78 Tests of General Relativity
Earth1 AU
Sun1 ×^10(^9) m
60 °
20 °
Figure 4.4eLISA orbits in the Solar System [4]. Reproduced with permission of Pau Amaro-
Seoane and the LISA consortium. (See plate section for color version.)
from each arm are combined by on-board computers to perform the multiple-arm
interferometry required to cancel the phase-noise common to all three arms. Fluctu-
ations in the optical paths between the test masses can be measured to sub-angstrom
precision, which, when combined with the large separation of the spacecraft, allowed
LISA to detect gravitational-wave strain down to a level of order 10−^23 in one year of
observation, with a signal to noise ratio of 5.
LISA targets high-priority astronomy such as massive black holes, stellar evolution,
the high-redshift universe and cosmology. By 2011 LISA had identified 20 000 individ-
ual sources, almost all resolvable binary white dwarfs, but no black hole—black hole
mergers and no signals of gravitational radiation [4]. Its follower, the European-led
variant named eLISA (see Figure 4.4 [4]) is scheduled to be launched before 2022.
Problems
- Calculate the gravitational redshift in wavelength for the 769.9nm potassium
line emitted from the Sun’s surface [1]. - Derive the deflection angle [Equation (4.4)] using Equations (4.26) and (4.5).
- Derive Equation (4.12). What are the amplification of the individual images [4]?
- Draw the bend angle due to gravitational potential훼gand lens geometry훼l,and
zero (dashed line) for a lens of mass푀≈ 7. 2 × 1011 푀⊙, with퐷S≈ 1 .64Gpc (푧=
3 .4 in a FRW cosmology with훺m= 0 .3and훺휆= 0 .7),퐷L≈ 1 .67 Gpc(푧= 0. 803 ),
퐷LS≈ 0 .96 Gpc and훼S≈ 0. 13 ′′. (Note that since distances are angular diameter
distances,퐷S≠퐷LS+퐷L.) Take the lens to be (a) a point mass and (b) a spher-
ical mass distribution with density given by Equation (4.11) for an ideal galaxy.
This roughly corresponds to the parameters associated with the ‘Einstein Cross’
lensed quasar found by the Hubble telescope, HST 14176+5226.