Effect in which a visible or invisible mass acts as a lens to "focalize" the light coming from a background source. (see bending of light) On the image above, arcs are images distorted and demultiplied, by the galaxy cluster, of more distant objects.
.
In general relativity, light follows the curvature of Space-Time induced by the mass M and undergoes a deviation of angle
.
This effect has been verified by Eddington in 1919 during an eclipse. The measurement of the deviation angle predicted by GR constituted the first experimental verification of Einstein's theory.
(where h is Planck's constant),
c (vacuum speed of light) and G (gravitation or Newton
constant) to obtain a quantity having the dimension of a length.
This microscopic length constitutes in ScR an unexceedable, unreachable (it is an asymptot, an horizon) and invariant under dilation, scale (or resolution). Even if it is only very small, it has all the physical properties of an infinitely small!
It is the smallest scale of Nature. The fact that it is not null has enormous repercussions on physics, like the finiteness of the speed of light.
The Planck mass MP and Planck Energy associated to it, EP = MP c2, are also computed from the 3 fundamental constants G, h and c: MP= (hbar c / G)^(1/2)
by:
Second length invariable under dilation that appears in ScR. It is the counterpart of the Planck length at large scale.
Is is the largest scale of Nature
but non-differentiable 
curve (function). In general these two limits are even not defined, i.e.
one cannot define the slope of the curve in any point.An example of topological dimension 1 is the curve:
5th iteration from the generator
: a (too) common error is to define a fractal curve or object as an object possessing the property to keep the same appearance under successive zooms. It applies only to self-similar fractals like the preceeding curve!
The word fractal refers, in ScR, to the appearance of a priori new structures along successive zooms toward small or large scales, which leads to a divergence for resolutions tending towards zero (towards Planck's scale in special ScR) and the infinity (cosmological scale in SScR).
In relativity and quantum mechanics, an event involving 1 body is characterized by 4 and only 4 coordinates (x,y,z,t). One puts apart properties like charge, spin, etc. and it does not mean that knowing only one (x,y,z,t) is sufficient to determine the evolution of a system but that the necessary quantities only depend on x, y, z, and t.
Space-Time constitutes the set of all possible (x,y,z,t) quadruplets and their transformations
Relativity's Space-Time is continuous and curved (the case were it is flat being not excluded a priori) and differentiable. This kind of Space cannot account for the quantum properties of matter.
Quantum Mechanics' Space-Time is in principle flat, minkovskian and differentiable. The contradiction with relativity as well as other considerations leads to attempts at including another kind of Space-Time in quantum mechanics, that have been up to now unfruitful..
In Scale Relativity, a criticism of measurement theory in physics, in particular of the role played by the resolutions, (c.f. the importance of the analysis of how we make measures of length and instant in the premisses of Einstein's theory) led L. Nottale to give up the (implicit) hypothesis of the differentiability of Space-Time, which implies its fractal and curved nature. The fractal Space-Time, explicitely dependent on resolutions, can be reduced to the definition of a "Space-Time-Zoom" with 5 dimensions (x,y,z,t,D). It is the fractal dimension, became a variable, which plays the role of a 5th dimension for scale laws (just as relativistic motion laws are implemented by the interpretation of time as a 4th dimension).
The concept of fractal space-time has also been independently introduced by Garnet Ord as a geometric analogue of relativistic quantum mechanics.
The deviation 
follows
an exponential law:
is called the chaos time of the
system. The inverse of the chaos time is the Lyapunov exponent.
If, having measured as precisely as we want (but not enough to distinguish
between the 2 initial conditions) the system's parameters (initial conditions)
at a time t0, we try to determine its state after a time interval
much greater than , we realize that the
deviation between the 2 solutions is now so big that one cannot say anything on
the state in which the system really is: knowing that there are, in fact, an
infinity of possible initial conditions between the two that we have
considered, the system can lie in an infinity of states very different one from
another. One has a sort of predictibility horizon which, like the
one we are familiar with (due to the roundness of the Earth), is not absolute:
short term prediction (before the horizon) remains entirely
possible!
Remarks:
- At the present time, in "chaos theory", one analyses the behaviour of a system whose differential equations, describing its evolution, are "classical". In such a frame, chaotic behaviour (diverging errors / deviations) only appears when the equations are non-linear. In Scale Relativity, the situation is very different in that sense that the fractal nature of space-time induces an other type of chaos, that could be called intrinsic, inherent to space-time. This has the effect that a behaviour of the chaotic type appears at large time scale even in an ultra-simple like one or 2-body problems (test particle orbiting around a much more massive one or 2 bodies orbiting around their barycentre). Such a behaviour is completely incomprehensible in the frame of current theories.
- Of course, if we could measure all the parameters with an infinite precision at the same time, the problem would disappear. Unfortunately, besides the imprecisions of the measuring apparatus, we have learned from quantum mechanics (Heisenberg inequalities) that it is physically impossible (I mean that the impossibility does not lie in the instruments but in Physics itself) to measure simultaneously to the desired precision the position and the speed of a particle, for instance. In that case, the knowledge of the exact position of a particle implies an absolute uncertainty on its velocity. However, it is not forbidden to measure individually some quantity with an absolute precision. In Scale Relativity, the Planck length constitutes an impassable (because it is an asymptot) lower limit to the resolution with which one can measure a position or a length. At this length/resolution, the concepts of position and of length become degenerated: the whole Universe, measured at Planck's resolution, as well as all it can contain, is of extension equal to the Planck length itself!!
To make it short , the chaotic behaviour is unavoidable!
Strangely, order happens in chaos! (attractors, etc.) How does this happen? Scale Relativity brings some elements of an answer to this question
to be continued ...!