# Can a Free Falling Interferometer (eg: LISA) Detect the Changing Schwarzschild Field from Nearby Masses?

My first inclination was to imagine LISA being in a freely falling elevator. If the elevator is sufficiently small so that tidal forces can be neglected, then the GR Equivalence Principle says there is no experiment we can do in the elevator that can determine if we are in a gravitational field or not. If LISA’s interference pattern changed due to the Earth, Sun, or Moon being moved closer, it would violate this principle. Unfortunately, as the elevator is made smaller to avoid tidal effects, LISA becomes smaller and less sensitive.

Maybe the size of the elevator has nothing to do with the answer, hence the question. It is expected LISA will be able to detect gravitational waves which strain the x and y positions of the mirrors. However, will the change in a Schwarzschild field which strains r (spatial radius) and t (time) effect the LISA interference pattern?

Asked a different way, would the interference pattern change with time as a freely falling LISA (in any orientation and started with zero velocity wrt Earth) fell radially toward the center of the Earth?

• Do you mean like was done by http://science.nasa.gov/missions/grace/? – honeste_vivere Aug 5 '16 at 21:06
• @honeste - Yes, GRACE could be relevant to the answer. If the two GRACE satellites formed one arm of the interferometer and there were a third satellite to make the second perpendicular arm, then a measurement of the two arm lengths and their changing in time would be very useful. – Gary Godfrey Aug 5 '16 at 21:19
• elisascience.org – CuriousOne Aug 5 '16 at 21:29

## 1 Answer

Grace would give you something, for the two current satellites, as Gary Godfrey says, but there is no third, and they are not that accurate. They measure gravity's acceleration to about 1 Gal accuracy, or $10^{-6}g$. To see the earth Sun gravitational wave (and the moon is negligible) you need to move them to be about 10% of a light year away, and see a strain amplitude or differential change in distance, it translates to the parts in g approximately, of $10^{-25}$. A lost cause.

As to interferometer effects from the Schwarzchild metric, in the pseudo-static regime, yes you can probably see something, but it's not more than a Doppler effect. The metric's $g_{tt}$ and $g_{rr}$ components are the only ones affected, and you could then see a Doppler or time dilation effect radially (both depend only on r). The interferometer could thus have its 'pattern' changed, two slightly different freqs one way vs the other way with two satellites (no plans for 3, see below), so you'd have to be able to record/measure and compare going the other way. Elisa plans to do that.

Other effects would not be very promising.

You could get that time dilation from Grace, if changed to be freely falling and you could get better accuracies than the time dilation measured by the current orbiting GPS satellites due to the Schwarzchild metric. If they are 1000 Kms apart falling freely in radially you could get the tidal gravity change due to distance and different acceleration, which would be about 4% (1000/4000 square), and you could see the different Doppler shift coming in or out, but you'd have to figure a way to do that -- ie, measure and record and then compare.

You need 3 satellites, 2 arms, to compare the real time interferometer effect, without recording and making sure you control things, as you then do it the other way.

You'd have to do some math to see if there's any other effects (like the perihelion advance measured for Mercury) on the earth Sun system from general relativity. But perihelion changes are more optical measurements. You can try doing a post-Newtonian GR approximation (or find one of many already done) to see what other effects there might be.

If you could wait 20 years or so and make sure the Elisa (the evolution of Lisa from 3 to two satellites, to make it more affordable) configuration was launched you'd do a little better, but still not so good. Right now the test satellite is just one, testing measurements and accuracies possible. If and when launched the separations would be 1 million km, be about 50 million Kms from the earth, and with maximum sensitivities of $10^{-22}$ in relative strain. Thus it would still not see the earth sun gravitational radiation, and anyway it would also be too close to see the radiation field, it would be in the near zone. Elisa is planned to detect gravitational waves from supermassive black holes, from inside neutron stars, from further out and longer while black holes are merging, from cosmological objects and earlier than the CMB, and others.

Stil, it might be possible, you'd have to calculate if it could get a greater accuracy on the pseudo-static gravity as the earth sun going around each other than seen so far - I'm guessing probably.

Most of those satellite systems will be built for gravitational wave astronomy, don't think the earth or sun effects will be much part of them. You'd have to figure out a way to piggyback on them for some experiment you propose. Nobody (that I've heard) cares much if Schwarzchild is totally accurate except in the strong field regime, i.e., black hole or neutron star or cosmology.