# Remarks on the Compatibility of Opposite Arrows of Time II

## Abstract

**:**

## 1 Introduction

## 2 Wet carpets and detective stories

## 3 Phase space volume

^{3}

^{N}, where 3N is the number of degrees of freedom. In my arguments I had raised the question if a two-times boundary condition of low entropy would possibly reduce phase space to less than one cell – thus signalling dynamical inconsistence in the case that the conditions can be regarded as statistically independent. Schulman replied that a typical phase space volume for a gas would contain something like 10

^{1020.28}phase space cells. A reduction of the spatial volume by a factor of 64, say, would reduce the second exponent only in the second figure after the decimal point, such that this reduction could easily be applied twice.

^{20}particles. Under customary conditions here on earth, the phase space for each particle in a gas is of the order 10

^{10}h

^{3}. The resulting N -particle phase space (10

^{10})

^{N}h

^{3}

^{N }= 10

^{1021}h

^{3}

^{N}is slightly larger than Schulman’s choice. Reducing the single-particle phase space by a factor of 64 would indeed lead to a very “small” change, given by (10

^{10}/64)

^{N}= 10

^{1020.916}, although it repesents a reduction of phase space by a factor of (1/64)

^{1020}. Even forgetting Gibbs’ paradox and enlarging phase space by the enormous factor of N! ≈ N

^{N}would lead to “no more” than (10

^{20}10

^{10})

^{N}= 10

^{1021.477}. The entropy is therefore almost exclusively determined by the particle number N.

^{5}(that is, a reduction from 10

^{10}to 10

^{5}) in an irreversible cosmic process appears quite conservative, and would be far from requiring degenerate matter. Applied once, it reduces total phase space to (10

^{5})

^{N}= 10

^{1020.699}– apparently not drastically different from the numbers given above, but applied twice (at two sufficiently distant times) it leads to (10

^{0})

^{N}= 1 (independent of N)! A similar consistency problem would arise in Wheeler and Feynman’s timesymmetric absorber theory (see Chap. 2 of [6]).

## 4 Quantum measurement

## 5 Gravity

^{−}

^{7}K (or weaker) Hawking radiation. In order to get rid of gravitating objects for a cosmic “time reversal” in the not extremely distant future, one would need advanced (incoming coherent) radiation to reverse their gravitational contraction [8]. These objects are, therefore, strongly coupled to the general arrow of time, and a further indication that the arrow cannot vary from place to place.

## References

- Schulman, L.S. Time’s arrows and quantum mechanics. Cambridge University Press, 1997. [Google Scholar]
- Schulman, L.S. Phys. Rev. Lett.
**1999**, 83, 5419. - Schulman, L.S. Time’s Arrow, Quantum Measurements and Superluminal Behavior; Mugnai, D., Ranfagni, A., Schulman, L.S., Eds.; Consiglio Nazionale delle Ricerche: Roma, 2001. [Google Scholar]
- Zeh, H.D. Entropy
**2005**, 7(4), 199. http://www.mdpi.org/entropy/list05.htm#issue4. - Schulman, L.S. Entropy
**2005**, 7(4), 208. http://www.mdpi.org/entropy/list05.htm#issue4. - Zeh, H.D. The Physical Basis of the Direction of Time. Springer: Berlin, 2001. [Google Scholar]
- Wheeler, J.A.; Feynman, R.P. Rev. Mod. Phys.
**1949**, 21, 425. - Kiefer, C.; Zeh, H.D. Phys. Rev.
**1995**, D51, 4145. - Wigner, E.P. Am. J. Phys.
**1963**, 31, 6.Zeh, H.D. Found. Phys.**1970**, 1, 69.both reprinted in Wheeler, J.A., and Zurek W.H., “Quantum Theory and Measurement” (Princeton UP 1983). - Penrose, R. Quantum Gravity II; Isham, C.J., Penrose, R., Sciama, D.W., Eds.; Clarendon Press, 1981. [Google Scholar]
- Zeh, H.D. Phys. Lett.
**2005**, A347, 1.

^{1}It would be illustrative to generalize this model to a set of local degrees of freedom with local interactions in order to study the propagation of the distortion in space, and the resulting “causal structures”.

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**MDPI and ACS Style**

Zeh, H.D.
Remarks on the Compatibility of Opposite Arrows of Time II. *Entropy* **2006**, *8*, 44-49.
https://doi.org/10.3390/e8010044

**AMA Style**

Zeh HD.
Remarks on the Compatibility of Opposite Arrows of Time II. *Entropy*. 2006; 8(2):44-49.
https://doi.org/10.3390/e8010044

**Chicago/Turabian Style**

Zeh, H. D.
2006. "Remarks on the Compatibility of Opposite Arrows of Time II" *Entropy* 8, no. 2: 44-49.
https://doi.org/10.3390/e8010044