Let be the Cartesian coordinate vector joining one end of a laser interferometer arm to another; and let this interferometer be freely-falling in a weakly curved spacetime
Practically all the pedagogical literature on gravitational physics tell us the distortion of this arm due to the presence of a gravitational wave is proportional to the transverse-traceless part of the metric perurbation :
But, what does “transverse-traceless” (TT) actually mean here? The field theorist reader would likely think that the must be the gauge-invariant massless spin-2 graviton, which obeys
Massless Spin-2 The helicity–2 character is the result of these TT conditions; for, each Fourier mode, it is always possible to find a basis of polarization tensors , namely
such that under a rotation along the axis through an angle ,
The may be viewed as the eigenvalues of the generator of rotation on the plane perpendicular to .
Gauge-Invariance Next, by gauge-invariance, I mean here that, under an infinitesimal change in coordinates
the TT character of this gravitational wave ensures it remains unaltered:
Now, this gauge-invariance is often invoked as a criterion for physical observability: for, if some observable is expressed in terms of the gauge dependent components in eq. (1), how does one know if the physical effect at hand cannot be rendered trivial simply by choosing an infinitesimally different coordinate system? However, the main point of this post is — the converse most certainly does not hold:
Gauge invariance does not imply physical observability.
The reason is simple: even though is gauge-invariant, it is acausal. More specifically, within the linearized approximation of General Relativity, this massless spin-2 gravitational wave (GW) admits the solution
where is the Green’s function of the TT GW and is the stress-energy tensor. Through a direct calculation, in arXiv: 1902.03294, Yen-Wei Liu and I showed that is non-zero outside the past light cone of the observer at . In other words, the signal receives contributions from portions of that are spacelike separated from the observer — and therefore cannot be a standalone observable.
Tidal Forces & GW Strain So, what is one to make of the formula in eq. (2) then? To this end, we first recall that — if describes the displacement between a pair of infinitesimally nearby timelike geodesics (a pair of freely-falling test masses, for instance); the fully covariant acceleration of this displacement vector is driven by the Riemann tensor:
(The is the unit norm timelike vector tangent to one of the two geodesics.) In a flat spacetime, the Riemann tensor is exactly zero; i.e., a pair of parallel lines will remain parallel because their relative acceleration is zero. Now, at first order in the perturbation , both sides of eq. (5) must be gauge-invariant since their ‘background value’ (evaluated on ) is zero. This in turn means we can choose any gauge we wish. Synchronous gauge, where the perturbations are strictly spatial
is particularly pertinent in this context of 2 infinitesimally close-by free falling test masses. For, if the of the synchronous-gauge coordinate system refers to the proper times of these free falling objects, their spatial coordinates are then automatically time independent, and
If we assume the clocks on this pair of test masses are synchronized at some initial time , then one may demonstrate using eq. (5) they will continue to remain so for later times; namely, if . Employing eq. (5”), the spatial tidal forces described by the geometrically induced relative acceleration is now
with the notation denoting the components of the linearized Riemann tensor.
Within the synchronous gauge, the proper distance between two free falling test masses and at a given time (accurate to first order in perturbations) is
from which, we see that the fractional distortion is
Moreover, the linearized Riemann in the synchronous gauge reads
Remember the linearized Riemann is gauge-invariant, so it ought to be possible to re-express in terms of gauge-invariant metric perturbation variables. (More on this below.) In fact, what Yen-Wei and I argued in arXiv:1902.03294 was that, in the far zone where (observer-source distance)/(characteristic timescale of source),
Therefore, in frequency space
that the linearized Riemann is gauge-invariant allows us to equate (8) and (8′) to conclude — for finite frequencies —
Important aside By placing at the center-of-mass of the material source of gravity, in the same far-zone limit, the transverse-traceless GW reduces to
Namely, the far-zone massless spin-2 GW is the de Donder gauge gravitational perturbation projected locally-in-space transverse to the propagation direction. But the de Donder gauge graviton is in fact causally dependent on the stress tensor; in the far zone, in particular,
This means the far zone TT GW in eq. (9′) is causal, even though its full form in eq. (4) is not. The reason is, the acausal portions begin at higher order in . In the GW literature, the local-in-space projection in eq. (9′) — what Ashtekar and Bonga, referenced below, dubbed to distinguish it from in equations (2′) and (2”) — is actually the one that is employed, not the tranverse-traceless one subject to equations (2′) and (2”). We see, the reason why it is possible to get away with mixing these two distinct notions of transverse-traceless projections is that they coincide when ; i.e., in the far zone. (Note: Racz and Ashtekar-Bonga, whose papers can be found below, have correctly complained that the GW literature wrongly mixes ‘tt’ versus ‘TT’.)
Summary Let us sum up the discussion within this section. In the far zone, the fractional distortion of the proper distance between the pair of free-falling test masses and at a given time is
This formula is to be understood as valid only for finite frequencies — for instance, LIGO is built to be sensitive to a limited bandwidth centered roughly at 100Hz. Otherwise, equating (8) and (8′), which was what led to eq. (9)-(10), actually misses the initial and its time derivative; in frequency space these initial conditions correspond to zero- Dirac function terms. In the limit where the wavelength of the GW is long compared to , so is approximately constant between and , eq. (10) then reduces to
This is equivalent to eq. (1); but to arrive at it we have assumed the following.
- The GW detector is in the far zone.
- The GW detector is only sensitive to finite gravitational wave frequencies.
- The GW detector’s proper size is much smaller than the gravitational wavelength.
Dynamical Degrees-Of-Freedom vs. Physical Observables In field theory speak, one often hears the statement that “4D Einstein-Hilbert gravity has only 2 dynamical degrees-of-freedom”. In its linearized form, we shall see this statement amounts to:
Of all the gauge-invariant variables formed from the metric perturbation in eq. (1) — the transverse-traceless tensor ; the transverse vector ; and the scalars and — only the tensor obeys a wave equation.
To build and out of the perturbation , refer to equations (A10), (A15) and (A16) of arXiv: 1611.00018. (Put ; remove the over-bars and note that .) What I wish to highlight here are the (3+1)D version of the equations-of-motion in (A25) and (A26):
The transverse-traceless conditions of equations (2′) and (2”) tell us, of the components of , only are independent. However despite this “2 d.o.fs” assertion regarding the TT GW, as I have already pointed out above its solution is acausal and cannot possibly be a standalone physical observable. In eq. (11) the is in fact a non-local functional of the spatial components of the stress tensor — heuristically, is smeared out over all space in such a manner that the resulting object obeys the constraints .
What, then, is one to make of this acausality; as well as the gauge-invariant content of linearized gravitation? A partial answer is offered by the spatial tidal forces exerted by geometric curvature, encoded within the discussed above. Yen-Wei and I showed that, even though the TT GW and its acceleration are acausal, the vector and scalars and appear in in such a way to precisely cancel out the acausal contributions from the tensor; with the end result yielding tidal forces that are strictly causally dependent on the material stress tensor:
Therefore, the tidal squeezing and stretching of a Weber bar or of a laser interferometer’s arms is not to be attributed to the entire spin-2 massless graviton — because of its acausal character — but to only the causal part of its acceleration. Even in the (quasi-)static limit where the in eq. (12) all appear to become negligible; say, for instance, the contribution to the tides on Earth due to differential gravitational tugs from either the Moon or the Sun; we should not attribute the rising and ebbing of the oceans to the second derivatives of the Newtonian-like potential in eq. (11). Rather, on grounds that physical tidal forces ought to be causal, according to eq. (12), it still has to be attributed to the causal part of the TT tensor perturbation’s acceleration.
Micro-causality in QFT If you have taken a course on Quantum Field Theory, you might have been told that the amplitude for a particle to propagate from to is given by the vacuum expectation value (for scalar particles ). However, a direct calculation for non-interacting scalars would reveal this object is non-zero for spacelike separated and ; i.e., a particle has a non-zero quantum mechanical amplitude to propagate outside the light cone. See discussion in of Peskin and Schroeder (P&S) for instance. P&S goes on to assert
To really discuss causality, however, we should ask not whether particles can propagate over spacelike intervals, but whether a measurement performed at one point can affect a measurement at another point whose separation from the first is spacelike. The simplest thing we could try to measure is the field , so we should compute the commutator ; if this commutator vanishes, one measurement cannot affect the other. In fact, if the commutator vanishes for , causality is preserved quite generally, … [truncated] — Chapter 2, page 28
At the quantum level, are the transverse massless spin-1 photon (subject to ) or spin-2 graviton field physically observable? That is, can we perform a direct measurement on them? P&S does not tell us, but although the commutators of free scalar fields vanish outside the light cone — they obey micro-causality — the helicity-1 and -2 photons and gravitons do not.
This is simply because their commutators are proportional to the difference between their corresponding retarded and advanced Green’s functions. As already alluded to after eq. (4), these retarded/advanced transverse Green’s functions are in fact non-zero outside the light cone. I end with the following question:
Can this violation of micro-causality by massless spin-1 photons be exploited within a physical setup?
Note added: I forgot to mention an interesting related discussion that took place over at Distler’s blog regarding micro-causality. My sense is, he knows a whole lot more than I do — but, sadly, he has closed his comments section for the post.
- I. Racz, “Gravitational radiation and isotropic change of the spatial geometry,” arXiv:0912.0128 [gr-qc]
- A. Ashtekar and B. Bonga, “On the ambiguity in the notion of transverse traceless modes of gravitational waves,” Gen. Rel. Grav. 49, no. 9, 122 (2017) doi:10.1007/s10714-017-2290-z [arXiv:1707.09914 [gr-qc]]
- A. Ashtekar and B. Bonga, “On a basic conceptual confusion in gravitational radiation theory,” Class. Quant. Grav. 34, no. 20, 20LT01 (2017) doi:10.1088/1361-6382/aa88e2 [arXiv:1707.07729 [gr-qc]]
- Y. Z. Chu and Y. W. Liu, “The Transverse-Traceless Spin-2 Gravitational Wave Cannot Be A Standalone Observable Because It Is Acausal,” arXiv:1902.03294 [gr-qc].
- Y. Z. Chu, “More On Cosmological Gravitational Waves And Their Memories,”
Class. Quant. Grav. 34, no. 19, 194001 (2017) doi:10.1088/1361-6382/aa8392
- S. Weinberg, “Photons and gravitons in perturbation theory: Derivation of Maxwell’s and Einstein’s equations,” Phys. Rev. 138, B988 (1965).