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Irreversible Physics Along Closed Timelike Curves

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Irreversible Physics Along Closed Timelike Curves Can we travel to the past? Irreversible physics along closed timelike curves Carlo Rovelli∗ Aix-Marseille University, Universit´ede Toulon, CNRS, CPT, 13288 Marseille, France. Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada. The Rotman Institute of Philosophy, 1151 Richmond St. N London, Ontario N6A 5B7, Canada. (Dated: December 10, 2019) The Einstein equations allow solutions containing closed timelike curves. These have generated much puzzlement and suspicion that they could imply paradoxes. I show that puzzlement and paradoxes disappears if we discuss carefully the physics of the irreversible phenomena in the context of these solutions. I. INTRODUCTION Notice that since s(0) = 0 but s(1) = S, ds is not an exact one-form on γ: that is, it is not the differential of Traveling to the distant future is possible in principle: a continuous function s on γ. it suffices to travel fast enough, or to spend little time A clock is a mechanical device including for instance sufficiently near to a black hole horizon. But can we a harmonic oscillator beating proper time. Let θ be the travel to the past? angular variable describing the position of the hand of General relativity appears to suggest that this might the clock, and ! = 2π=T its frequency. The position of be possible in principle, because it allows solutions with the hand of the clock along the path is given by closed timelike curves [1,2]. But the idea of traveling θ(τ) = θ(0) + ! s(τ); modulo 2π (3) to the past has always raised puzzlement. It is com- monly said that traveling to the past would generate log- and the number of oscillations since the start is the (real) ical paradoxes, such as the possibility of killing our own number parents before we were born. ! s(τ) n(τ) = s(τ) = : (4) Here I argue that this and similar paradoxes disappear 2π T if we examine with care the full thermodynamical and statistical physics along the closed timelike curves (see If the two physical variables gab(x) and θ(τ) satisfy also [3]). The paradoxes disappear, without any need of the equations of motion it follows necessarily that θ(0) = resorting to quantum physics, as suggested for instance θ(1). That is, the physical equations impose that the in [4,5]. hand of the clock \comes back at the end of the loop" to the same position as it was at the start of the loop. In other words, the total number of oscillations around the line, N = n(1) = S=T , must be integer. If N is not an II. A CLOCK AROUND A CLOSED TIMELIKE integer, the equations of motion are not satisfied, and the CURVE theory tells us that this is an impossible state of affairs. This observation might seem to trivially eliminate any Consider a clock coupled to the gravitational field and paradox but it appears to be strange: intuitively it seems whose worldline follows a closed timelike curve. Let that if I follow a closed timelike curve I get back to the 2 a b ds = gab(x)dx dx be a solution of the Einstein equa- initial spacetime point but I can have a memory of hav- tions (with the clock) containing a closed timelike curve ing being around, therefore there should be something γ. Here a; b = 0; 1; 2; 3 and I assume signature [+,-,-,-] for different in my final configuration, with respect to my de- simplicity of notation. The proper time along the entire parting configurations, after going round the loop. This, R timelike curve is S = γ ds: Writing the curve explicitly after all, is what intuitively mean by \traveling back to in coordinates, γ : τ 7! γa(τ); τ 2 [0; 1], the proper time the past". grows along the curve grows as To investigate this, let us focus simply on the total amount S = NT of proper time along the closed timelike Z τ q 0 0 a 0 b 0 line γ. This is measured by the integer N. Any clock s(τ) = dτ gab(γ(τ ))γ _ (τ )_γ (τ ) (1) 0 capable of measuring N would indeed get back to the initial spacetime location with a record of having been whereγ _ a = dγa/dτ. A closed timelike curve satisfies around. γ(1)=γ(0) and its normal is everywhere timelike, that is Can the clock keep track of the number N of its oscil- lations? Physical clocks do that regularly: they not only a b gabγ_ γ_ > 0: (2) beat the period, but have also a device that records the number of oscillations. However, now comes the main point of this paper: any mechanical clock that counts oscillations dissipates en- ∗Electronic address: [email protected] ergy. That is, any clock is ultimately thermodynamical. 2 Entropy The escapement of a pendulum clock for instance cannot Proper time work without friction. This fact has been emphasized for instance by Eddington [6]. It is also discussed in a won- derful lecture by Feynman [7]. This is the key observation Thermodynamical time of this paper. Let us see what it implies, disregarding, Thermodynamical time for the moment, the statistical mechanics underpinning γ<latexit sha1_base64="HZsI6crMQgxF7BWk5N6VylP7rA8=">AAAB7XicbVDLSgNBEOz1GeMr6tHLYBA8hV0V9Bj04jGCeUCyhN7JbDJmZnaZmRVCyD948aCIV//Hm3/jJNmDJhY0FFXddHdFqeDG+v63t7K6tr6xWdgqbu/s7u2XDg4bJsk0ZXWaiES3IjRMcMXqllvBWqlmKCPBmtHwduo3n5g2PFEPdpSyUGJf8ZhTtE5qdPooJXZLZb/iz0CWSZCTMuSodUtfnV5CM8mUpQKNaQd+asMxasupYJNiJzMsRTrEPms7qlAyE45n107IqVN6JE60K2XJTP09MUZpzEhGrlOiHZhFbyr+57UzG1+HY67SzDJF54viTBCbkOnrpMc1o1aMHEGqubuV0AFqpNYFVHQhBIsvL5PGeSW4qPj3l+XqTR5HAY7hBM4ggCuowh3UoA4UHuEZXuHNS7wX7937mLeuePnMEfyB9/kDh5WPFw==</latexit> thermodynamics (to which I return later). Spacetime To have a clock counting oscillations, we need dissi- pation, hence entropy to grow. Let S(τ) be a measure of local entropy along the closed loop γ. For the clock Figure 1: The direction of increasing proper time around a to work all along γ, registering the number of its oscil- closed timelike curve does not agree with the future direction lations, we need dS(τ)/dτ > 0 everywhere going around of the thermodynamical arrow of time. the loop. But since γ is a loop, S(τ) cannot grow monotonically as we go round. Therefore it is impossible for a clock to forward passage of time, cannot loop back onto itself, count its own oscillations along a closed timelike curve. because it is a gradient. There is no physical way for a clock to count its oscilla- In other words, the proper time measured by a re- tions along a closed timelike curve. This conclusion has versible periodic device is mathematically described by far reaching consequences, discussed below. a closed one-form ds which can be well defined along a closed path. But the time that distinguishes the past from the future is a thermodynamical quantity dS that is III. TRAVEL TO THE PAST IS the differential of a state function, and therefore is exact, THERMODYNAMICALLY IMPOSSIBLE and therefore cannot grow uniformly along a circle. What happens in the case in which along a closed The above conclusion is in fact far more general than timelike curve the entropy increases and decreases in τ? clock oscillation counting. For instance, if we want to Consider the simplest possibility where dS/dτ > 0 for travel to the past and arrive to the past keeping our τ 2 γ+ = [0; τ^] and dS/dτ < 0 for τ 2 γ− = [^τ; 2π]. Then memory of events happened in the future, we need some the above discussion immediately clarifies the physics of device (like our brain) capable of memory. But memory this solution: for everything that concerns irreversible is an irreversible phenomenon (we remember the past not phenomena such as memory, decisions and keeping track the future) and, like all irreversible phenomena, is based of the past, the effective direction of time is towards in- solely on the only fundamental irreversible law: the sec- creasing τ in γ+ and towards decreasing τ in γ−. See ond principle of thermodynamics dS(τ)/dτ ≥ 0. Along Figure 1. All paradoxes disappear. a closed timelike loop γ the only possibility of having The above discussion clarifies the physics of closed dS/dτ ≥ 0 everywhere is having dS/dτ = 0. But this timelike curves and their relation with our intuition that means that all the processes around γ are reversible, and \time moves only forward". Our intuition is based on the therefore there can be no memory. fact that all phenomena that we see as characteristic of The phenomenology that we commonly associate to \forward moving time" are thermodynamical phenomena the forward passage of time (memory, decisions, cumula- where entropy grows. Since entropy cannot grow con- tive counting of the oscillations of a periodic device....) stantly along a closed loop, we cannot travel forward in depends entirely on the second principle of thermody- time and return to a previous space time location in this namics. This implies that the future direction is deter- sense, even if spacetime admits closed timelike curves. mined by the derivative of a state function, the entropy. Since no function can uniformly increase around a circle, we can never \travel to the past" in the sense of arriving IV. THE STATISTICAL MECHANICS PICTURE to the past having memory of the future, having counted the oscillations of our clock, being in a position of act- Thermodynamics is nothing but mechanics restricted ing different that what we did, or similar. These are all to a relatively small number of \macroscopic" (\coarse- thermodynamical phenomena (that require irreversibil- granined") variables, and under the condition that the ity), and thermodynamics does not permit travel to the entropy defined by this coarse-graining was low in some past in this sense. region, denoted \past". Any thermodynamical statement General relativity allows closed timelike curves, and can therefore be in principle translated into a statistical this is not incompatible with anything.
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