---
abstract: 'We respond to N. Szpak’s comment \[arXiv: 0907.5146v2\] on our paper “Late-time tails of a self-gravitating massless scalar field, revisited”, Class.  Quantum Grav. ** 26**, 175006 (2009).' 
author:
- Piotr Bizoń
- Tadeusz Chmaj
- Andrzej Rostworowski
title: |
    Reply to the comment by Szpak on leading order asymptotics\
    of late-time tails for a self-gravitating massless scalar field
---

In a recent paper [@bcr5] we revisited the problem of long-time behavior of a spherically symmetric self-gravitating massless scalar field.  We emphasized that the asymptotic convergence to a static equilibrium (Minkowski or Schwarzschild) is an essentially nonlinear phenomenon which cannot, despite many assertions to the contrary in the literature, be properly described by the theory of linearized perturbations on a fixed static asymptotically flat background (Price’s tails).  To substantiate this claim in the case of small initial data we computed the late-time tails (both the decay rate and the amplitude) in four and higher even spacetime dimensions using nonlinear perturbation theory and we verified the results numerically.  The reason for considering this problem in higher dimensions was motivated by the desire to demonstrate an accidental and misleading character of equality of decay rates of linear and nonlinear tails in four dimensions. 

Recently, Szpak [@szpak] showed that most (but not all) of our results can be reproduced by an asymptotic calculation which keeps track of only leading order terms in the perturbation equations.  This is an interesting observation which provides insight into the mechanism of some cancelations in the asymptotic expansions.  Delighted as we are to see someone going through the details of our calculations (and confirming the results), we wish to make a couple of clarifying remarks:

-   From a practical viewpoint, the calculation by Szpak is only superficially more efficient.  Although his asymptotic equations are simpler to deal with, this fact is counterbalanced by an extra effort which goes into proving that the terms dropped from the equations are really irrelevant. 

-   Using only leading order asymptotics Szpak could not obtain our interpolating formula (Eq.(41) in [@bcr5]) for the tail in $d+1$ dimensions (for odd $d\geq
    5$).  Note that this formula contains significantly more information about the asymptotic behavior of solutions than its limiting cases at time and null infinities. ( Nota bene, the first version of Szpak’s comment \[arXiv: 0907.5146v1\] contained a critical remark about the validity of our results at null infinity; this remark, based on a misunderstanding, was withdrawn from the final version of the comment so we are glad not to have to respond to it.) 

-   The identification of a dominant term in the perturbation equations is useful only if this term gives rise to a nonzero tail in the leading order (as it luckily happens in the case at hand).  If there were further cancelations, it would be impossible to see them at the level of equations and an asymptotic calculation of [@szpak] would not work.  Such a situation arises, for example, for self-gravitating wave maps [@bcr6]. 

[10]{}

P. Bizoń, T. Chmaj, and A. Rostworowski, Class.  Quantum Grav. ** 26**, 175006 (2009). 

N. Szpak, arXiv: 0907.5146

P. Bizoń, T. Chmaj, A. Rostworowski, and S. Zajac, arXiv: 0906.2919
