Consider a $d$-dimensional smooth Lorentzian manifold $(M,g)$ (we assume $d\geq 3$, $M$ to be Hausdorff, paracompact and connected, hence second-countable, and that the signature convention of $g$ is $(+-\cdots-)$, i.e. the signature of $g$ is $2-d$) with Levi-Civita connection $\nabla$, and consider the Klein-Gordon equation on $(M,g)$: $$\tag{1}\label{e1}\Box_g u+m^2 u=0\ ,\quad\Box_g=g^{-1}\nabla^2\ .$$ We assume $(M,g)$ to be globally hyperbolic: there is $\tau\in\mathscr{C}^\infty(M,\mathbb{R})$ surjective such that $\mathrm{d}\tau$ is a future-directed, timelike covector field and $\Sigma_t=\tau^{-1}(t)$ is a Cauchy hypersurface for $(M,g)$ (i.e. $M=$ the Cauchy development of $\Sigma_t$ in $(M,g)$) for all $t\in\mathbb{R}$. We then say that $\tau$ is a Cauchy time function for $(M,g)$, and one can show that $M$ is then diffeomorphic to $\mathbb{R}\times\Sigma_t$ for every $t\in\mathbb{R}$ (it suffices to follow the maximal orbits of a future-directed timelike vector field $T$ - e.g. $T=g^\sharp(\mathrm{d}\tau)$ -, which must intersect $\Sigma_t$ for all $t\in\mathbb{R}$ since they are all Cauchy hypersurfaces by hypothesis). We can (and shall) use a Cauchy time function $\tau$ to provide us a global time coordinate.

If $(M,g)=(\mathbb{R}^d,\eta)$ is Minkowski space-time with $\tau=$ the standard (Cartesian) time coordinate $x^0$, the stationary phase method (among others) allows us to show that if $u$ is a smooth solution of (1) with initial data of compact support and $m\neq 0$, then $$\tag{2}\label{e2}\|u(x^0,\cdot)\|_{L^\infty(\mathbb{R}^{d-1})}=O\Big(|x^0|^{-\frac{d-1}{2}}\Big)\ .$$ If $m=0$, we have instead $$\tag{3}\label{e3}\|u(x^0,\cdot)\|_{L^\infty(\mathbb{R}^{d-1})}=O\Big(|x^0|^{-\frac{d-2}{2}}\Big)\ .$$

On the other hand, if $M=\mathbb{R}\times\mathbb{S}^{d-1}\ni(t,\theta)$ and $g=\mathrm{d}t\otimes\mathrm{d}t-h$, where $h$ is the standard round metric on $\mathbb{S}^{d-1}$ (i.e. $(M,g)=$ the so-called Einstein cylinder or Einstein static universe), then in both cases we have no decay at all. Intuitively, this is due to the fact that all light rays are trapped in a compact spatial region due to the simple fact that $(M,g)$ has compact Cauchy hypersurfaces. For cases in between (e.g. Schwarzschild and Kerr geometries, etc.) with non-compact Cauchy hypersurfaces (which is the only case I am interested in from now on), there are two possible (not completely independent) phenomena due to focusing of null geodesics which slow down global decay of solutions of \eqref{e1} as compared with \eqref{e2} and \eqref{e3}:

  1. Trapping. This means that at least some null geodesics remain within a compact spatial region (often lower-dimensional). This is the case of static black hole space-times such as Schwarzschild - light rays with initial position at the radius $r=3M$ (in $d=4$) and purely angular initial velocity stay forever in the photosphere $r=3M$;

  2. Caustics. This happens whenever (say, future-directed) null geodesics starting at a point $p\in M$ go beyond the (future) null conjugate locus of $p$.

Typically, trapping has a stronger effect in reducing global decay than caustics. On the other hand, caustics are usually stable against small perturbations of $g$, whereas trapping need not be. A lot of work has been dedicated by several people (Alinhac, Baskin, Sogge, Tataru, etc.) to investigate decay of solutions of \eqref{e1} when $(M,g)$ is either free of 1. and 2. or has a particular kind of trapping.

Having set up our context and framework, I am now finally able to pose my question. Consider the Klein-Gordon equation \eqref{e1} in a globally hyperbolic $(M,g)$ endowed with a steep Cauchy time function $\tau$ (i.e. $g^{-1}(\mathrm{d}\tau,\mathrm{d}\tau)\geq 1$). Since the set of all globally hyperbolic smooth Lorentzian metrics on $M$ is Whitney-$\mathscr{C}^0$-open in the space of all smooth Lorentzian metrics on $M$, we are able to perform small perturbations $g'$ of $g$ in that topology while keeping the globally hyperbolic character of $g'$. Moreover, if the perturbation is suitably small in that topology, a steep Cauchy time function remains a Cauchy time function with respect to $g'$.

Question: Suppose that smooth solutions to \eqref{e1} in $(M,g)$ with compactly supported initial data in $\Sigma_0$ have a time decay rate of the form $$\|u\|_{L^\infty(\Sigma_t)}=O(f(|t|))\ ,\quad t>0$$ for some $\delta>0$, where $f:(0,+\infty)\to(0,+\infty)$ is a strictly decreasing function satisfying $\lim_{t\to+\infty}f(t)=0$. Assume that the Cauchy hypersurfaces of $(M,g)$ are not compact. For a "generic" small perturbation $g'$ within the above conditions, what is the time decay rate for solutions $u'$ of the same kind to $$\tag{1'}\label{e1a}\Box_{g'} u'+m^2 u'=0\ ,\quad\Box_{g'}=g'{}^{-1}\nabla'{}^2$$ (here $\nabla'$ is the Levi-Civita connection associated to $g'$)? Is it faster than $O(f(|t|))$ for $g'$ "generic" but $g$ not so?

In other words, which is the best "structurally stable" global time decay rate we can expect from smooth solutions of the Klein-Gordon and wave equation with compactly supported initial data if $(M,g)$ has non-compact Cauchy hypersurfaces (to prevent situations where we know for sure there is no decay at all)?

I expect that loss of decay due to trapping should "generically" disappear but loss of decay due to caustics (which is supposedly milder) should "generically" remain, but I am not sure if this intuition is correct.

(Remark: Müller and Sánchez (Lorentzian Manifolds Isometrically Embeddable in $\mathbb{L}^N$, Trans. Amer. Math. Soc. 363 (2011) 5367-5379, arXiv:0812.4439 [math.DG]) and Minguzzi (On the Existence of Smooth Cauchy Steep Time Functions, Class. Quantum Grav. 33 (2016) 115001, arXiv:1601.05932 [gr-qc]) have shown that any globally hyperbolic $(M,g)$ admits a steep Cauchy time function)

  • $\begingroup$ Pedro, just to clarify, your (1') should have the equation $\Box_{g'} u'+m^2 u'=0$, with the prime on both $u$'s, right? $\endgroup$ Jun 22, 2016 at 11:26
  • $\begingroup$ Ooops, of course, just fixed that... Thanks Igor! $\endgroup$ Jun 22, 2016 at 13:17

1 Answer 1


A longish bunch of remarks: there's a big leap that doesn't make sense between your question and your motivation.

  1. On the maximally extended Schwawrzschild solution there is no decay to the wave equations: the space-time "ends" in finite proper time at the singularity, and there's not enough time (compare to the not-enough-space scenario you mentioned with compact Cauchy surface) for solutions to decay.

  2. Compact Cauchy surface does not rule out decay. Wave equation on de Sitter space is known to decay (strictly speaking, not in $L^\infty$ due to the constant solutions; but in $\mathring{W}^{1,\infty}$).

  3. The results concerning Schwarzschild and Kerr mostly deal with their outer domain of dependence (with some authors considering a little sliver that pokes into the black hole). If you take a steep time function on the outer domain of Schwarzschild, for example (and an example would be the function $t$ in the traditional Schwarzschild coordinates), the solutions to wave and Klein-Gordon equations have absolutely no decay even for compactly supported initial data, as a wave packet can fall into the black hole very quickly, but the black hole boundary is "at $t = \infty$".

    If you actually read the papers on Schwarzschild and Kerr decays, you will see that instead of using a Cauchy time function, their "time function" when measuring decay are never Cauchy. It usually intersects the black hole boundary (and sometimes also $\mathscr{I}^+$) transversely.

  4. In fact, the only cases where I can think of where decay of wave/Klein-Gordon equation has been proven for a globally hyperbolic Lorentzian manifold with respect to a steep time function are (1) background metrics which are perturbations of Minkowski space [where generic perturbations cannot improve things since Minkowski is as about as good as it comes]; (2) cosmological scenarios [independently of whether you have an open or closed Cauchy hypersurface] where the energy decay is driven by the spatial expansion and so is independent of trapping or caustics (which capture failure of "dispersion to infinity"); and (3) works on static product metrics where one can do resolvent estimates.

    I admit not being up to date in the results of type (3), but it seems that the type of hypotheses usually assumed for obtaining results of type (3) are generally stable under perturbations.

  5. The focus on effects of trapping and caustics that you mentioned in your question is mainly due to them being obstacles for (global and local) dispersion. So your question really only makes sense when the decay effects are dispersion dominated (see my comment in the previous point about cosmological solutions). As such you probably want your manifold to admit global dispersion in a suitable sense, which you can guarantee by, e.g., imposing some sort of asymptotic simplicity or asymptotic flatness.

  6. On the other hand, for wave equations in particular, it has been long understood that the decay parametrized by a time function $\tau$ does not capture the real underlying phenomenon where the decay should be understood more along the line of something like "parametrized by affine parameters of a null geodesic congruence",[insert place holder here; on second reading I am not 100% sure about my word choices there; until I can come up with a better way to say it succinctly, I'll leave this at "something technical to describe"] which incidentally also more closely resembles the result from the stationary phase argument if you look at the geometric optics. This is related to why in point 3 above, the "time function" used to measure decay in modern literature are not Cauchy time functions.

  7. I am also not sure what you mean by that trapping should disappear under generic perturbations. Consider the class of (connected) globally hyperbolic Lorentzian manifolds with two asymptotically flat ends. This class is stable under small perturbations. But such manifolds must have trapped null geodesics.

  • $\begingroup$ Dear Willie, your answer gave me a lot of food for thought. It was already clear to me that several aspects of my question (which has been on the back of my mind for quite some time) need to be further qualified - I've read some of the papers on the subject quite a while ago, so certain pitfalls in them eluded my memory. I'll probably need some time to process everything you wrote and reread those papers. Then I'll return to comment on your points and make the necessary improvements to the question to make it better posed. Thank you for having taken the time to explain all the above issues! $\endgroup$ Jun 22, 2016 at 20:56

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