4 metric. Making several further choices (like e.g., Ko- dama vector in the absence of Killing vector to define energy) one can relate Einstein’s equations with horizon thermodynamics, as has been done in the literature [6] in many ways. Reversing these steps, one can obtain Ein- stein’s equations from the thermodynamics of horizons, provided one makes specific choices. For a general Friedmann model, however, we will not have a coordinate system in which one can define hori- zon, effective Unruh temperature etc. A possible way of using Eq. (8) in this context is suggested by the fact its derivation in ref.[7] used the Einstein’s equations in the form ∇ $_{µ}$a µ ∝ ¯ T$_{ab}$u $^{a}$ub where a i was the proper ac- celeration of observers with x $^{µ}$= constant. This will not hold in the case of a general time-dependent metric; in fact, for Friendman universe the proper acceleration of comoving observers is zero. But we know that the rela- tion ∇ $_{µ}$a µ ∝ ¯ T$_{ab}$u $^{a}$ub will continue to hold in any space- time, if we take a µ to be the proper acceleration of the geodesic deviation vector and the expressions are evalu- ated in the comoving frame. This suggests that one could try to interpret Eq. (8) for Friedmann universe with a µ identified with the acceleration of the geodesic separation vector. While the details of such choices differ significantly, this programme provides the roadmap for linking the cos- mic arrow of time with the thermodynamic arrow of time in four steps: First, we connect the cosmic arrow of time to the dynamical evolution governed by Einstein’s equa- tions, which was the main scope of this paper. Second, we link the gravitational field equations to the thermo- dynamics of horizons for which we have fair amount of results in general [5, 8] as well as in the context of cosmol- ogy [6]. But in the latter context one needs to understand better the physical origin of various choices, in a time de- pendent scenario. Third, we link horizon thermodynam- ics to ‘normal’ matter entropy using the physical pro- cesses version of first law of horizon and a suitable form of generalized second law. This is again nontrivial. The rig- orous results are only available for blackhole spacetimes with asymptotic flatness, stationarity etc. while we need them in the case of time-dependent, observer-dependent, cosmological horizons. One requires a local formulation which does not exist at present. Finally we link the nor- mal entropy and second law of thermodynamics to the thermodynamic arrow of time. Successful completion of these steps will provide a link between the two arrows of time. Choosing ∂ V to be a two-sphere of fixed proper ra- dius R = ra ( t ), it seems reasonable to set E = ( ρ + 3 p )(4 πR $^{3}$/ 3) and S = A/ 4 L 2 P where A ∝ a 2 is the proper area of ∂ V . (This gives ˙ S/S = 2˙ a/a for each surface and is consistent with the expansion of the universe leading to increase in gravitational entropy, but in a rather naive way.) If we further identify T with the Unruh tempera- ture corresponding to the magnitude of the acceleration | (¨ a/a ) | R of the geodesic deviation vector, then Eq. (8) will reduce to Eq. (2) as far as magnitude of the acceler- ation is concerned. (The r drops out of the relation and hence ∂ V could have been chosen to have special prop- erties, like e.g, location of horizon etc. if desired.) One cannot determine the (crucial) fact that the universe is decelerating for ( ρ +3 p ) > 0 without further assumptions since Unruh temperature only depends on the magnitude of the acceleration. Unfortunately, there is no simple jus- tification for defining T using the acceleration of geodesic deviation vector and hence the success of the procedure must be considered fortuitous at this stage. Acknowledgements : These ideas were first presented in my lecture at ICTS-IUCAA Programme on Cosmol- ogy with CMB and LSS, July-Aug 2008 and I thank the participants for their feedback. I also thank H.Nicolai and R.Penrose for useful discussions on this topic during the ‘Foundations of Space and Time’ meeting at Cape Town, 10-14 August 2009. More rigorously, one can express any Friedmann model in a spherically symmetric form with a time-dependent [1] One source which mentions “expansion problem” explic- itly is Peacock J.A., Cosmological Physics (Cambridge University Press, 1999), page 324. [2] DeWitt B.S., Phys.Rev ., 160 , 1113 (1967) has one of the earliest explorations. [3] See e.g., Guth A.H. and So-Young Pi, (1985), Phys.Rev. , D32 , 1899. [4] Penrose R. in Proceedings of the First Marcel Grossmann meeting in general Relativity , edited by R.Ruffini (North Holland, 1977) p. 173; also see, Penrose R., The Road to Reality , (Alfred A Knoff, NY, 2004). 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