{'fraction_non_alphanumeric': 0.028403316394757957, 'fraction_numerical': 0.009414281893554427, 'mean_word_length': 4.4475524475524475, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Interferometric Measurement of Acceleration at Relativistic Speeds

We show that an interferometer moving at a relativistic speed relative to a point source of light offers a sensitive probe of acceleration. Such an accelerometer contains no moving parts, and is thus more robust than conventional"mass-on-a-spring"accelerometers. In an interstellar mission to Alpha-Centauri, such an accelerometer could be used to measure the masses of planets around other stars as well as the mass distribution of the Milky Way Galaxy.


INTRODUCTION
The Terrell effect (Terrell, J. 1959;Penrose, R. 1959) implies that a spherical source will always appear circular to a moving camera, in seeming contradiction to the naive expectation from Lorentz contraction in special relativity. In addition, for the simplest case of a sphere that extends across a small solid angle, the sphere will appear rotated.
Following this realization, much work has been done in generalizing the Terrell effect to cases where the solid angle is not necessarily small. It was found that while the spherical source is still viewed as a circle, more complex transformations than rotations are required to map the surface of the sphere to the one in the photographic plane (Hollenbach, D. 1976;Suffern, K. 1988;Scott, G. D. and van Driel, H. J. 1970). Here we will restrict our attention to the simple case of a small solid angle, and focus on the temporal aspect of the Terrell effect. Even if the sphere is not featureless, one cannot infer whether the sphere is moving and Lorentz contracted or stationary but merely rotated just by taking a snapshot of it using a conventional camera. However, each point in the photographic plane can be traced back to the source through null geodesics. Due to the finite speed of light, geodesics corresponding to different points in the circular photograph travel for differing amounts of time. The timing information encoded by each light ray conveys the true nature of the sphere's motion.
Whereas previous considerations of the temporal Terrell effect were based on an abstract construction involving a lattice of clocks (Sheldon, E. 1989), we propose a more natural way to detect the temporal Terrell effect based on interferometry, where the temporal information is encoded in phase measurements of light. In addition, we show that the temporal Terrell effect allows a measurement of acceleration with high precision due to a relative motion between the source and the camera.
Recently, an interstellar travel mission, Breakthrough Starshot (Breakthrough Starshot 2016), was initiated. The project aims to send a large number of gram mass microchips equipped with cameras to the nearest star system, Alpha-Centauri (4.2 light years away from Earth), at 20% the speed of light. These microchips will aim to take close-up images of the planets in the star system. Furthermore, if during the travel these chips pchristian@cfa.harvard.edu; aloeb@cfa.harvard.edu emit light pulses at known intervals, they can be used as standard-clock beacons much like pulsars. A passing gravitational wave will modulate the time between pulses, allowing the detection of long wavelength gravitational radiation.
One other science benefit of such a mission could be to measure the masses of both exoplanets and their host stars as well as to test theories of modified gravity. Launching a spacecraft to a relativistic speed, however, requires a high acceleration that could potentially damage conventional mechanical accelerometers. A feasible alternative is to use the Doppler shift of the signal between the spacecraft and Earth to measure the accelerations. However, the total shift in speed for a spacecraft passing at 0.2c at a distance of 1 AU from an Earth mass planet is ∼ 4 × 10 −3 cm s −1 . The best Doppler sensitivity for bright stars is 4 orders of magnitude weaker (Schneider et al. 2011) and it is difficult to imagine performance that is much better for a source that is much fainter than a star.
In particular, an Earth-mass planet was recently discovered in the habitable zone (11.2 days orbit) around Proxima Centauri, a 0.12M star in the three-star system of Alpha Centauri (Anglada-Escude, G. et al 2016). The radial velocity technique by which the planet was detected allows to set only a minimum value on its mass, 1.3 Earth masses. Using a fleet of spacecrafts equipped with cameras and laser communication devices to the vicinity of this planet, Breakthrough Starshot will aim to address the question whether the planet hosts life by taking color photographs and potentially measuring its mass.
Our proposed interferometric accelerometer possesses a sensitivity that is superior to the Doppler technique. Furthermore, due to the interferometric nature of the proposed accelerometer, an array of N 1 cameras would provide N 2 independent measurements instead of N if used merely for standard Doppler measurements.
The outline of the paper is as follows. In §2 we describe the Terell effect. In §3 we discuss the application of the temporal Terrell effect in the realm of interferometry. In §4 we propose a method of measuring accelerations using such an interferometer. In §5 we describe the uncertainties inherent in the measurements and in §6 we describe various scientific utilizations of such accelerometers. Finally, we summarize our conclusions in §7.

THE TERRELL EFFECT
In the reference frame of a camera, located a distance y 0 R away from a radiating circle of rest frame radius R, the shape of the circular source is an ellipse given by where x, y, z, and t are the camera frame 4−coordinates, v the relative velocity between the camera and the source, and γ is the corresponding Lorentz factor. Due to the finite speed of light, c, the emission time t is given by where T is the arrival time of the photons to the camera. Without loss of generality, we can choose a time coordinate for which T = 0, yielding, This equation can be solved for t(x), the time at which the photons are emitted along the circle parameterized as a function of the x coordinate, where the ± in the square-root corresponds to light emitted from the far and near ends of the circle. The Terrell effect amounts to the fact that the (x, y) values yielding a real t solution inscribe a circle in the (x, y) plane. However, the two end points of the circle corresponding to the equation are rotated from that of a circle at rest by an angle (see Figure 2), θ T = arcsin β .
3. THE TEMPORAL TERRELL EFFECT AND INTERFEROMETRY Next, we shift from the point of view of the camera to that of the source. We consider an interferometric array of cameras arranged in a circle of radius R in its rest frame. This array is moving at a speed cβ in the x direction at a large distance y 0 R from a point source. Because electromagnetism is symmetric with respect to time reversal, this set up is exactly the same as the Terrell configuration described in the previous section, where now the "camera" is the source and the "radiating circle" is the array.
In the frame of the point source, the array is distorted into the ellipse of equation (1). However, due to the Terrell effect, the situation is equivalent to where the array stays circular but is rotated, as seen in Figure 1. This is simply equivalent to relativistic aberration: a moving source will appear as if it is oriented at a different angle.
An advantage of an interferometer array over a "photon counting" camera is that it collects phase information. This allows us to leverage a lesser known aspect of the Terrell effect. A moving circle is mapped to a rotated circle; if the circle has no features this renders Lorentz  (b) shape of a square interferometric array in the frame of the source. The square array is Lorentz contracted, resulting in the objective shape depicted in panel (a). However, due to the Terrell effect, the apparent shape of the array is rotated by an angle θ T = arcsin β. If the array would take an image of the source, the moving source would appear as if it is located at a different angle on its sky due to relativistic aberation.
contraction unphotographable. Temporal effects stemming from the Lorentz time dilation and the light travel time are, however, detectable by an interferometer.
The two end points of the circular array (where t(x) has only one real solution; see Figure 2) are given by equation (5), which can be solved to yield two equations for x as a function of y 0 and β. Plugging these values of x back to equation (1) gives the arrival time of photons at these points if emitted from y 0 at t = 0. Labelling the leftmost point A and the rightmost point B, we can calculate the time difference between the photons arriving at A and at B (see Figure 2), Photons arriving at B have to travel for an extra time 2βR/c compared to photons arriving at A. The phase difference due to this extra travel time for electromagnetic waves with a frequency ν = ω/2π and a wavelength λ = c/ν is For an interferometric array where R/λ is of order unity, this extra phase is also of order unity at relativistic speeds with β ∼ 1. However, the larger the array is (in units of wavelength), the larger would this extra phase be (modulus 2π, as with any phase measurements). Since the ratio R/λ is known, this phase measurement can be used to infer the speed, β. Combined with a measurement of angular position change, one can use this to measure the distance between the array and the source, y 0 .

THE TERRELL ACCELEROMETER
Based on the derivations in §3, we are now at a position to consider an accelerometer on board a spacecraft that contains no mechanical parts and is usable as long  (5). In the source frame, these points label the first photon (B) and the last photon (A) received, forming the apparent shape of the array. Photons emitted after B miss the array as it moves to the right, while photons emitted prior to A miss the array as it is not there yet. In the frame of the array the photons are emitted at an angle θ T = arcsin β. Every pair of points other than A and B reflect two solutions of equation (4), for positive and negative values of y corresponding to the far and near sides of the array.
as the spacecraft's antennae are operational. If our interferometric array passes close to an object, it would be subject to gravitational acceleration. This induces a time derivative,β, to the array's speed that could be measured. Here, we will assume the mildly relativistic regime and expand all our equations to leading order in β. Moreover, we will assume that the acceleration is small compared to the velocity during the encounter, in the sense that cβ |βy 0 /c|. Furthermore, note that β is oriented initially radially away from the observer.
Since the Terrell effect respects the fact that information travels at most at the speed of light, c, in reality the gravitational field should also be expanded to leading order in β. In a relativistic theory, there is no action at a distance, and the gravitational changes are transmitted at speed c. Since the gravitational field scales as 1/d 2 , where d denotes the distance between the source and the array, we find that to first order in β, where n is the unit vector between the array and the object, M the object's mass, G is Newton's constant, and the equation is evaluated at the retarded time t r = (t − d/c). In electrodynamics, this is the familiar equation for the electric field of a moving body as given by the Lienard-Wiechert potentials, taken to first order in β and setting the acceleration term to be small compared to the velocity term (Jackson, J. D. 1998). To lowest order, general relativity reduces to the gravitoelectric and gravitomagnetic fields. The n term in equation (9) is the standard Newtonian acceleration, while the term proportional to β is a special relativistic term. In a flyby encounter, where the image is taken when the interferometric array and the gravitating object are at closest approach, n − β evaluated at t r points directly towards the instantaneous position of the object and is perpendicular to the instantaneous β.
For the sake of clarity, we examine the limit of Newtonian gravity, where t r → t and only the n term is present. We consider a flyby where the array is passing the gravitating mass near closest approach, i.e. the x component of the velocity is much larger than the y component, |β x | |β y |. Furthermore, we assume that the array is already near closest approach, where the x displacement of the object to be much smaller than its y displacement, |x 0 | |y 0 |. In this case, the x component of the acceleration is given as, The first derivative of equation (8) yields, yielding a relation between the rate of change of the phase and the acceleration of the system. This gives, As mentioned in section III, the value of y 0 can be inferred from combining the measurement of the phase, φ, which directly probes β x ≈ β with measurements of angular acceleration,θ, The phase change due to the acceleration is given by the integral of equation (12) with respect to time, where τ is the observation time and τ 0 is an arbitrary starting time. This phase change is parabolic with respect to time; it is negative as the spacecraft approaches the gravitating body, and positive as it moves away from it. If we define T 0 = 0 at the point where the phase change is zero (when the spacecraft is directly in front of the gravitating body), then the mass of the body is given by For an array of N antennae, this procedure can be repeated for every single baselines in the array, resulting in N (N − 1)/2 independent measurements of the mass, M .

NOISE AND SYSTEMATICS
To assess the signal's detectability, we compare them to the noise inherent in the interferometer. Free from atmosphere induced errors plaguing ground based interferometers, the fundamental phase noise of the interferometer is (Thompson, Moran, and Swenson 1986), where T S and T A are respectively the system and antenna temperatures, B the bandwidth, N the number of baselines, and τ the observation time. The system temperature is given in terms of the physical temperature T of the detector by (Thompson, Moran, and Swenson 1986), where h and k B are the Planck and Boltzmann constants, respectively. The antenna temperature T A is related to the source's intensity by where A is the area of the antenna and S the flux density of the source. For our purposes, the point source chosen for the measurement is the Sun (as seen from the Alpha-Centauri system), N is taken to be 100, A to be 1 cm 2 , the physical temperature T to be the temperature 1 AU from a sunlike star (270 K), and the bandwidth B to be 10 percent of the frequency of observation. For such parameters, equation (19) implies a phase noise level at a wavelength of 1µm of Substituting the passage time τ ≈ d/βc as the observation time, we obtain that σ φ ≈ 10 −22 for d being the Earth-Moon distance. Within a single chip, thermal expansion causes the length of the baselines to change. Baselines between antennae located onboard different spacecrafts could also change due to orbital drifts. These phenomena generates systematic uncertainties that could be controlled by monitoring the relative positions of the antennae. This can be done by transmitting light signals between the antennae. Timing these signals allows for the distances and relative velocities between antennae to be known. In the swarm configuration where each antennae is on board a different spacecraft, either laser or the same transmission that is used to communicate back to Earth can be utilized for this purpose. In the case of multiple antennae on a single chip, optical fibers can be used as an alternative.
6. SCIENCE UTILIZATION 6.1. Weighting Stellar and planet masses At a distance of 1 AU from a Sun-like star, the total change of phase for an array moving at β = 0.2 is given by where in this case we equated the observation time to the crossing time, τ ≈ d/βc. For an array of antennae onboard the spacecrafts envisioned for the Breakthrough Starshot initiative with R ∼ 1 m and λ ∼ 1µm, this phase change is ∆φ ∼ 0.3, which is large compared to the noise described by equation (19). One could also send many antennae on separate spacecrafts, and the interferometry process can be conducted across different spacecrafts. In this case, R would be the distance between spacecrafts, allowing the use of longer wavelengths to produce the same amount of phase change.
If the spacecraft passes sufficiently close to a planet, the gravitational pull of the planet would dominate over that of its parent star. For a planetary system similar to the Sun-Earth system, this would occur when the planetspacecraft separation is about 3×10 10 cm, or roughly the Earth-Moon distance. In such a flyby, the total change of phase at β = 0.2 is given by which gives ∆φ ∼ 4.7 × 10 −4 for the configuration with R ∼ 1 m and λ ∼ 1µm. This phase difference is large compared to the noise described by equation (19). This will complement an alternative measurement of the planet's mass that is enabled by sending a spacecraft to the system, such as by determining the orbital inclination of the planet by resolving the planet's orbits as the spacecraft approaches the planetary system.
We note that it is not necessary to use the star or planet as the light source. If the array is focused on some distant point source like the Sun, the acceleration of the detector will still be apparent. However, if the star or planet is used as the light source, its resolved image can be viewed as a collection of point sources, and each of them will suffer the same temporal drift in phase due to the acceleration. The observer would see each pixel in the u-v plane of the interferometer drifting in time. This will also enable a measurement of the acceleration.

Measuring density profiles of planetary systems
Approaching the target star system with antennae allows one to measure the enclosed mass of the system as a function of distance from the star. In particular, this allows to measure the cumulative mass of Kuiper belt analogs in the target system. At a distance of 30 AU, a Kuiper belt analog of combined mass ∼ 30M ⊕ produces an extra acceleration of which translates into an additional phase change of For the configuration with R ∼ 1 m and λ ∼ 1µm, this phase change becomes ∼ 10 −6 , which is above the noise implied by equation (19).

Measuring the Milky Way mass
Travel to the closest star from Earth requires an extended period of time for the journey. At β = 0.2, it will take roughly 20 years to travel from Earth to Alpha-Centauri. During this travel time, one can use the phase