66 research outputs found

### A time machine for free fall into the past

Inspired by some recent works of Tippett-Tsang and Mallary-Khanna-Price, we
present a new spacetime model containing closed timelike curves (CTCs). This
model is obtained postulating an ad hoc Lorentzian metric on $\mathbb{R}^4$,
which differs from the Minkowski metric only inside a spacetime region bounded
by two concentric tori. The resulting spacetime is topologically trivial, free
of curvature singularities and is both time and space orientable; besides, the
inner region enclosed by the smaller torus is flat and displays geodesic CTCs.
Our model shares some similarities with the time machine of Ori and Soen but it
has the advantage of a higher symmetry in the metric, allowing for the explicit
computation of a class of geodesics. The most remarkable feature emerging from
this computation is the presence of future-oriented timelike geodesics starting
from a point in the outer Minkowskian region, moving to the inner spacetime
region with CTCs, and then returning to the initial spatial position at an
earlier time; this means that time travel to the past can be performed by free
fall across our time machine. The amount of time travelled into the past is
determined quantitatively; this amount can be made arbitrarily large keeping
non-large the proper duration of the travel. An important drawback of the model
is the violation of the classical energy conditions, a common feature of many
time machines. Other problems emerge from our computations of the required
(negative) energy densities and of the tidal accelerations; these are small
only if the time machine is gigantic.Comment: 40 pages, 10 figures; the final version accepted for publication. In
comparison with version v2, some references added (see [4,21,35]) and
commented on in the Introductio

### On the constants for some fractional Gagliardo-Nirenberg and Sobolev inequalities

We consider the inequalities of Gagliardo-Nirenberg and Sobolev in R^d,
formulated in terms of the Laplacian Delta and of the fractional powers D^n :=
(-Delta)^(n/2) with real n >= 0; we review known facts and present novel
results in this area. After illustrating the equivalence between these two
inequalities and the relations between the corresponding sharp constants and
maximizers, we focus the attention on the L^2 case where, for all sufficiently
regular f : R^d -> C, the norm || D^j f||_{L^r} is bounded in terms of || f
||_{L^2} and || D^n f ||_{L^2} for 1/r = 1/2 - (theta n - j)/d, and suitable
values of j,n,theta (with j,n possibly noninteger). In the special cases theta
= 1 and theta = j/n + d/2 n (i.e., r = + infinity), related to previous results
of Lieb and Ilyin, the sharp constants and the maximizers can be found
explicitly; we point out that the maximizers can be expressed in terms of
hypergeometric, Fox and Meijer functions. For the general L^2 case, we present
two kinds of upper bounds on the sharp constants: the first kind is suggested
by the literature, the second one is an alternative proposal of ours, often
more precise than the first one. We also derive two kinds of lower bounds.
Combining all the available upper and lower bounds, the Gagliardo-Nirenberg and
Sobolev sharp constants are confined to quite narrow intervals. Several
examples are given.Comment: LaTex, 63 pages, 3 tables. In comparison with version v2, just a few
corrections to eliminate typo

### On the constants for multiplication in Sobolev spaces

For n > d/2, the Sobolev (Bessel potential) space H^n(R^d, C) is known to be
a Banach algebra with its standard norm || ||_n and the pointwise product; so,
there is a best constant K_{n d} such that || f g ||_{n} <= K_{n d} || f ||_{n}
|| g ||_{n} for all f, g in this space. In this paper we derive upper and lower
bounds for these constants, for any dimension d and any (possibly noninteger) n
> d/2. Our analysis also includes the limit cases n -> (d/2) and n -> +
Infinity, for which asymptotic formulas are presented. Both in these limit
cases and for intermediate values of n, the lower bounds are fairly close to
the upper bounds. Numerical tables are given for d=1,2,3,4, where the lower
bounds are always between 75% and 88% of the upper bounds.Comment: LaTeX, 45 page

### Quantitative functional calculus in Sobolev spaces

In the framework of Sobolev (Bessel potential) spaces H^n(\reali^d, \reali
{or} \complessi), we consider the nonlinear Nemytskij operator sending a
function x \in \reali^d \mapsto f(x) into a composite function x \in
\reali^d \mapsto G(f(x), x). Assuming sufficient smoothness for $G$, we give a
"tame" bound on the $H^n$ norm of this composite function in terms of a linear
function of the $H^n$ norm of $f$, with a coefficient depending on $G$ and on
the $H^a$ norm of $f$, for all integers $n, a, d$ with $a > d/2$. In comparison
with previous results on this subject, our bound is fully explicit, allowing to
estimate quantitatively the $H^n$ norm of the function $x \mapsto G(f(x),x)$.
When applied to the case $G(f(x), x) = f^2(x)$, this bound agrees with a
previous result of ours on the pointwise product of functions in Sobolev
spaces.Comment: LaTex, 37 pages. Final version, differing only by minor typographical
changes from the versions of May 23, 2003 and March 8, 200

### Smooth solutions of the Euler and Navier-Stokes equations from the a posteriori analysis of approximate solutions

The main result of [C. Morosi and L. Pizzocchero, Nonlinear Analysis, 2012]
is presented in a variant, based on a C^infinity formulation of the Cauchy
problem; in this approach, the a posteriori analysis of an approximate solution
gives a bound on the Sobolev distance of any order between the exact and the
approximate solution.Comment: Author's note. Some overlaps with our previous works arXiv:1402.0487,
arXiv:1310.5642, arXiv:1304.2972, arXiv:1203.6865, arXiv:1104.3832,
arXiv:1009.2051, arXiv:1007.4412, arXiv:0909.3707, arXiv:0709.1670; these
overlaps aim to make the paper self-contained and do not involve the main
results. Final version to appear in Nonlinear Analysi

### Local zeta regularization and the scalar Casimir effect III. The case with a background harmonic potential

Applying the general framework for local zeta regularization proposed in Part
I of this series of papers, we renormalize the vacuum expectation value of the
stress-energy tensor (and of the total energy) for a scalar field in presence
of an external harmonic potential.Comment: Some overlaps with our works arXiv:1104.4330, arXiv:1505.00711,
arXiv:1505.01044, arXiv:1505.03276. These overlaps aim to make the present
paper self-contained, and do not involve the main result

### Local Casimir Effect for a Scalar Field in Presence of a Point Impurity

The Casimir effect for a scalar field in presence of delta-type potentials
has been investigated for a long time in the case of surface delta functions,
modelling semi-transparent boundaries. More recently Albeverio, Cacciapuoti,
Cognola, Spreafico and Zerbini [9,10,51] have considered some configurations
involving delta-type potentials concentrated at points of $\mathbb{R}^3$; in
particular, the case with an isolated point singularity at the origin can be
formulated as a field theory on $\mathbb{R}^3\setminus \{\mathbf{0}\}$, with
self-adjoint boundary conditions at the origin for the Laplacian. However, the
above authors have discussed only global aspects of the Casimir effect,
focusing their attention on the vacuum expectation value (VEV) of the total
energy. In the present paper we analyze the local Casimir effect with a point
delta-type potential, computing the renormalized VEV of the stress-energy
tensor at any point of $\mathbb{R}^3\setminus \{\mathbf{0}\}$; to this purpose
we follow the zeta regularization approach, in the formulation already employed
for different configurations in previous works of ours (see [29-31] and
references therein).Comment: 20 pages, 6 figures; the final version accepted for publication. In
the initial part of the paper, possible text overlaps with our previous works
arXiv:1104.4330, arXiv:1505.00711, arXiv:1505.01044, arXiv:1505.01651,
arXiv:1505.03276. These overlaps aim to make the present paper
self-contained, and do not involve the main result

### On the expansion of the Kummer function in terms of incomplete Gamma functions

The expansion of Kummer's hypergeometric function as a series of incomplete
Gamma functions is discussed, for real values of the parameters and of the
variable. The error performed approximating the Kummer function with a finite
sum of Gammas is evaluated analytically. Bounds for it are derived, both
pointwisely and uniformly in the variable; these characterize the convergence
rate of the series, both pointwisely and in appropriate sup norms. The same
analysis shows that finite sums of very few Gammas are sufficiently close to
the Kummer function. The combination of these results with the known
approximation methods for the incomplete Gammas allows to construct upper and
lower approximants for the Kummer function using only exponentials, real powers
and rational functions. Illustrative examples are provided.Comment: 21 pages, 6 figures. To appear in "Archives of Inequalities and
Applications

### Local zeta regularization and the scalar Casimir effect IV. The case of a rectangular box

Applying the general framework for local zeta regularization proposed in Part
I of this series of papers, we compute the renormalized vacuum expectation
value of several observables (in particular, of the stress-energy tensor and of
the total energy) for a massless scalar field confined within a rectangular box
of arbitrary dimension.Comment: Some overlaps with our works arXiv:1104.4330, arXiv:1505.00711,
arXiv:1505.01044, arXiv:1505.01651. These overlaps aim to make the present
paper self-contained, and do not involve the main results. In comparison with
version v3, reference [26] adde

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