From Gravitational Waves to Supergravity

S. Al Saleh A. Mahrousseh and L.A. Al Asfar

Department of Physics and Astronomy , College of Science King Saud University

February 28, 2016 Is Gravity a Force ?

Isaac Newton showed that the ’ force’ driving the apple down to earth, is the same as the force in the ’ heavens’ that lets the moon orbits the earth. This was a great triumph for modern science ! Mm F = G . (1) g N r2 This famous equation is known for almost all secondary school students, it resembles the universal law of gravitation. N1 A particle that is not affected by any force will move uniformly in a straight line . N2 Any deviation from the straight line, uniform motion of the particle . Is made by a force affecting that particle.

However, if we read Newton’s laws of motion carefully, we find there is something not quit right about (1). N2 Any deviation from the straight line, uniform motion of the particle . Is made by a force affecting that particle.

However, if we read Newton’s laws of motion carefully, we find there is something not quit right about (1). N1 A particle that is not affected by any force will move uniformly in a straight line . However, if we read Newton’s laws of motion carefully, we find there is something not quit right about (1). N1 A particle that is not affected by any force will move uniformly in a straight line . N2 Any deviation from the straight line, uniform motion of the particle . Is made by a force affecting that particle. Laplace was the first to think about this issue seriously. He proposed to ditch the concept of gravity as a force, instead, thinking of it as a prescription for background geometry.

However, N1 cannot be true in a universe with gravity and with more than one particle. Because there will always be a gravitational force affecting any particle in that universe, no matter how small. Hence, fundamentally N1 is meaningless if gravity is a force ! However, N1 cannot be true in a universe with gravity and with more than one particle. Because there will always be a gravitational force affecting any particle in that universe, no matter how small. Hence, fundamentally N1 is meaningless if gravity is a force !

Laplace was the first to think about this issue seriously. He proposed to ditch the concept of gravity as a force, instead, thinking of it as a prescription for background geometry. Geometric Gravity

Laplace read N1 differently, he considered it as a physical way of defininga straight line in space. Thus gravity could be the description of the underlying geometry. Like an ant moving as straight as it could on the curved surface of a sphere.

Figure: An ant moving on the surface of the ball would locally experience it as flat, but globally the surface is -in fact- curved . A particle that is not affected by any force will move uniformly in a straight line.

He missed the fact that curvature should be in spacetime not just space!

Gravity - in Lplace’s conjecture- is merely a curvature in space created by mass. He spent a long time trying to prove his conjecture, however he could not succeed . He did not posses the sophisticated mathematics to describe curved spaces. More importantly, he missed an important part of N1: He missed the fact that curvature should be in spacetime not just space!

A particle that is not affected by any force will move uniformly in a straight line. Gravity - in Lplace’s conjecture- is merely a curvature in space created by mass. He spent a long time trying to prove his conjecture, however he could not succeed . He did not posses the sophisticated mathematics to describe curved spaces. More importantly, he missed an important part of N1:

A particle that is not affected by any force will move uniformly in a straight line.

He missed the fact that curvature should be in spacetime not just space! The Birth of General Relativity

David Hilbert, Einstein and others, started working on the generalisation of special relativity for accelerated observers. Einstein discovered the principle of strong equivalence. Making gravity involved in this generalisation. In 1915, he succeeded in publishing series of papers demonstrating a geometric theory of gravitation.

Figure: The principle of (strong) equivalence. The Field Equations

One of the magnificent products of GR is the field equations :

1 8πGN Rµν − Rgµν = Tµν (2) | {z2 } c4 |{z} Matter/ Energy Geometry

These equations tell spacetime how to curve, and one the same footing tells matter how to move. Along with the geometry, causal structure is imposed on spacetime, i.e. the relation between different events. GR is confirmed by many observations, particularly gravitational lensing, resulting from the curvature of spacetime near massive objects.

Figure: Gravitational lensing Bending of light by an angle θ by a massive object with mass M at distance D is derived from Einstein field equations: √ 4G M δθ = N . (3) c²D

Figure: Bending of star-light by solar gravity Another example of gravitational lensing

Figure: Einstein Ring The Metric

Since gravity is a geometric theory, it is described by a mathematical object called the metric g. The metric gives rise to shapes of surfaces - or spaces- as it defines lengths . For example, a vector X in the spacetime can have a norm by the metric in the following way: ||X|| = g(X, X) (4) The field equations gives rise to the metric, hence the shape ofthe spacetime in the presence of matter/energy. When energy is absent, the spacetime admits a flat metric η. Consider a small perturbation over the flat metric, small energy added to the flat spacetime. One can solve Einstein filed equations by assuming the new metric could be written as:

g = η + h (5)

Where h is the small geometric fluctuations resulting from perturbing the flat spacetime. It turns out that h satisfies the wave equation ! Like water waves forming from perturbing the calm water surface .

λ ikλx hµν = Aµν e (6) Gravitational Waves The metric h is a gravitational wave metric, in other words, it describes the geometry behaving like a wave. Lengths in the spacetime would increase and decrease in a periodic way as the gravitational wave passes.

Figure: Gravitational waves in general relativity and outside GR. Like electromagnetic waves, gravitational waves are produced when the (field) is disturbed in a certain way. For example: • The big bang. • Blackhole formation. • Binary blackholes (mergers). • Rotating binary neutron stars. • Supernovae. • Rotating asymmetric massive objects. …and others.

Gravitational waves are ’ripples’ in the geometry propagating above the background geometry . They are very important prediction of GR, like electromagnetic waves are a crucial aspect of Maxwell’s electromagnetic theory. • The big bang. • Blackhole formation. • Binary blackholes (mergers). • Rotating binary neutron stars. • Supernovae. • Rotating asymmetric massive objects. …and others.

Gravitational waves are ’ripples’ in the geometry propagating above the background geometry . They are very important prediction of GR, like electromagnetic waves are a crucial aspect of Maxwell’s electromagnetic theory. Like electromagnetic waves, gravitational waves are produced when the (field) is disturbed in a certain way. For example: • The big bang. • Blackhole formation. • Binary blackholes (mergers). • Rotating binary neutron stars. • Supernovae. Gravitational waves are ’ripples’ in the geometry propagating above the background geometry . They are very important prediction of GR, like electromagnetic waves are a crucial aspect of Maxwell’s electromagnetic theory. Like electromagnetic waves, gravitational waves are produced when the (field) is disturbed in a certain way. For example: • The big bang. • Blackhole formation. • Binary blackholes (mergers). • Rotating binary neutron stars. • Supernovae. • Rotating asymmetric massive objects. …and others. Since gravity is a very weak interaction, it requires a huge disturbance in energy to produce a measurable gravitational wave. For example, the amplitude of gravitational waves can measured as a strain ∆L L .The strain of a wire with 10 Kg of weights hung on it - like the ones in 10X labs- is about ∼ 10−2, in the case of the gravitational waves detected it was 1 × 10−21 ; that is 19 orders of magnitudes difference!! Figure: The spectrum of gravitational waves By NASA Goddard Space Flight Center - http://science.gsfc.nasa.gov/663/research/index.html. The Big News !

LIGO and Virgo collaborations had published one of the most influential papers maybe for this century in Physical Review Letters . Confirming the detection of gravitational waves from about 1 billion light years away binary blackholes merging. This is a tremendous advancement on both theoretical physics and observational astrophysics. The incident had energy 50 time greater that the all energy produced by all the stars in the observable universe in milliseconds ; as the two blackholes have spun near the speed of light till they merged .

The Discovery

The source of gravitational waves was a binary blackhole system (GW150914), The blackholes have masses of 36 and 29 solar masses respectively, merging to form a spinning ( Kerr) blackhole with mass 62 solar masses. As they approached each other, they starting to whirl and produce gravitational waves. The detector measured strain characterised by changing the distance between the detector is moved by about the 1/1000 th of the diameter of the proton per metre. The Discovery

The Data The two collaborations had demonstrated a very matching data, and both agree with numerical GR calculations:

The LIGO detector Theoretical Importance

This discovery does not ’prove’ general relativity, as the latter was already proven by many previous observations, as we have seen earlier. But it is considered the final and strongest indication that gravity is indeed a geometrodynamical theory ; this is slightly different from GR. That is, the formulation of superspace in which spacetime could be the phase space of the 3-space. ( Wheeler, 1960).

Why geometrodynamics ? Wheeler describes geometrodynamics as the realisation of mass without mass, charge without charge and a field without a field. That is the split spacetime into space+time and study how curvature varies with time. This plays an important rôle in quantising gravity. The Graviton

In 1905, the idea of electromagnetic field quantum, the photon ,had emerged from the study of the photoelectric effect. The photon is now considered as ”quantisation ” of the Maxwell’s electromagnetic waves. One can think of a hypothetical quantisation for the gravitational waves, and call them ’gravitons’; unlike photons, they have a 2 units of spin instead of one, and mediate the gravitational interaction. However, it is much harder to apply the same quantisation done for photons on gravitons; even when gravitational waves are discovered. Infinities Everywhere!

Physicists tried to apply typical quantisation techniques used for the electromagnetic field ( and others) ; this process involves a mathematical technique called Renormalisation. Although it was useful for getting rid of infinities that appear in quantum fields, it failed completely todosowith the gravitational field.

Hence gravity is called non-renormalisable theory. Supergravity is a particle theory aiming to quantise gravity by applying the principles of supersymmetry to gravity.

Symmetry ( Could Be) the Answer

If one insists on using the typical methods of quantisation, adding more symmetries ( and dimensions) could force gravity to be renormalisable . In particular, the concept of Supergraity. Symmetry ( Could Be) the Answer

If one insists on using the typical methods of quantisation, adding more symmetries ( and dimensions) could force gravity to be renormalisable . In particular, the concept of Supergraity.

Supergravity is a particle theory aiming to quantise gravity by applying the principles of supersymmetry to gravity. Supergravity with extra dimensions, could explain why gravity is so weak, see the figure above. Gravitons could have the freedom to travel to higher dimensions, thus diluting its effect.

Supersymmetry states that there exist a transformation ( supertransformation), that could transform a Boson to Fermion and vice versa. Same goes for the graviton ( a boson) having a fermionic superpartner ; the Gravitino. Supersymmetry states that there exist a transformation ( supertransformation), that could transform a Boson to Fermion and vice versa. Same goes for the graviton ( a boson) having a fermionic superpartner ; the Gravitino.

Supergravity with extra dimensions, could explain why gravity is so weak, see the figure above. Gravitons could have the freedom to travel to higher dimensions, thus diluting its effect. • General relativity passes the hardest test ever, giving hope for more advanced theories of gravitation; like geometrodynamics. • Detecting gravitational waves is a huge supporting evidence for the urgent need to quantising gravity. • Spacetime might have more dimensions. Moreover, it could have a fermionic aspect of it as supergravity predicts.For example gravitino could be the component of dark matter

Conclusion

• Gravitational waves are promising for futuristic observational astrophysics, it could tell us a lot about the big bang and other dramatic astrophsical events; like a hearing tool when seeing is not possible. • Detecting gravitational waves is a huge supporting evidence for the urgent need to quantising gravity. • Spacetime might have more dimensions. Moreover, it could have a fermionic aspect of it as supergravity predicts.For example gravitino could be the component of dark matter

Conclusion

• Gravitational waves are promising for futuristic observational astrophysics, it could tell us a lot about the big bang and other dramatic astrophsical events; like a hearing tool when seeing is not possible. • General relativity passes the hardest test ever, giving hope for more advanced theories of gravitation; like geometrodynamics. • Spacetime might have more dimensions. Moreover, it could have a fermionic aspect of it as supergravity predicts.For example gravitino could be the component of dark matter

Conclusion

• Gravitational waves are promising for futuristic observational astrophysics, it could tell us a lot about the big bang and other dramatic astrophsical events; like a hearing tool when seeing is not possible. • General relativity passes the hardest test ever, giving hope for more advanced theories of gravitation; like geometrodynamics. • Detecting gravitational waves is a huge supporting evidence for the urgent need to quantising gravity. Conclusion

• Gravitational waves are promising for futuristic observational astrophysics, it could tell us a lot about the big bang and other dramatic astrophsical events; like a hearing tool when seeing is not possible. • General relativity passes the hardest test ever, giving hope for more advanced theories of gravitation; like geometrodynamics. • Detecting gravitational waves is a huge supporting evidence for the urgent need to quantising gravity. • Spacetime might have more dimensions. Moreover, it could have a fermionic aspect of it as supergravity predicts.For example gravitino could be the component of dark matter Conclusion

• Gravitational waves are promising for futuristic observational astrophysics, it could tell us a lot about the big bang and other dramatic astrophsical events; like a hearing tool when seeing is not possible. • General relativity passes the hardest test ever, giving hope for more advanced theories of gravitation; like geometrodynamics. • Detecting gravitational waves is a huge supporting evidence for the urgent need to quantising gravity. • Spacetime might have more dimensions. Moreover, it could have a fermionic aspect of it as supergravity predicts.For example gravitino could be the component of dark matter Thank you !

It was a great honour to be able to give this lecture, it is such an amazing time for gravity we happen to witness! I hope this lecture answered some of your questions, and better yet let you ask even more !

References

• Abbott, B. P., et al. ”Observation of Gravitational Waves from a Binary Black Hole Merger.” Physical Review Letters 116.6 (2016): 061102. • Wheeler, John Archibald. ”Geometrodynamics.” (1962). • Freedman, Daniel Z., and Antoine Van Proeyen. Supergravity. Cambridge University Press, 2012. • Misner, Charles W., Kip S. Thorne, and John Archibald Wheeler. Gravitation. Macmillan, 1973. • Hassan, Sayed Fawad. ”ASPECTS OF STRING THEORY.” (1995). • Arnowitt, Richard, Stanley Deser, and Charles W. Misner. ”Dynamical structure and definition of energy in general relativity.” Physical Review 116.5 (1959): 1322.

• De Boer, Jan. ”Six-dimensional supergravity on S 3× AdS 3 and 2d conformal field theory.” Nuclear Physics B 548.1 (1999):

139-166. Why Gravity is fundementally non-renormalisable ? - Bonus Slide-

In particle physics, one can think of infinite number of possible interaction between two particles (fields).

However, the ’closer’ we get the more irrelevant these interaction are ! In renormalisation theory, we simply neglect such irrelevant ’details’, and consider the interactions relevant to our energy scale. This is NOT the case with gravity, because of the dimension of its coupling constant. The closer you get, the more relevant interactions become. Gravity becomes significant near the Plank scale, thus we cannot throw away the ’details’.

Hence, supergravity, could take away this property of gravity and allows it to be renormalised. Sadly, this doesn’t work in 4 dimensions.