A Gravitational Energy-Conservation Paradox in General Relativity?, revised draft in light of Max Li’s very thoughtful and detailed analysis of the initial draft

Consider Spherical-Shell Experimenter A (SSEA) at rest at Schwarzschild-coordinate radial distance rA > rS (rS = the Schwarzschild radius) and Spherical-Shell Experimenter B (SSEB) at rest at Schwarzschild-coordinate radial distance rB >> rA from the center of a spherically-symmetrical nonrotating gravitator of mass M. (This gravitator could be either a black hole or a non-black-hole. If a black hole rB >> rA > rS; if a non-black-hole of Schwarzschild-coordinate radius R, rB >> rA > R > rS.) Radial ruler distance is related to Schwarzschild-coordinate radial distance by dl = [1 – (rS/r)]−1/2dr [1]. [In accordance with Rindler [1], we employ r to denote Schwarzschild-coordinate radial distance [1] (which is also radial area distance and radial distance from apparent size [1]) and l to denote radial ruler distance [1].] The gravitational field strength at rA is gA = [GM/(rA)2][1 – (rS/rA)]−1/2 [2], and that at rB is gB = [GM/(rB)2][1 – (rS/rB)]−1/2 [2].

Let the small mass m > rA > rS if a black hole and rB >> rA > R > rS if a non-black-hole of Schwarzschild-coordinate radius R, f = [1 – (rS/rA)]1/2 and hence dEB = mgBdl = [GMm/(rA)2][1 – (rS/rA)]−1/2dr very nearly.} Of course, real strings are not massless, but SSEB can recover the energy (over and above dEB required to lift the mass m) expended to lift the strings by then lowering the mass mand hence also the strings by the same distance that they were lifted. This raising/lowering cycle can be repeated.

Here is the paradox: After SSEB has lifted the mass m located at rA by ruler distance dl, let SSEA lower it by the same ruler distance dl. This yields SSEA energy of dEA = mgAdl = (GMm/r2)[1 – (rS/rA)]−1dr[3] = dEB/f. Thus prima facie it seems that while neither SSEA nor SSEB can contravene the law of energy conservation if acting alone, SSEB can enable creation of energy for SSEA— but only for SSEA. This cycle can then be repeated, with net energy creation of dEnet = dEA − dEB= dEA(1 – f) per cycle — but only for SSEA. The gravitational redshift reduces by the factor f the energy cost dEB that SSEB requires to lift the mass m but not the energy yield dEA that SSEA obtains by lowering it. (Of course, prima facie it also seems that if SSEA lifts the mass m, expending energy dEA, but then lets SSEB lower it, owing to the gravitational-redshift loss SSEB obtains energy dEB = fdEA < dEA; but destroying energy seems less useful than creating it.)

Here is a more extreme version of the paradox: SSEB must expend energy ∆EB = mc2 — the entire energy content of the mass m — to lift it from rS to rB >> rS with massless strings [4]. Obviously, SSEB can obtain energy ∆EB = mc2— the entire energy content of the mass m — by lowering the mass m from to rB >> rS to rS with massless strings [4]. [Of course, real strings are not massless, but SSEB can recover the energy (over and above ∆EB required to lift the mass m) expended to lift the strings by then lowering the mass m and hence also the strings by the same distance that they were lifted. This raising/lowering cycle can be repeated.] Now: What if SSEB lifts the mass m from to rS to rB >> rS, expending energy ∆EB = mc2 — but then lets SSEA lower the mass m from rB >> rS to rS? Then SSEA — but only SSEA — obtains energy ∆EA= ∆EB/f = mc2/f > mc2. This cycle can then be repeated, with net energy creation of ∆Enet = ∆EA − ∆EB = ∆EA(1 – f) = mc2[(1/f) – 1] per cycle — but only for SSEA. The gravitational redshift reduces by the factor f the energy cost ∆EB that SSEB requires to lift the mass m but not the energy yield ∆EA that SSEA obtains by lowering it. (Of course, prima facie it also seems that if SSEA lifts the mass m, expending energy ∆EA = mc2/f, but then lets SSEB lower it, owing to the gravitational-redshift loss SSEB obtains energy ∆EB = f∆EA = mc2 < ∆EA; but destroying energy seems less useful than creating it.)

Four points in reply to Max Li’s very thoughtful and detailed analysis of the three-step cycle previously considered in the initial draft (Experimenter A lowers a mass m, Experimenter A transmits the energy thereby obtained to Experimenter B who receives it with a gravitational-redshift loss, Experimenter B lifts the mass m): (i) In this revised two-step cycle (SSEB lifts a mass m, SSEA lowers the mass m), we take not only the gravitator, but also the mass m (not only when stationary but also when being lifted or lowered) and both SSEA and SSEB, to be spherically symmetrical (which implies also nonrotating). (ii) In this revised two-step cycle (SSEB lifts a mass m, SSEA lowers the mass m), there is no transmission of energy from SSEA to SSEB (or vice versa). So energy costs associated with changes of the spacetime geometry accompanying such energy transmissions do not occur. (iii) Since not only the gravitator of mass M, but also the mass m (not only when stationary but also when being lifted or lowered) and both SSEA and SSEB, are spherically symmetrical (which implies also nonrotating), by Birkhoff’s Theorem [5] and by spherical symmetry [5] the mass M of the gravitator and hence also the spacetime geometry at r < rm, where rm is the Schwarzschild radial coordinate of the spherical-shell mass m, do not change via SSEB raising the mass m or via SSEA lowering it [5]. And it is only M and the associated spacetime geometry at r < rm that determine the energy cost imposed on SSEB for raising m and the energy yield obtainable by SSEA upon lowering m. (iv) This seems to be the most important point: This prima facie creation of energy is of a restricted, local, and non-universal nature. It benefits only SSEA. SSEA cannot export even the slightest smidgen of this prima facie nascent energy to r ≥ rB, because the gravitational redshift would take all of it away. Hence with respect to the external, entire, Universe, energy is strictly conserved. (If instead SSEA lifts the mass m and then lets SSEB lower it, the same gravitational-redshift loss with respect to the external, entire, Universe is imposed this way: SSEB then receives simply the standard red-shifted energy. Furthermore, since rB >> rA > rS if a black hole and rB >> rA> R > rS if a non-black hole, SSEB is effectively in the external, entire, Universe. Consequently, also the external, entire Universe then receives simply the standard red-shifted energy, and hence with respect to the external, entire, Universe, energy is strictly conserved.)

Aside 1: We note that with respect to the entire Universe, energy is strictly conserved even in cases where this is sometimes questioned. For example, it is sometimes stated that the redshift of the frequency ν (be it the average frequency, the peak frequency, or the frequency at any point on the near-blackbody spectrum) of the thermal cosmic background radiation (CBR) as the Universe expands and its radius of curvature ℝ (assuming for simplicity that it is a 3-sphere) increases [ν(ℝ2)/ν(ℝ1) = ℝ1/ ℝ2] entails a diminution of energy. [This is mentioned in the second sentence of the second paragraph of Section 4 (Discussion) of a very thought-provoking arXiv paper [6].] But this decrease in the CBR’s thermal kinetic energy can be construed as being counterbalanced by its increased gravitational potential energy if the Universe’s gravitational potential within the 3-sphere (i.e., of the entire 3-sphere Universe) increases as the Universe expands as per ν2/ν1 = exp[−(Φ2– Φ1)/c2] [7] or equivalently Φ2 – Φ1= c2 ln(ν1/ν2) [7], which in this case yields Φ(ℝ2) – Φ(ℝ1) = c2 ln[ν(ℝ1)/ν(ℝ2)] = c2 ln(ℝ2/ℝ1) [7]. (If the Universe oscillates, as it contracts the increase in the CBR’s thermal kinetic energy can likewise be construed as being counterbalanced by its decreased gravitational potential energy.) Similarly, for thermal radiation trapped at the outer edge of a 2-sphere shell of mass M whose gravity is sufficiently weak that the Newtonian potential Φ(ℝ) = −GM/ℝ is sufficiently accurate, ν(ℝ2)/ν(ℝ1) = exp[−(GM/c2)(1/ℝ2 − 1/ℝ1)] ≈ 1 − [(GM/c2)(1/ℝ2 − 1/ℝ1)] [7]. Thus the total (thermal kinetic plus gravitational potential) energy of thermal radiation is conserved for both the 3-sphere Universe and the 2-sphere “universe” when they expand or contract: its decreased thermal kinetic energy counterbalanced by increased gravitational energy during expansion, and vice versa during contraction.

Aside 2: A distinction between what happens with respect to a specific, special, reference frame and what happens with respect to the entire Universe [recall Item (iv) above] perhaps may obtain concerning the Second Law of Thermodynamics as well as [again recall Item (iv) above] concerning the First Law (energy conservation). For example, if closed timelike curves exist and if they are traversable, an experimenter who traverses one and returns to the past experiences an enormous decrease in the entropy of the entire Universe with respect to the experimenter’s own reference frame: randomized thermal energy restored to nuclear energy for all the stars in the Universe! By repeatedly traversing a closed timelike curve, at least prima facie it seems that the experimenter can keep all the stars in general and our Sun in particular alive forever with respect to the experimenter’s own reference frame and own proper time (at least in an ever-expanding Universe: a Big Crunch, even if only to a finite density and followed by a Big Bounce, might destroy the experimenter’s machinery). By contrast, in the comoving frame of an ever-expanding Universe, all the stars will have burned out in ≈ 1013 years (our Sun in less than a mere 1010 years) of cosmic time [8] (the Universe’s proper time [8]). A violation of the Second Law seems avoidable only if — over and above all other requirements that the laws of physics impose for traversing a closed timelike curve (if they allow it to be traversed at all) — there is furthermore imposed a negentropy and free-energy cost at least sufficient to compensate for recharging all the stars in the Universe! And all other entropy-increasing processes as well! (Unless, perhaps, if increases in entropy and hence the Second Law of Thermodynamics are only apparent, order being “folded up” — i.e., converted from sensible to latent — rather than destroyed [9]?)

[1] Wolfgang Rindler, Essential Relativity: Special, General, and Cosmological, Second Edition (Oxford University Press, Oxford, UK, 2006), Sections 11.1 and 11.5, and pp. 384−385.

[2] Rindler, op. cit., Section 11.2.

[3] Rindler, op. cit., Sections 1.16, 11.2, 11.6, and 12.2.

[4] Rindler, op. cit., Section 12.2.

[5] Rindler, op. cit., Sections 11.2B, 14.4, and 16.1H (especially Section 11.2B).

[6] D. P. Sheehan and V. G. Kriss, Energy Emission by Quantum Systems in an Expanding FRW Metric. arXiv:astro-ph/0411299v1 (11 Nov. 2004).

[7] Rindler, op. cit., Section 1.16, especially Equation 1.11.

[8] Rindler, op. cit., p. 359.

[9] P. C. W. Davies, The Physics of Time Asymmetry, First Paperback Edition (University of California Press, Berkeley, California, 1977), David Joseph Bohm’s hypothesis cited in the fourth paragraph of Section 7.4.