The other day I had to give a semi-popular talk about "Special Relativity". I aimed

to explain E=mc^2 in simple terms. To my horror I could not find a simple

enough explanation. Of course I can derive it with 4 - momentum or photon momenta.

But this is not what I was looking for. So for a change I post a question here:

Who can provide me with the most simple explanation: Why is E=mc^2 ?

Find below the slides I used for my talk. The crucial point of special relativity

is really to rethink the meaning of time and how to define the simulaneity of two events.

Well at the end I give a derivation for E=mc^2, which goes back to a 1906 paper by Einstein, but it uses the momentum of a photon. I would like to see it purely from mechanics.

October 27, 2005 at 03:17 AM | Permalink | Comments (13)

We just submitted a paper where we probe a modified gravity model

with the help of Supernova observations [astro-ph/0510453].

This was a numerical Tour de Force, since the equations which usually

govern the dynamical evolution of the Universe and are usually very

simple to solve turn into highly non-trivial, non-linear higher order

differential equations. It took as about four month to come up with a

satisfactory solution. But the work is already sparking some interest

in the community. We have basically shown that you can restrict the

free parameters appearing in these models with observations of distant

Supernovae and the expansion rate of the Universe.

What is the possible consequence of these models ?

Well essentially the outcome is that we do NOT need a dark energy

component. We just have to modify gravity in an appropriate way

[see my previous blog]. And the interesting thing is, it really works.

Unfortunately we could not push these models so far to also get rid of

dark matter, but we are currently thinking about this :).

Of course there is a plethora of other cosmological probes like the cosmic

microwave background, clusters of galaxies and large scale structure.

However since we work in a modified gravity background we have to

carefully recalculate all the theoretical assumptions made for these

probes, in order to make sure that we pick up any possible new effects

emerging in modified gravity scenarios. This is a very hard task, but it

is fun.

Here is a nice picture about the constraints derived in the paper

October 20, 2005 at 07:12 AM | Permalink | Comments (2)

I wrote the following piece for the "Einstein: big ideas" blog:

Einstein introduced his theory of General Relativity in his seminal

paper in 1917. Building up on Special Relativity, which describes a

dynamical system under the assumption that no body can move faster

than the speed of light, General Relativity was meant to include the

forces of gravity and accelerated frames into Special Relativity. One

of its basic assumption was the equivalence principle, i.e. that the

instance of mass as a source of the gravitational field, is the same

as the instance of mass as the measure of "resistance" a body exerts

if one tries to distort it from linear unaccelerated motion. The onset of Special

Relativity is that light propagates with a constant speed along the

shortest distance between two points A and B. This is for our everyday

experience a straight line, however in the presence of massive bodies

light propagates along geodesics in a curved space. This is just like

for long distance flights which follow the shorter parts of great circles

along the surface of the earth.

Hence, according to General Relativity, if light seemingly deviates from a straight path,

space must be curved. This happens in the close vicinity of large masses,

and the effect is a valuable tool in modern astrophysics, known as

gravitational lensing.

From it's beginning one of the amazing results was that General

Relativity seemed to describe a dynamically evolving Universe. The

dynamics of the large scale motion of the Universe is described by

Einstein's equation of gravity, which relates the geometry of the

Universe to its energy (matter) contents. Einstein, who firmly

believed at the time that the Universe should be a static entity,

could only accommodate this by introducing a term which carefully

balanced the other energy constituents. He called this term the

cosmological constant, because the energy associated with it, has the

peculiar feature that even if the Universe expands, it's density stays*constant*.

In 1929 Hubble discovered his famous law, that the further away

galaxies are from us the faster they are receding. This was later

confirmed by Sandage and others and is continued to be tested by the

Hubble Space Telescope to unprecedented distances.

Einstein realized that the Universe

must be expanding and the the Universe is *NOT *static. He

threw the cosmological constant out of the window, and according to Gamow

Einstein called the introduction of the

cosmological constant its greatest blunder. Nevertheless the inclusion

of a cosmological constant in the equations of General Relativity is

mathematically sound and no fudge factor.

In 1933 Fritz Zwicky observed the Coma Cluster and noticed that

outlying galaxies where moving much faster than would be expected for

the sum of the mass of galaxies further inside. He hence estimated

that the cluster must consist of 90% dark matter which we can not

observe directly.

At the beginning of the 1970s Vera Rubin made the

discovery that the velocity of hydrogen clouds in galaxies does not

decrease with distance from the center, as would be expected from the

distribution of visible stars. Hence, galaxies, like clusters must

also contain some form of dark matter.

In 1997 Cosmology experienced another surprising turn. With the

observation of imploding stars (Type Ia Supernovae for the experts),

which are as bright as an entire galaxy, the Supernovae Cosmology

Project and the High-z Supernovae Search Team announced, that they

found evidence that the expansion of the Universe is speeding up.

What was measured was the receding velocity of the host galaxies of

the Supernovae, i.e. their redshift and the brightness of the

Supernovae. The brightness can be related to the distance of the

Supernovae, with the simple fact that more distance objects appear

fainter. Hence we can plot a distance - redshift diagram. Since the

receding velocity (or redshift) is related to its distance, via

Einstein's equations of General Relativity, and hence to the energy

(matter) contents in the Universe, one can probe cosmological models

in this way. The outcome was shocking:

A cosmology with a cosmological constant seem to be a much better

explanation of the data. In a sense it was discovered that expansion

rate of the Universe at early times was slower than it is now, hence

we require some source for this cosmic speed up. Einstein's balancing

term was swiftly remembered. What, if we actually over-balance the

ordinary matter terms with the cosmological constant ? Instead of

obtaining a static Universe, we obtain one which expands in an

accelerated fashion.

The early findings of the Supernovae teams are now confirmed by

observations of the cosmic microwave background, together with large

scale structure and clusters of galaxies (Wilkinson Microwave Anisotropy Probe ,

2 degree Field Galaxy Redshift Survey , Sloan Digital Sky Survey , Chandra Satellite).

However we are now faced with the fact that 95% (!) of the Universe

are made of a form of energy, or matter, which we have hardly an

inkling what the stuff actually is made of. Only 5% is made of matter

we have observed directly in Nature. 25% reside in form of

dark matter, which is only very weakly, if at all, interacting with

"natural" forms of matter. However fundamental physics, like

Supersymmetry, predicts a plethora of particles, which are valid dark

matter candidates. The situation is far more grim for the 70% which

are responsible for cosmic acceleration and are usually called dark

energy. There are some interesting description, like scalar field

Quintessence models, but most of them are made up more or less ad hoc

and have no sound footing in fundamental physics.

We are in a situation now, that if we believe

Einstein's Equation of Gravity, 95% of the Universe are unknown to us.

This reminds me of the fairy tale by Hans Christian Anderson, "The

Emperor's New Clothes", where everybody admires the new suit of the

emperor, which according to the crooked tailors is only visible to

smart people. Until a little kid shouts: "But he [the emperor] has nothing

on at all".

This is setting the scene for a new approach to gravity. The earliest

published approaches to an alternative description of General

Relativity were done by Pasqual Jordan in 1945. In 1961 Brans and

Dicke studied a similar theory, which effectively resulted in a theory

of General Relativity with a varying gravitational constant.

Since Brans and Dicke wrote their paper about an extension of Einstein

gravity with higher order curvature terms, theorists in the back of their

minds are aware that Einstein's theory of gravity might not be the

final answer. However Einstein's general relativity is an incredible

successful theory, which so far has stood all direct, albeit local

tests, like the perihel precession of mercury. Hence any valid

extension of gravity has to pass these tests as well.

The gravity in galaxies and clusters of galaxies is mainly governed by

Newtonian mechanics, however with the addition of a dark matter

halo as mentioned above. Milgrom [1983] however introduced a

different explanation. What if Newtonian dynamics is modified on large

scales and hence explains the rotation curves of galaxies without the

introduction of dark matter. These theories are usually called MOND

(the German word for 'moon' incidently), for Modified Newtonian

Dynamics. However, the problem is that Einstein's General Relativity

reproduces in it's low mass limit Newton's dynamics. That is why it is

incredibly hard to experience any effect of General Relativity on

Earth. Up until two years ago their was no mathematically

sound theory which produced a relativistic extension of MOND. However,

Bekenstein introduced a mathematically consistent theory of MOND in

the beginning of 2004. This theory is able to reproduce the galactic

rotation curves

without introducing dark matter. But it is significantly different

from Einstein's General Relativity. An even bolder, but nevertheless,

valid step is to try and explain the accelerated expansion of the

Universe by modifying the laws of gravity. These approaches contain

higher orders of the curvature of the

Universe. An interesting class of these was recently suggested by

Carroll and collaborators. In this case the *inverse* of the curvature

is included into the set up of the equations which describe

gravity. In this case, the dominant effect is at large scales where

the Universe is apparently flat and there is zero curvature. It can be

shown that some of these models lead to accelerated expansion and also

pass solar system tests. A different approach is to try and explain these

modifications of gravity in our four dimensional space-time with

higher dimensional world models. So called Brane Worlds which could appear

from compactifications of 11-dimensional theories to our four

dimensions could lead in fact to modifications of of the equations

which govern the the dynamical evolution of a 4-dimensional world. In

this case a straight forward extension of General Relativity holds at

the higher dimensions and our low dimensional Physics is just some

effective theory as a result of the compactification.

The exciting prospects for the future are that there is a hand full of

missions and observations planned which would allow the

distinction between dark energy and modified gravity. One such probe

is the "Dark Energy Survey" which is led by Fermilab scientist, with

collaborators from other US institutions and Europe. The "Dark Energy

Survey" tries to test dark energy models simultaneously with four

different astrophysical probes. Some of these respond differently to

modified gravity than to dark energy. The combination of all four

probes might be able to distinguish between modified gravity models

and dark energy approaches. In the end the "Dark Energy Survey" might

be able to show if Einstein is in fact naked or has indeed a multi

layered suit whose fabric needs to be revealed.

October 02, 2005 at 02:40 PM | Permalink | Comments (3)