pointing from the observer to the source and

\mathbf{x} is the unit vector pointing from the observer to the gravitating mass M

See Dot product.

The above formula can be rearranged like this:

c\Delta t=-\frac{2GM}{c^2}\ln(1-\mathbf{R}\cdot\mathbf{x})

\Delta x= -R_s \ln(1-\mathbf{R}\cdot\mathbf{x})

Which is the extra distance the light has to travel.

Where:

Rs is the Schwarzchild radius.

This is the same as:

\Delta t = -\frac{R_s}{c} \ln(1-\mathbf{R}\cdot\mathbf{x})

[edit] Special cases

[edit] Shapiro delay and interplanetary probes

Shapiro delay must be considered along with ranging data when trying to accurately determine the distance to interplanetary probes such as the Voyager and Pioneer spacecraft (see the Voyager program, the Pioneer program, and the Pioneer anomaly).

[edit] Quote by Einstein

"In the second place our result shows that, according to the general theory of relativity, the law of the constancy of the velocity of light in vacuo, which constitutes one of the two fundamental assumptions in the special theory of relativity and to which we have already frequently referred, cannot claim any unlimited validity. A curvature of rays of light can only take place when the velocity of propagation of light varies with position. Now we might think that as a consequence of this, the special theory of relativity and with it the whole theory of relativity would be laid in the dust. But in reality this is not the case. We can only conclude that the special theory of relativity cannot claim an unlimited domain of validity ; its results hold only so long as we are able to disregard the influences of gravitational fields on the phenomena (e.g. of light)." - Albert Einstein (The General Theory of Relativity: Chapter 22 - A Few Inferences from the General Principle of Relativity)

[edit] References

  • "Boost for General Relativity." Nature. 12 July 2001.
  • Relativity : the Special and General Theory by Albert Einstein., available at Project Gutenberg.
  • Irwin I. Shapiro (December 1964). "Fourth Test of General Relativity". Physical Review Letters 13: 789-791. 
  • Irwin I. Shapiro, Gordon H. Pettengill, Michael E. Ash, Melvin L. Stone, William B. Smith, Richard P. Ingalls, and Richard A. Brockelman (May 1968). "Fourth Test of General Relativity: Preliminary Results". Physical Review Letters 20: 1265–1269. 
  • d'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Clarendon Press. ISBN 0-19-859686-3.  See Section 15.6 for an excellent advanced undergraduate level introduction to the Shapiro effect.
  • Will, Clifford M. (2001). "The Confrontation between General Relativity and Experiment". Living Rev. Rel. 4: 4-107.  gr-qc/0103036 A graduate level survey of the solar system tests, and more.
  • John C. Baez, Emory F. Bunn (2005). "The Meaning of Einstein's Equation". Amer. Jour. Phys. 73: 644-652.  gr-qc/0103044

[edit] See also