The ESA reports on this spectacular lensed quasar image obtained by the Hubble space telescope. This is the first time a 5 times lensed object has been discovered.
Gravitational lensing results in an odd number of images - but some images are very weakly magnified and difficult to see, especially if the image is close to the lensing object. In this case the geometry is quite good and the fifth image is discernable from the lensing object. The five images of the distant quasar are marked by blue circles.
Also shown are three lensed images of a background galaxy, marked by red circles, and a distant supernova marked by the yellow circle.
Full story:European Space Agency
Some papers on gravitational lensing from the arXiv and elsewhere
I seem to get a fair number of hits querying the basic formulas for velocity, so I’ve decided to post them.
Let [tex]\cal{O}[/tex] be an object moving along a path. The position of the object [tex]\cal{O}[/tex] is a function of the time [tex]t[/tex]. We denote the position vector [tex]\bold{x}(t)[/tex].
The time derivative of the position vector [tex]\bold{x}(t)[/tex] is the velocity vector [tex]\bold{v}(t)[/tex] of the object [tex]\cal{O}[/tex].
[tex]\bold{v}(t) \equiv \frac{d\bold{x}(t)}{dt} = \bold{\dot{x}}(t)[/tex].
The time derivative of the velocity vector [tex]\bold{v}(t)[/tex] is called the acceleration vector [tex]\bold{a}(t)[/tex] of the object [tex]\cal{O}[/tex] .
[tex]\bold{a}(t) \equiv \frac{d\bold{v}(t)}{dt} = \frac{d^2\bold{x}(t)}{dt^2} = \bold{\ddot{x}}(t) [/tex].
Integrate this last equation with respect to [tex]t[/tex],
[tex]\int \bold{\ddot{x}}(t) \ dt=\int \bold{a}(t) \ dt \implies \bold{\dot{x}}(t) =\bold{a}( t)t + \bold{C}[/tex],
where [tex]\bold{C}[/tex] is a constant vector to be determined by the initial conditions. Now, [tex]\bold{\dot{x}}(t) =\bold{v}(t)[/tex] so that
[tex]\bold{v}(t) =\bold{a}(t) t + \bold{C}[/tex]
We assume that at time [tex]t =0[/tex] the object [tex]\cal{O}[/tex] is moving with initial velocity [tex]\bold{v}(0) =\bold{v_0}[/tex]
[tex] \bold{v}(0) = \bold{v_0} =\bold{a}(0) \ 0 + C \implies C = \bold{v_0}[/tex]
and substitution gives the familiar formula for the velocity of an object
[tex]\bold{v}(t) = \bold{v_0} + \bold{a}(t) t[/tex].
We can integrate the equation
[tex]\bold{\dot{x}}(t) =\bold{v_0} + \bold{a}(t) t[/tex] with respect to [tex]t[/tex] a second time
[tex]\int \bold{\dot{x}}(t) \ dt = \int \bold{v_0} + \bold{a}(t) t \ dt[/tex] to give
[tex] \bold{x}(t) = \bold{v_0} t+ \frac{1}{2}\bold{a}(t) t^2 + \bold{K} [/tex], where [tex]\bold{K}[/tex] is a constant vector to be determined from the initial conditions.
We assume that at [tex]t =0[/tex] the object [tex]\cal{O}[/tex] has position vector [tex]\bold{x}(0)=\bold{x_0}[/tex]
[tex] \bold{x}(0) = \bold{x_0} = \bold{v_0} t+ \frac{1}{2}\bold{a}(0) 0^2 + \bold{K} \implies \bold{K}=\bold{x_0}[/tex]
and we have the familiar formula for the displacement (position) of the object at time [tex]t[/tex]
[tex] \bold{x}(t) = \bold{x_0} + \bold{v_0} t+ \frac{1}{2}\bold{a}(t) t^2 [/tex].
In a first treatment we often simplify things a bit. Firstly, to simplify notation we don’t write in the explicit time dependence for the position [tex]\bold{x}(t)[/tex] , so that it becomes [tex]\bold{x}[/tex] and similarly for [tex]\bold{v}[/tex] and [tex]\bold{a}[/tex]. Secondly we assume that the object starts from the origin rather than from some arbitrary position, thus we take [tex]\bold{x_0}=0[/tex], (which is really just a change of co-ordinates). This gives the familiar form of the equations of motion from high school text books:
[tex] \bold{x} = \bold{v_0} t+ \frac{1}{2}\bold{a} t^2 [/tex]
[tex]\bold{v} = \bold{v_0} + \bold{a} t[/tex].
These equations are commonly written as
[tex] s = u t+ \frac{1}{2} a t^2 [/tex]
[tex] v = u+ a t[/tex],
where we have used the alternate notation [tex]s \equiv x[/tex] for the position and [tex]u \equiv v_0[/tex] for the intial velocity. Note that we drop the bold face symbols [tex]\bold{v} \rightarrow v[/tex] which are used to denote vectors, since in a first treatment we consider only the one dimensional or scalar case.
Squaring the last equation gives
[tex]v^2 = (u+ a t)^2 = u^2 + 2 u a t + a^2t^2 = u^2 + 2a (ut+\frac{1}{2}at^2) [/tex]
[tex]\phantom{v^2}= u^2 + 2as[/tex]
another commonly used equation in high school physics texts.
The average velocity over the time interval [tex] (t_1,t_2) \equiv (t_i,t_f)[/tex] is given by
[tex]v_{av} = \frac{s_2 -s_1}{t_2-t_1} = \frac{s_f-s_i}{t_f-t_i} = \frac{\Delta(s)}{\Delta(t)}[/tex]
where, [tex] s_1 \equiv s_i[/tex] is the intial displacement and [tex]s_2 \equiv s_f[/tex] is the final displacement. Finally, in words: it is the change in position [tex]\Delta(s)[/tex] divided by the change in time [tex]\Delta(t)[/tex].
Einstein’s letters credit Greek maths expert’s work, Greece and Israel say from PhysOrg.com
ATHENS (AFP) - Greece has received copies of letters by Albert Einstein which suggest that the work of an unheralded Greek mathematician helped shape some of his theories, Greek and Israeli officials said.
Israel’s ambassador to Athens, Ram Aviram, presented the Greek foreign ministry with copies of 10 letters between Einstein and Greek mathematician Constantine Karatheodoris, part of a long correspondence which lasted from 1916 to 1930.
According to experts at the National Archives of Israel — custodians of the original letters — the mathematical side of Einstein’s physics theory was partly substantiated through the work of Karatheodoris, Aviram told AFP.
“The correspondence between the two mathematicians is intensive and quite close,” Aviram said. “At a certain moment, they called themselves in private names.”
The son of a Greek-born diplomat who served as the Ottoman Empire’s ambassador to Berlin, Karatheodoris who was born in 1873 and died in 1950 taught mathematics at four German universities — including those of Munich and Goettingen — and also worked on physics and archaeological engineering.
His scientific papers are in the collection of Goettingen University, and have never been translated into Greek, though a number of American universities have copies of his theories, said deputy foreign minister Evripidis Stylianidis.
The Greek authorities intend to create a museum honouring Karatheodoris in Komotini, a major town of the northeastern Greek region where his family came from.
Interesting that they call Einstein a mathematician, as I’ve always thought of him as a theoretical physicist, rather than a mathematical physicist. The only thing mathematical I associate with Uncle Albert is the Einstein summation convention.