Monday, February 28, 2011

Ground, Wall, Pipe Penetrating Radar

Definitely one of the coolest uses of engineering technology is to look through surfaces and see things without breaking those surfaces. We can do it on the surface, we can do it from orbit. All of it thanks to Physics and the Mathematics that Physicists love so much. So thank you Babylonians and Pythagoras and Euclid and Newton and Euler, where ever you are.

Dig, we must?

Not necessarily .......

Ground-penetrating radar (GPR) is a geophysical method that uses radar pulses to image the subsurface. This non-destructive method uses electromagnetic radiation in the microwave band (UHF/VHF frequencies) of the radio spectrum, and detects the reflected signals from subsurface structures. GPR can be used in a variety of media, including rock, soil, ice, fresh water, pavements and structures. It can detect objects, changes in material, and voids and cracks.[1]

GPR uses transmitting and receiving antennas or only one containing both functions. The transmitting antenna radiates short pulses of the high-frequency (usually polarized) radio waves into the ground. When the wave hits a buried object or a boundary with different dielectric constants, the receiving antenna records variations in the reflected return signal. The principles involved are similar to reflection seismology, except that electromagnetic energy is used instead of acoustic energy, and reflections appear at boundaries with different dielectric constants instead of acoustic impedances.

The depth range of GPR is limited by the electrical conductivity of the ground, the transmitted center frequency and the radiated power. As conductivity increases, the penetration depth decreases. This is because the electromagnetic energy is more quickly dissipated into heat, causing a loss in signal strength at depth. Higher frequencies do not penetrate as far as lower frequencies, but give better resolution. Optimal depth penetration is achieved in ice where the depth of penetration can achieve several hundred meters. Good penetration is also achieved in dry sandy soils or massive dry materials such as granite, limestone, and concrete where the depth of penetration could be up to 15 m. In moist and/or clay-laden soils and soils with high electrical conductivity, penetration is sometimes only a few centimetres.

Ground-penetrating radar antennas are generally in contact with the ground for the strongest signal strength; however, GPR air launched antennas can be used above the ground.

Cross borehole GPR has developed within the field of hydrogeophysics to be a valuable means of assessing the presence and amount of soil water.


Ground penetrating radar survey of an archaeological site in Jordan.
GPR has many applications in a number of fields. In the Earth sciences it is used to study bedrock, soils, groundwater, and ice. Engineering applications include nondestructive testing (NDT) of structures and pavements, locating buried structures and utility lines, and studying soils and bedrock. In environmental remediation, GPR is used to define landfills, contaminant plumes, and other remediation sites, while in archaeology it is used for mapping archaeological features and cemeteries. GPR is used in law enforcement for locating clandestine graves and buried evidence. Military uses include detection of mines, unexploded ordnance, and tunnels.

Before 1987 the Frankley Reservoir in Birmingham, England UK was leaking 540 litres of drinking water per second. In that year GPR was used successfully to isolate the leaks.[2]

Borehole radars utilizing GPR are used to map the structures from a borehole in underground mining applications. Modern directional borehole radar systems are able to produce three-dimensional images from measurements in a single borehole.

One of the other main applications for ground penetration radars to locate underground utilities, since GPR is able to generate 3D underground images of pipes, power, sewage and water mains.

Three-dimensional imaging

Individual lines of GPR data represent a sectional (profile) view of the subsurface. Multiple lines of data systematically collected over an area may be used to construct three-dimensional or tomographic images. Data may be presented as three-dimensional blocks, or as horizontal or vertical slices. Horizontal slices (known as "depth slices" or "time slices") are essentially planview maps isolating specific depths. Time-slicing has become standard practice in archaeological applications, because horizontal patterning is often the most important indicator of cultural activities.


The most significant performance limitation of GPR is in high-conductivity materials such as clay soils and soils that are salt contaminated. Performance is also limited by signal scattering in heterogeneous conditions (e.g. rocky soils).

Other disadvantages of currently available GPR systems include:
  • Interpretation of radargrams is generally non-intuitive to the novice.
  • Considerable expertise is necessary to effectively design, conduct, and interpret GPR surveys.
  • Relatively high energy consumption can be problematic for extensive field surveys.
Recent advances in GPR hardware and software have done much to ameliorate these disadvantages, and further improvement can be expected with ongoing development.

Power regulation

In 2005, the European Telecommunications Standards Institute introduced legislation to regulate GPR equipment and GPR operators to control excess emissions of electromagnetic radiation [3]. The European GPR association (EuroGPR) was formed as a trade association to represent and protect the legitimate use of GPR in Europe.

Similar technologies

Ground penetrating radar uses a variety of technologies to generate the radar signal, these are impulse, stepped frequency, FMCW and noise. Systems on the market in 2009 also use DSP to process the data, while survey work is being carried out rather than off line.

GPR is used on vehicles for close-in high speed road survey and landmine detection as well as in stand-off mode.

Pipe Penetrating Radar (PPR) is an application of GPR technologies applied in-pipe where the signals are directed through pipe and conduit walls to detect pipe wall thickness and voids behind the pipe walls.

Wall-penetrating radar can read through walls and even act as a motion sensor for police.[citation needed]
GPR is used for hand held landmine detection to reduce the false alarms experienced by the standard metal detector and systems are available off the shelf (Vallon and L3Com Cyterra)

The "Mineseeker Project" seeks to design a system to determine whether landmines are present in areas using ultra wideband synthetic aperture radar units mounted on blimps.


  1. ^ Daniels DJ (ed.) (2004). Ground Penetrating Radar (2nd ed.). Knoval (Institution of Engineering and Technology). pp. 1–4. ISBN 978-0-86341-360-5. 
  2. ^ Penguin Dictionary of Civil Engineering p347 (Radar)
  3. ^ ETSI EG 202 730 V1.1.1 (2009-09), "Electromagnetic compatibility and Radio spectrum Matters (ERM); Code of Practice in respect of the control, use and application of Ground Probing Radar (GPR) and Wall Probing Radar (WPR) systems and equipment
Borchert, Olaf: Receiver Design for a Directional Borehole Radar System Dissertation, University of Wuppertal, 2008, [1]

External links

Sunday, February 27, 2011

1776. My Favorite Patriotic Film

Athenian Democracy ended when Athens got too big for its britches, when it stuck its nose where it didn't belong, into a small conflict between a colony of Corinth's on the west coast of Greece, and a colony of that colony. The result was the Peloponnesian War, with the lesser city-states Sparta and Corinth teaming up to defeat mighty Athens, which they did.

So died Democracy, for the next 2000 years.

This is the story of how it came back:

John Adams, newlywed Martha Jefferson, and Ben Franklin, in Philadelphia, Pennsylvania, in June of 1776


The film version of the Broadway musical comedy of the same name. In the days leading up to July 4, 1776, Continental Congressmen John Adams and Benjamin Franklin coerce Thomas Jefferson into writing the Declaration of Independence as a delaying tactic as they try to persuade the American colonies to support a resolution on independence. 

As George Washington sends depressing messages describing one military disaster after another, the businessmen, landowners and slave holders in Congress all stand in the way of the Declaration, and a single "nay" vote will forever end the question of independence. Large portions of spoken and sung dialog are taken directly from the letters and memoirs of the actual participants.  

Written by Dave Heston
Despite or because of the state of the Revolutionary War led by General George Washington, the Second Continental Congress, meeting in Philadelphia, has long skirted the issue of independence from Great Britain, much to the chagrin of its chief proponents, Massachusetts Congressman John Adams and Pennsylvania Congressman Dr. Benjamin Franklin. Adams knows that much of the debate is against him as a person, many who see him as being obnoxious and a blow-hard. 

He decides a more judicious approach may be to work behind the scenes rather than be front and center in the fight as he has been. 

On June 7, 1776, Adams gets Virginia Congressman Richard Henry Lee to propose a motion in Congress to debate the issue, which finally passes. However when the vote for independence finally looks like it will pass, its chief opponent, Pennsylvania Congressman John Dickinson, manages to pass a motion that any vote for independence needs to be unanimous. 

As a delay tactic, Adams initiates a successful motion to postpone the vote for three weeks to July 2, 1776 until they can vote on the actual text for a declaration of independence - his assertion is how can they vote on something that does not exist. Adams and Franklin talk a reluctant Virginia Congressman Thomas Jefferson to be the one to draft the document. Jefferson's reluctance is that he has other more personal issues on his mind. 

As Jefferson takes to his writing duties, Adams and Franklin and their supporters know they only have three weeks to convince the six opposing colonies to support independence. As Franklin states, it may take some improvisation and some compromise. Written by Huggo

Left to right - John Dickinson, Martha Jefferson, Thomas Jefferson, Richard Henry Lee

And 200+ years later ... we get  ...

JOAN JETT  !! <== Click to play

Colonizing the Moon

You know you want to go there. You know we will. Why wait? What's stopping us, other than ourselves?

Random Graphs, Event Symmetry, and Quantum Graphity

When I was a lad in the 1960's, I became fascinated with the "new math" of nodes and lines known as "Network Analysis."

I always wondered whatever happened to it. Well, I wonder no more!

Apparently it changed its name to Graph Theory, and since my main hobby is Quantum Gravity, that reminded me of  Fotini Markopoulou-Kalamara's Quantum Graphity.

So, what is that? It's this (from 2 Wiki articles):

In mathematics, a random graph is a graph that is generated by some random process.[1] The theory of random graphs lies at the intersection between graph theory and probability theory, and studies the properties of typical random graphs.

Random graph models

A random graph is obtained by starting with a set of n vertices and adding edges between them at random. Different random graph models produce different probability distributions on graphs. Most commonly studied is the one proposed by Edgar Gilbert, denoted G(n,p), in which every possible edge occurs independently with probability p. A closely related model, the Erdős–Rényi model denoted G(n,M), assigns equal probability to all graphs with exactly M edges. The latter model can be viewed as a snapshot at a particular time (M) of the random graph process \tilde{G}_n, which is a stochastic process that starts with n vertices and no edges, and at each step adds one new edge chosen uniformly from the set of missing edges.

If instead we start with an infinite set of vertices, and again let every possible edge occur independently with probability p, then we get an object G called an infinite random graph. Except in the trivial cases when p is 0 or 1, such a G almost surely has the following property:
Given any n + m elements a_1,\ldots, a_n,b_1,\ldots, b_m \in V, there is a vertex c\in V that is adjacent to each of a_1,\ldots, a_n and is not adjacent to any of b_1,\ldots, b_m.
It turns out that if the vertex set is countable then there is, up to isomorphism, only a single graph with this property, namely the Rado graph. Thus any countably infinite random graph is almost surely the Rado graph, which for this reason is sometimes called simply the random graph. However, the analogous result is not true for uncountable graphs, of which there are many (nonisomorphic) graphs satisfying the above property.

Another model, which generalizes Gilbert's random graph model, is the random dot-product model. A random dot-product graph associates with each vertex a real vector. The probability of an edge uv between any vertices u and v is some function of the dot product uv of their respective vectors.
The network probability matrix models random graphs through edge probabilities, which represent the probability pi,j that a given edge ei,j exists for a specified time period. This model is extensible to directed and undirected; weighted and unweighted; and static or dynamic graphs.

For M \simeq pn the two most widely used models, G(n,M) and G(n,p), are almost interchangeable[2].
Random regular graphs form a special case, with properties that may differ from random graphs in general.

Properties of random graphs

The theory of random graphs studies typical properties of random graphs, those that hold with high probability for graphs drawn from a particular distribution. For example, we might ask for a given value of n and p what the probability is that G(n,p) is connected. In studying such questions, researchers often concentrate on the asymptotic behavior of random graphs—the values that various probabilities converge to as n grows very large. Percolation theory characterizes the connectedness of random graphs, especially infinitely large ones.

(threshold functions, evolution of G~)
Random graphs are widely used in the probabilistic method, where one tries to prove the existence of graphs with certain properties. The existence of a property on a random graph can often imply, via the famous Szemerédi regularity lemma, the existence of that property on almost all graphs.

Random trees

A random tree is a tree or arborescence that is formed by a stochastic process. Types of random trees include uniform spanning tree, random minimal spanning tree, random binary tree, treap, rapidly-exploring random tree, Brownian tree, and random forest.


Random graphs were first defined by Paul Erdős and Alfréd Rényi in their 1959 paper "On Random Graphs"[3] and independently by Gilbert in his paper "Random graphs"[4].

See also

The term event symmetry refers to invariance principles that have been used in some discrete approaches to quantum gravity where the diffeomorphism invariance of general relativity can be extended to a covariance under any permutation of spacetime events.[1]

The principle of event symmetry

What it means

Since general relativity was discovered by Albert Einstein in 1915 it has been demonstrated by observation and experiment to be an accurate theory of gravitation up to cosmic scales. On small scales the laws of quantum mechanics have likewise been found to describe nature in a way consistent with every experiment performed, so far. To describe the laws of the universe fully a synthesis of general relativity and quantum mechanics must be found. Only then can physicists hope to understand the realms where both gravity and quantum come together. The big bang is one such place.

The task to find such a theory of quantum gravity is one of the major scientific endeavours of our time. Many physicists believe that string theory is the leading candidate, but string theory has so far failed to provide an adequate description of the big bang, and its success is just as incomplete in other ways. That could be because physicists do not really know what the correct underlying principles of string theory are, so they do not have the right formulation that would allow them to answer the important questions. In particular, string theory treats spacetime in quite an old fashioned way even though it indicates that spacetime must be very different at small scales from what we are familiar with.

General relativity by contrast, is a model theory based on a geometric symmetry principle from which its dynamics can be elegantly derived. The symmetry is called general covariance or diffeomorphism invariance. It says that the dynamical equations of the gravitational field and any matter must be unchanged in form under any smooth transformation of spacetime coordinates. To understand what that means you have to think of a region of spacetime as a set of events, each one labelled by unique values of four coordinate values x,y,z, and t. The first three tell us where in space the event happened, while the fourth is time and tells us when it happened. But the choice of coordinates that are used is arbitrary, so the laws of physics should not depend on what the choice is. It follows that if any smooth mathematical function is used to map one coordinate system to any other, the equations of dynamics must transform in such a way that they look the same as they did before. This symmetry principle is a strong constraint on the possible range of equations and can be used to derive the laws of gravity almost uniquely.

The principle of general covariance works on the assumption that spacetime is smooth and continuous. Although this fits in with our normal experience, there are reasons to suspect that it may not be a suitable assumption for quantum gravity. In quantum field theory, continuous fields are replaced with a more complex structure that has a dual particle-wave nature as if they can be both continuous and discrete depending on how you measure them. Research in string theory and several other approaches to quantum gravity suggest that spacetime must also have a dual continuous and discrete nature, but without the power to probe spacetime at sufficient energies it is difficult to measure its properties directly to find out how such a quantised spacetime should work.

This is where event symmetry comes in. In a discrete spacetime treated as a disordered set of events it is natural to extend the symmetry of general covariance to a discrete event symmetry in which any function mapping the set of events to itself replaces the smooth functions used in general relativity. Such a function is also called a permutation, so the principle of event symmetry states that the equations governing the laws of physics must be unchanged when transformed by any permutation of spacetime events.

How it works

It is not immediately obvious how event symmetry could work. It seems to say that taking one part of space time and swapping it with another part a long distance away is a valid physical operation and that the laws of physics have to be written in such a way that this is supported. Clearly this symmetry can only be correct if it is hidden or broken. To get this in perspective consider what the symmetry of general relativity seems to say.

A smooth coordinate transformation or diffeomorphism can stretch and twist spacetime in any way so long as it is not torn. The laws of general relativity are unchanged in form under such a transformation. Yet this does not mean that objects can be stretched or bent without being opposed by a physical force. Likewise, event symmetry does not mean that objects can be torn apart in the way the permutations of spacetime would make us believe. In the case of general relativity the gravitational force acts as a background field that controls the measurement properties of spacetime. In ordinary circumstances the geometry of space is flat and Euclidean and the diffeomorphism invariance of general relativity is hidden thanks to this background field. Only in the extreme proximity of a violent collision of black holes would the flexibility of spacetime become apparent. In a similar way, event symmetry could be hidden by a background field that determines not just the geometry of spacetime, but also its topology.

General relativity is often explained in terms of curved spacetime. We can picture the universe as the curved surface of a membrane like a soap film which changes dynamically in time. The same picture can help us understand how event symmetry would be broken. A soap bubble is made from molecules which interact via forces that depend on the orientations of the molecules and the distance between them. If you wrote down the equations of motion for all the molecules in terms of their positions, velocities and orientations, then those equations would be unchanged in form under any permutation of the molecules (which we will assume to be all the same). This is mathematically analogous to the event symmetry of spacetime events. The equations may be different, and unlike the molecules on the surface of a bubble, the events of spacetime are not embedded in a higher dimensional space, yet the mathematical principle is the same.

Physicists do not presently know if event symmetry is a correct symmetry of nature, but the example of a soap bubble shows that it is a logical possibility. If it can be used to explain real physical observations then it merits serious consideration.

Maximal Permutability

American philosopher of physics John Stachel has used permutability of spacetime events to generalize Einstein's hole argument[2]. Statchel uses the term quiddity to describe the universal qualities of an entity and haecceity to describe its individuality. He makes use of the analogy with quantum mechanical particles, that have quiddity but no haecceity. The permutation symmetry of systems of particles leaves the equations of motion and the description of the system invariant. This is generalised to a principle of maximal permutability[3] that should be applied to physical entities. In an approach to quantum gravity where spacetime events are discrete, the principle implies that physics must be symmetric under any permutations of events, so the principle of event symmetry is a special case of the principle of maximal permutability.

Stachel's view builds on the work of philosophers such as Gottfried Leibniz whose monadology proposed that the world should be viewed only in terms of relations between objects rather than their absolute positions. Ernst Mach used this to formulate his relational principle which influenced Einstein in his formulation of general relativity. Some quantum gravity physicists believe that the true theory of quantum gravity will be a relational theory with no spacetime. The events of spacetime are then no longer a background in which physics happens. Instead they are just the set of events where an interaction between entities took place. Characteristics of spacetime that we are familiar with (such as distance, continuity and dimension) should be emergent in such a theory, rather than put in by hand.

Quantum Graphity and other random graph models

In a random graph model of spacetime, points in space or events in spacetime are represented by nodes of a graph. Each node may be connected to any other node by a link. In mathematical terms this structure is called a graph. The smallest number of links that it takes to go between two nodes of the graph can be interpreted as a measure of the distance between them in space. The dynamics can be represented either by using a Hamiltonian[disambiguation needed] formalism if the nodes are points in space, or a Lagrangian formalism if the nodes are events in spacetime. Either way, the dynamics allow the links to connect or disconnect in a random way according to specified probability rule. The model is event-symmetric if the rules are invariant under any permutation of the graph nodes.

The mathematical discipline of random graph theory was founded in the 1950s by Paul Erdős and Alfréd Rényi [4]. They proved the existence of sudden changes in characteristics of a random graph as parameters of the model varied. These are similar to phase transitions in physical systems. The subject has been extensively studied since with applications in many areas including computation and biology. A standard text is "Random Graphs" by Béla Bollobás[5].

Application to quantum gravity came later. Early random graph models of space-time have been proposed by Frank Antonsen (1993)[6], Manfred Requardt (1996)[7] and Thomas Filk (2000)[8]. Tomasz Konopka, Fotini Markopoulou-Kalamara, Simone Severini and Lee Smolin of the Canadian Perimeter Institute for Theoretical Physics introduced a graph model that they called Quantum Graphity[9],[10][11] . An argument based on quantum graphity combined with the holographic principle can resolve the horizon problem and explain the observed scale invariance of cosmic background radiation fluctuations without the need for cosmic inflation[12].

In the quantum graphity model, points in spacetime are represented by nodes on a graph connected by links that can be on or off. This indicates whether or not the two points are directly connected as if they are next to each other in spacetime. When they are on the links have additional state variables which are used to define the random dynamics of the graph under the influence of quantum fluctuations and temperature. At high temperature the graph is in Phase I where all the points are randomly connected to each other and no concept of spacetime as we know it exists. As the temperature drops and the graph cools, it is conjectured to undergo a phase transition to a Phase II where spacetime forms. It will then look like a spacetime manifold on large scales with only near-neighbour points being connected in the graph. The hypothesis of quantum graphity is that this geometrogenesis models the condensation of spacetime in the big bang.

Event symmetry and string theory

String theory is formulated on a background spacetime just as quantum field theory is. Such a background fixes spacetime curvature which in general relativity is like saying that the gravitational field is fixed. However, analysis shows that the excitations of the string fields act as gravitons which can perturb the gravitational field away from the fixed background, so string theory is actually a theory which includes dynamic quantised gravity. More detailed studies have shown that different string theories in different background spacetimes can be related by dualities. There is also good evidence that string theory supports changes in topology of spacetime. Relativists have therefore criticised string theory for not being formulated in a background independent way, so that changes of spacetime geometry and topology can be more directly expressed in terms of the fundamental degrees of freedom of the strings.

The difficulty in achieving a truly background independent formulation for string theory is demonstrated by a problem known as Witten's Puzzle[13]. Ed Witten asked the question "What could the full symmetry group of string theory be if it includes diffeomorphism invariance on a spacetime with changing topology?". This is hard to answer because the diffeomorphism group for each spacetime topology is different and there is no natural way to form a larger group containing them all such that the action of the group on continuous spacetime events makes sense. This puzzle is solved if the spacetime is regarded as a discrete set of events with different topologies formed dynamically as different string field configurations. Then the full symmetry need only contain the permutation group of spacetime events. Since any diffeomorphism for any topology is a special kind of permutation on the discrete events, the permutation group does contain all the different diffeomorphism groups for all possible topologies.

There is some evidence from Matrix Models that event-symmetry is included in string theory. A random matrix model can be formed from a random graph model by taking the variables on the links of the graph and arranging them in a N by N square matrix, where N is the number of nodes on the graph. The element of the matrix in the nth column and mth row gives the variable on the link joining the nth nodes to the mth node. The event-symmetry can then be extended to a larger N dimensional rotational symmetry.

In string theory, random matrix models were introduced to provide a non-perturbative formulation of M-Theory using noncommutative geometry. Coordinates of spacetime are normally commutative but in noncommutative geometry they are replaced by matrix operators that do not commute. In the original M(atrix) Theory these matrices were interpreted as connections between instantons (also known as D0-branes), and the matrix rotations were a gauge symmetry. Later, Iso and Kawai reinterpreted this as a permutation symmetry of space-time events[14] and argued that diffeomorphism invariance was included in this symmetry. The two interpretations are equivalent if no distinction is made between instantons and events, which is what would be expected in a relational theory. This shows that Event Symmetry can already be regarded as part of string theory.


Greg Egan's Dust Theory

The first known publication of the idea of event symmetry is in a work of science fiction rather than a journal of science. Greg Egan used the idea in a short story called "Dust" in 1992[15] and expanded it into the novel Permutation City in 1995. Egan used dust theory as a way of exploring the question of whether a perfect computer simulation of a person differs from the real thing. However, his description of the dust theory as an extension of general relativity is also a consistent statement of the principle of event symmetry as used in quantum gravity.

The essence of the argument can be found in chapter 12 of "Permutation City". Paul, the main character of the story set in the future, has created a copy of himself in a computer simulator. The simulation runs on a distributed network which is sufficiently powerful to emulate his thoughts and experiences. Paul argues that the events of his simulated world have been remapped to events in the real world by the computer in a way that resembles a coordinate transformation in relativity. General relativity only allows for covariance under continuous transformations whereas the computer network has formed a discontinuous mapping which permutes events like "a cosmic anagram". Yet Paul's copy in the simulator experiences physics as if it was unchanged. Paul realises that this is "Like […] gravity and acceleration in General Relativity — it all depends on what you can't tell apart. This is a new Principle of Equivalence, a new symmetry between observers."

Saturday, February 26, 2011

Relativistic Jets and Frame Dragging

The realm of the very large, of galaxies and the filaments of same and the voids between them have always fascinated me.

Perhaps chief among them are the jets, called polar jets when non-relativistic and relativistic when not, that shoot out from the poles of everything from neutron stars to stellar-sized black holes to galactic-sized black holes to quasars (active galaxies). 

Apparently there are 2 competing theories on the origin and mechanism of the faster relativistic ones, that I just became aware of. I'm curious if anyone knows how things are progressing to solving this interesting mystery, given the fantastic amount of new data the observatories on earth and in space are providing. 

These are also drawn into the mathematics of "Frame-dragging", an interesting field in its own right, which basically says that matter in a rotating body drags space-time due to the rotation, which was predicted in 1918 based on Einstein's field equations of General Relativity.

The Mathematics of General Relativity are pretty straight forward assuming one has taken the first four college courses in Calculus, which end with the extremely important PDE (Partial Differential Equations) in 4th-semester Calc IV.

Frame-dragging is an excellent application of same. From two different Wiki articles:

Relativistic jets are extremely powerful jets of plasma which emerge from presumed massive objects at the centers of some active galaxies, notably radio galaxies and quasars. Their lengths can reach several thousand[1] or even hundreds of thousands of light years.[2] The hypothesis is that the twisting of magnetic fields in the accretion disk collimates the outflow along the rotation axis of the central object, so that when conditions are suitable, a jet will emerge from each face of the accretion disk. If the jet is oriented along the line of sight to Earth, relativistic beaming will change its apparent brightness. The mechanics behind both the creation of the jets[3][4] and the composition of the jets[5] are still a matter of much debate in the scientific community; it is hypothesized that the jets are composed of an electrically neutral mixture of electrons, positrons, and protons in some proportion.

Elliptical Galaxy M87 emitting a relativistic jet, as seen by Hubble Space Telescope's WFPC2 in the visible spectrum.
Similar jets, though on a much smaller scale, can develop around the accretion disks of neutron stars and stellar black holes. These systems are often called microquasars. A famous example is SS433, whose well-observed jet has a velocity of 0.23c, although other microquasars appear to have much higher (but less well measured) jet velocities. Even weaker and less-relativistic jets may be associated with many binary systems; the acceleration mechanism for these jets may be similar to the magnetic reconnection processes observed in the Earth's magnetosphere and the solar wind.

The general hypothesis among astrophysicists is that the formation of relativistic jets is the key to explaining the production of gamma-ray bursts. These jets have Lorentz factors of ~100 (that is, speeds of roughly 0.99995c), making them one of the swiftest celestial objects currently known.

Rotating black hole as energy source

Because of the enormous amount of energy needed to launch a relativistic jet, some jets are thought to be powered by spinning black holes. There are two competing theories for how the energy is transferred from the black hole to the jet.
  • Blandford-Znajek process.[6] This is the most popular theory for the extraction of energy from the central black hole. The magnetic fields around the accretion disk are dragged by the spin of the black hole. The relativistic material is possibly launched by the tightening of the field lines.
  • Penrose mechanism.[7] This extracts energy from a rotating black hole by frame dragging. This theory was later proven to be able to extract relativistic particle energy and momentum,[8] and subsequently shown to be a possible mechanism for the formation of jets.[9]

Albert Einstein's theory of general relativity predicts that rotating bodies drag spacetime around themselves in a phenomenon referred to as frame-dragging. The rotational frame-dragging effect was first derived from the theory of general relativity in 1918 by the Austrian physicists Josef Lense and Hans Thirring, and is also known as the Lense–Thirring effect.[1][2][3] Lense and Thirring predicted that the rotation of an object would alter space and time, dragging a nearby object out of position compared with the predictions of Newtonian physics. The predicted effect is small—about one part in a few trillion. To detect it, it is necessary to examine a very massive object, or build an instrument that is very sensitive. More generally, the subject of field effects caused by moving matter is known as gravitomagnetism.

Frame dragging effects

Rotational frame-dragging (the Lense–Thirring effect) appears in the general principle of relativity and similar theories in the vicinity of rotating massive objects. Under the Lense–Thirring effect, the frame of reference in which a clock ticks the fastest is one which is rotating around the object as viewed by a distant observer. This also means that light traveling in the direction of rotation of the object will move around the object faster than light moving against the rotation as seen by a distant observer. It is now the best-known effect, partly thanks to the Gravity Probe B experiment.

Linear frame dragging is the similarly inevitable result of the general principle of relativity, applied to linear momentum. Although it arguably has equal theoretical legitimacy to the "rotational" effect, the difficulty of obtaining an experimental verification of the effect means that it receives much less discussion and is often omitted from articles on frame-dragging (but see Einstein, 1921).[4]

Static mass increase is a third effect noted by Einstein in the same paper.[5] The effect is an increase in inertia of a body when other masses are placed nearby. While not strictly a frame dragging effect (the term frame dragging is not used by Einstein), it is demonstrated by Einstein to derive from the same equation of general relativity. It is also a tiny effect that is difficult to confirm experimentally.

Experimental tests of frame-dragging

In 1976 Van Patten and Everitt[6][7] proposed to implement a dedicated mission aimed to measure the Lense–Thirring node precession of a pair of counter-orbiting spacecraft to be placed in terrestrial polar orbits with drag-free apparatus. A somewhat equivalent, cheaper version of such an idea was put forth in 1986 by Ciufolini[8] who proposed to launch a passive, geodetic satellite in an orbit identical to that of the LAGEOS satellite, launched in 1976, apart from the orbital planes which should have been displaced by 180 deg apart: the so-called butterfly configuration. The measurable quantity was, in this case, the sum of the nodes of LAGEOS and of the new spacecraft, later named LAGEOS III, LARES, WEBER-SAT. Although extensively studied by various groups,[9][10] such an idea has not yet been implemented. The butterfly configuration would allow, in principle, to measure not only the sum of the nodes but also the difference of the perigees,[11][12][13] although such Keplerian orbital elements are more affected by the non-gravitational perturbations like the direct solar radiation pressure: the use of the active, drag-free technology would be required. Other proposed approaches involved the use of a single satellite to be placed in near polar orbit of low altitude,[14][15] but such a strategy has been shown to be unfeasible.[16][17][18] In order to enhance the possibilities of being implemented, it has been recently claimed that LARES/WEBER-SAT would be able to measure the effects[19] induced by the multidimensional braneworld model by Dvali, Gabadaze and Porrati[20] and to improve by two orders of magnitude the present-day level of accuracy of the equivalence principle.[21] Such claims have been shown to be highly unrealistic.[22][23]

Limiting the scope to the scenarios involving existing orbiting bodies, the first proposal to use the LAGEOS satellite and the Satellite Laser Ranging (SLR) technique to measure the Lense–Thirring effect dates back to 1977–1978.[24][25] Tests have started to be effectively performed by using the LAGEOS and LAGEOS II satellites in 1996,[26] according to a strategy[27] involving the use of a suitable combination of the nodes of both satellites and the perigee of LAGEOS II. The latest tests with the LAGEOS satellites have been performed in 2004-2006[28][29] by discarding the perigee of LAGEOS II and using a linear combination[30][31][32][33][34][35] involving only the nodes of both the spacecraft. Although the predictions of general relativity are compatible with the experimental results, the realistic evaluation of the total error raised a debate.[36][37][38][39][40][41] Another test of the Lense–Thirring effect in the gravitational field of Mars, performed by suitably interpreting the data of the Mars Global Surveyor (MGS) spacecraft, has been recently reported.[42] There is also debate about this test.[43][44][45] Attempts to detect the Lense–Thirring effect induced by the Sun's rotation on the orbits of the inner planets of the Solar System have been reported as well:[46] the predictions of general relativity are compatible with the estimated corrections to the perihelia precessions,[47] although the errors are still large. However, the inclusion of the radiometric data from the Magellan orbiter recently allowed Pitjeva to greatly improve the determination of the unmodelled precession of the perihelion of Venus. It amounts to −0.0004 ± 0.0001 arcseconds/century, while the Lense–Thirring effect for the Venus' perihelion is just -0.0003 arcseconds/century.[48] The system of the Galilean satellites of Jupiter was investigated as well,[49] following the original suggestion by Lense and Thirring.

The Gravity Probe B experiment[50][51] is currently under way to experimentally measure another gravitomagnetic effect, i.e. the Schiff precession of a gyroscope,[52][53] to an expected 1% accuracy or better. Unfortunately, it seems that such an ambitious goal will not be achieved: indeed, first preliminary results released in April 2007 point toward a so far obtained accuracy of[54] 256–128%, with the hope of reaching about 13% in December 2007.[55] However, in 2008 the Senior Review Report of the NASA Astrophysics Division Operating Missions stated that it is unlikely that Gravity Probe B team will be able to reduce the errors to the level necessary to produce a convincing test of currently-untested aspects of General Relativity (including Frame-dragging).[56][57]

A 1% measurement of the Lense–Thirring effect in the gravitational field of the Earth could be obtained by launching at least two entirely new satellites, preferably with active mechanisms of compensation of the non-gravitational forces, in eccentric orbits, as stated in 2005 by Lorenzo Iorio.[58] Recently, the Italian Space Agency (ASI) has announced that the LARES satellite should be launched with a Vega rocket at the beginning of 2011.[59] The goal of LARES is to measure the Lense–Thirring effect to 1%, but there are doubts that this can be achieved,[60][61][62][63][64] mainly due to the relatively low-orbit which LARES should be inserted into bringing into play more mismodelled even zonal harmonics.[clarification needed] That is, spherical harmonics of the Earth's gravitational field caused by mass concentrations (like mountains) can drag a satellite in a way which may be difficult to distinguish from frame-dragging. Recently, an indirect test of the gravitomagnetic interaction accurate to 0.1% has been reported by Murphy et al. with the Lunar laser ranging (LLR) technique,[65] but Kopeikin questioned the ability of LLR to be sensitive to gravitomagnetism.[66]
In the case of stars orbiting close to a spinning, supermassive black hole, frame dragging should cause the star's orbital plane to precess about the black hole spin axis. This effect should be detectable within the next few years via astrometric monitoring of stars at the center of the Milky Way galaxy.[67] By comparing the rate of orbital precession of two stars on different orbits, it is possible in principle to test the no-hair theorems of general relativity, in addition to measuring the spin of the black hole.[68]

Astronomical evidence

Relativistic Jet. The environment around the AGN where the relativistic plasma is collimated into jets which escape along the pole of the supermassive black hole
Relativistic jets may provide evidence for the reality of frame-dragging. Gravitomagnetic forces produced by the Lense–Thirring effect (frame dragging) within the ergosphere of rotating black holes[69][70] combined with the energy extraction mechanism by Penrose[71] have been used to explain the observed properties of relativistic jets. The gravitomagnetic model developed by Reva Kay Williams predicts the observed high energy particles (~GeV) emitted by quasars and active galactic nuclei; the extraction of X-rays, γ-rays, and relativistic e-e+ pairs; the collimated jets about the polar axis; and the asymmetrical formation of jets (relative to the orbital plane).

Mathematical derivation of frame-dragging

Frame-dragging may be illustrated most readily using the Kerr metric,[72][73] which describes the geometry of spacetime in the vicinity of a mass M rotating with angular momentum J
c^{2} d\tau^{2} = 
\left( 1 - \frac{r_{s} r}{\rho^{2}} \right) c^{2} dt^{2} 
- \frac{\rho^{2}}{\Lambda^{2}} dr^{2} 
- \rho^{2} d\theta^{2}
- \left( r^{2} + \alpha^{2} + \frac{r_{s} r \alpha^{2}}{\rho^{2}} \sin^{2} \theta \right) \sin^{2} \theta \ d\phi^{2} 
+ \frac{2r_{s} r\alpha c \sin^{2} \theta }{\rho^{2}} d\phi dt
where rs is the Schwarzschild radius
r_{s} = \frac{2GM}{c^{2}}
and where the following shorthand variables have been introduced for brevity
\alpha = \frac{J}{Mc}
\rho^{2} = r^{2} + \alpha^{2} \cos^{2} \theta\,\!
\Lambda^{2} = r^{2} - r_{s} r + \alpha^{2}\,\!
In the non-relativistic limit where M (or, equivalently, rs) goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates
c^{2} d\tau^{2} = 
c^{2} dt^{2} 
- \frac{\rho^{2}}{r^{2} + \alpha^{2}} dr^{2} 
- \rho^{2} d\theta^{2}
- \left( r^{2} + \alpha^{2} \right) \sin^{2}\theta d\phi^{2}
We may re-write the Kerr metric in the following form
c^{2} d\tau^{2} = 
\left( g_{tt} - \frac{g_{t\phi}^{2}}{g_{\phi\phi}} \right) dt^{2}
+ g_{rr} dr^{2} + g_{\theta\theta} d\theta^{2} + 
g_{\phi\phi} \left( d\phi + \frac{g_{t\phi}}{g_{\phi\phi}} dt \right)^{2}
This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius r and the colatitude θ
\Omega = -\frac{g_{t\phi}}{g_{\phi\phi}} = \frac{r_{s} \alpha r c}{\rho^{2} \left( r^{2} + \alpha^{2} \right) + r_{s} \alpha^{2} r \sin^{2}\theta}
In the plane of the equator this simplifies to:[74]
\Omega = \frac{r_{s} \alpha c}{r^{3} + \alpha^{2} r + r_{s} \alpha^{2}}
Thus, an inertial reference frame is entrained by the rotating central mass to participate in the latter's rotation; this is frame-dragging.

The two surfaces on which the Kerr metric appears to have singularities; the inner surface is the spherical event horizon, whereas the outer surface is an oblate spheroid. The ergosphere lies between these two surfaces; within this volume, the purely temporal component gtt is negative, i.e., acts like a purely spatial metric component. Consequently, particles within this ergosphere must co-rotate with the inner mass, if they are to retain their time-like character.
An extreme version of frame dragging occurs within the ergosphere of a rotating black hole. The Kerr metric has two surfaces on which it appears to be singular. The inner surface corresponds to a spherical event horizon similar to that observed in the Schwarzschild metric; this occurs at
r_{inner} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2}}}{2}
where the purely radial component grr of the metric goes to infinity. The outer surface is not a sphere, but an oblate spheroid that touches the inner surface at the poles of the rotation axis, where the colatitude θ equals 0 or π; its radius is defined by the formula
r_{outer} = \frac{r_{s} + \sqrt{r_{s}^{2} - 4\alpha^{2} \cos^{2}\theta}}{2}
where the purely temporal component gtt of the metric changes sign from positive to negative. The space between these two surfaces is called the ergosphere. A moving particle experiences a positive proper time along its worldline, its path through spacetime. However, this is impossible within the ergosphere, where gtt is negative, unless the particle is co-rotating with the interior mass M with an angular speed at least of Ω. However, as seen above, frame-dragging occurs about every rotating mass and at every radius r and colatitude θ, not only within the ergosphere.

Lense–Thirring effect inside a rotating shell

Inside a rotating spherical shell the acceleration due to the Lense–Thirring effect would be [75]
\bar{a} = -2d_1 \left( \bar{ \omega} \times \bar v \right) - d_2 \left[ \bar{ \omega} \times \left( \bar{ \omega} \times \bar{r} \right) + 2\left( \bar{ \omega}\bar{r} \right) \bar{ \omega} \right]
where the coefficients are
d_1 = \frac{4MG}{3Rc^2}
d_2 = \frac{4MG}{15Rc^2}
for MGRc2 or more precisely,
d_1 = \frac{4 \alpha(2 - \alpha)}{(1 + \alpha)(3- \alpha)}, \qquad \alpha=\frac{MG}{2Rc^2}
The space-time inside the rotating spherical shell will not be flat. A flat space-time inside a rotating mass shell is possible if the shell is allowed to deviate from a precisely spherical shape and the mass density inside the shell is allowed to vary.[76]

See also

External links

An early version of this article was adapted from public domain material from