Shapiro delay

The Shapiro delay , named after Irwin I. Shapiro , has the effect that in the reference system of an observer far away from centers of gravity (zero potential) the speed of propagation of light near a large mass is lower than the local speed of light . This is in line with general relativity .

The gravitational lensing effect , in which light is deflected by gravity , can be explained with the Shapiro delay. The deflection of the light results, similar to its refraction on lenses made of glass, from a local change in its speed of propagation.

effect

For weakly rotating, time-independent gravitational fields , the metric of the Schwarzschild solution in spherical coordinates is obtained as an approximation

${\displaystyle g={\begin{pmatrix}1+2\phi (r)&0&0&0\\0&{\frac {-1}{1+2\phi (r)}}&0&0\\0&0&-r^{2}&0\\0&0&0&-r^{2}\sin ^{2}{\theta }\end{pmatrix}}}$

left: local shell speed, right: Shapiro-delayed coordinate speed (click starts animation)

left: bundle of rays in flat space-time, right: shape-delayed and deflected rays in the presence of a mass (click to start animation)

The approximation can be z. For example, it can be used well on the surface of a star, but on the surface of a strongly rotating and much denser neutron star it is not so applicable and there are measurable deviations.

When applied to a star, this is the gravitational potential normalized by  c²

${\displaystyle \phi (r)=-{\frac {Gm}{c^{2}r}}=-{\frac {r_{G}}{r}},}$

in which

• ${\displaystyle r_{G}}$the radius of gravity
• G is the Newtonian gravitational constant
• m is the mass of the star
• c are the speed of light .

With this approximation, the deflection of light by gravitation can be clearly interpreted as a refraction effect. To do this, you have to consider what the local time is at a spacetime point. One defines for an infinitesimal time interval : ${\displaystyle \tau }$ ${\displaystyle \mathrm {d} \tau }$

${\displaystyle \mathrm {d} \tau ={\sqrt {g_{00}(r)}}{\frac {\mathrm {d} x^{0}}{c}}}$

with  x ° = ct as the time component, as the local or proper time measured by an observer at the spacetime point  x .

In addition, one has to take into account the radial length contraction and define the radial length  x near mass as

${\displaystyle \mathrm {d} x={\sqrt {-g_{11}(r)}}\mathrm {d} r}$.

If one now looks at a ray of light, its real local speed is the speed of light:

${\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} \tau }}=c,}$

and is its externally measured speed

${\displaystyle c_{n}=c{\frac {\mathrm {d} r}{\mathrm {d} x^{0}}}}$.

According to the definition of proper time above, they are related as follows:

• ${\displaystyle c_{n}=c{\frac {\mathrm {d} r}{\mathrm {d} x^{0}}}={\sqrt {\frac {g_{00}(r)}{-g_{11}(r)}}}{\frac {\mathrm {d} x}{\mathrm {d} \tau }}={c}({1+2\phi (x)})=c\left(1-{\frac {r_{s}}{r}}\right)}$ for radial movement relative to the mass,
• ${\displaystyle c_{n}=c{\sqrt {1-{\frac {r_{s}}{r}}}}}$ for transverse movement relative to the mass,

each with the Schwarzschild radius . ${\displaystyle r_{s}=2r_{G}}$

If you take into account that an attractive gravitational potential, i.e. negative, you can see that the measured speed of the light beam appears locally smaller than the speed of light: ${\displaystyle \phi }$

${\displaystyle \phi <0\Rightarrow c_{n}<c}$

In this view, one can interpret the gravitational field as a medium with the location-dependent refractive index :

${\displaystyle n(x)={\frac {c}{c_{n}}}={\frac {1}{1+2\phi (x)}}\approx 1-2\phi (x)>1}$.

Since light propagates along geodesics , this can also be formulated in such a way that near a mass the geodesics are curved in space, which can be explained by the contracted radius. In addition to the curvature of light, this also leads to light delay, which is called Shapiro delay after its discoverer. With k², the effect is  twice as strong as with the simple Lorentz factor  k if only gravitational forces were taken into account.

At the edge of the sun is what results as the refractive index . The effect is therefore very small compared to normal optical refraction. The angle of light deflection in the gravitational field is correspondingly small. ${\displaystyle \phi =-10^{-6}}$ ${\displaystyle n=1{,}000\,002}$

Status of experiments already carried out

The light delay was theoretically predicted by Irwin I. Shapiro in 1964 and measured for the first time in 1968 and 1971. Here the time difference was measured by means of radar signals reflected from Venus , while the latter was behind the sun from Earth, so that the radar waves had to pass close to the edge of the sun. The measurement uncertainty was initially several percent. With repeated measurements and later also with measurements with the help of space probes ( Mariner , Viking ) instead of Venus, the measurement accuracy could be increased to 0.1%.

The most accurate measurement of the effect to date was made in 2002 when the Cassini space probe was conjunct with the sun. Frequency measurements in the K _a band made it possible to determine the Shapiro delay with an accuracy of 0.001%.

Individual evidence

1. ↑ SI Blinnikov, LB Okun, MI Vysotsky: Critical velocities c / √3 and c / √2 in general theory of relativity
2. ↑ Irwin I. Shapiro: Fourth Test of General Relativity in Physical Review Letters 13 (1964), 789 - 791 doi : 10.1103 / PhysRevLett.13.789
3. ↑ Irwin I. Shapiro et al .: Fourth Test of General Relativity: Preliminary Results . In: Physical Review Letters 20, 1968, pp. 1265-1269
4. ↑ Irwin I. Shapiro et al .: Fourth Test of General Relativity: New Radar Result . In: Physical Review Letters 26, 1971, pp. 1132-1135
5. ↑ B. Bertotti, L. Iess, P. Tortora, A test of general relativity using radio links with the Cassini spacecraft , Nature 425 (2003), 374–376 online (PDF; 199 kB)

literature

• CM Will: Theory and experiment in gravitational physics. Cambridge University Press, Cambridge (1993). Standard work for the experimental review of the ART
• CM Will: Was Einstein Right ?: Putting General Relativity to the Test. Basic Books (1993). A popular science summary of the same
• CM Will: The Confrontation between General Relativity and Experiment , Living Reviews in Relativity. (2014). Shorter but more recent version of Theory and experiment in gravitational physics