Trouble for Alternate Theories of Gravity

[lpetrich]

Troubled Times for Alternatives to Einstein’s Theory of Gravity | Quanta Magazine

Only adding to what they have already suffered: The Confrontation between General Relativity and Experiment | SpringerLink. Here is a rundown.

An important feature of gravity is its independence of composition. This makes gravity look like acceleration, and Albert Einstein had proposed that they are fundamentally identical: his Equivalence Principle. This led him to conclude that gravity is due to space-time being curved, and he also developed a theory for the origin of that curvature: general relativity. Here are versions of the EP:

• Weak EP: all nongravitational mass makes the same gravity
• Einstein EP: the WEP with Local Lorentz Invariance (velocity invariance) and Local Position Invariance
• Strong EP: the EEP where all mass makes the same gravity, including gravitational self-energy

WEP (different composition): < 10^(-12), LLI (Michelson-Morley and its successors): < 10^(-21), LPI (gravitational redshift): < 10^(-6)

Violations of the EP may be interpreted as the result of a "fifth force", so these upper limits also place upper limits on such possible interactions.

Even if gravity is due to space-time curvature, there is still the question of how that curvature is related to mass density, or more precisely, mass/energy/momentum density/flux and pressure. General relativity is a simple possible relation, but still only one possible relation, and various physicists have devised a variety of others. These are all adjusted to give the correct Newtonian limit, but they differ when one expands in a series beyond that. The first step in this expansion is called post-Newtonian, the second step post-post-Newtonian, etc. One can capture the predictions of general relativity and most proposed alternatives by using 10 parameters in the "parametrized post-Newtonian" formalism.

Five of these parameters are for violation of conservation of energy, linear momentum, and angular momentum, and three of them for preferred-frame effects, with one parameter shared. Constraints on them vary from around 0.02 to 4*10^(-20), with a median of 4*10^(-5).

Here are the remaining three parameters.

Parameter (gamma) is the ratio of (space distortion by gravity) / (time distortion by gravity). For slow-moving objects, one only sees a time-distortion effect, and this effect only indirectly as Newton's law of gravity. This time-distortion effect is more readily observable as gravitational redshift. But for anything traveling at c, space distortion will have an effect on its travels that is (gamma) times the time-distortion effect, giving

(value) = (1 + (gamma)) * (Newtonian value)

General relativity predicts (gamma) = 1, and that is why GR's deflection of light is twice as large as Soldner's Newtonian value.

For deflection of visible light, this parameter's departure from 1 is less than 0.01. For deflection of radio waves, it is less than 10^(-4), and for delay of radio waves, it is less than 10^(-5).

(beta) is the relative amount of nonlinearity in the time-distortion effect as a function of the Newtonian gravitational potential. It enters into the pericenter precession of an object in an orbit. In the Solar System, Mercury has the largest effect of any planet, and one that was discovered as an excess over what was predicted from Newtonian gravity. It was first attributed to an intra-Mercurian planet or planets or asteroid belt, but such intra-Mercurian objects were never convincingly observed.

From Mercury's orbit, the departure of (beta) from GR's value, 1, is less than about 10^(-4).

(eta) is the relative size of the "Nordtvedt effect", the amount of violation of the Strong Equivalence Principle. It is the difference between the amount of gravity for gravitational self-energy and the amount of gravity for everything else. GR predicts zero difference, but many GR alternatives predict nonzero difference.

This effect has been searched for by tracking the motion of the Moon, and that gives an upper limit on |(eta)| of about 10^(-3).

-

So it's remarkable what we have been able to do with Solar-System tests of general relativity. The main surviving alternative is something called the Generalized Brans-Dicke theory. It features an extra field, a scalar field, one without any built-in direction, and that field makes the gravitational constant vary. How much it does that is a free parameter in the theory, and one that can be made arbitrarily small. The parameter limits give that parameter an upper limit of about 10^(-5). Thus, GBD would have to be very close to GR.

 

[lpetrich]

Solar-System tests have their limits. Those tests only test theories of gravity at only the next level of series expansion beyond the Newtonian limit, and those tests do not involve strong-field behavior or gravitational waves. Those waves are ripples of space-time distortion, and they are predicted by GR and by most of its alternatives. However, these waves have been a difficult and controversial subject in GR, with even Einstein himself doubting their existence on a few occasions.

It all changed in 1974, when Joseph Taylor and Russell Hulse discovered a pulsar that they called PSR 1913+16 ("pulsar", a sort of celestial longitude, a sort of celestial latitude). They noted that its pulses were alternately fast and slow, and they could fit an orbit to these observations. They found that this pulsar and its companion were slowly spiraling into each other, something most likely due to radiating gravitational waves. They then assumed that GR was correct to post-Newtonian order and they estimated the masses of the pulsar and its companion, and also the size, shape, and orientation of their orbit. With these results, they calculated how much G-wave energy that they system should be radiating, and thus how fast their orbit should be shrinking. That result agreed with observations to within a fraction of 1%.

Some other pulsars in close binary systems have been observed, and some of them also show good agreement with GR predictions for G-wave emission. PSR J0737−3039 is a binary pulsar where both pulsars have been observed, and PSR J1738+0333 is the first pulsar discovered to have a white-dwarf star orbiting it, and PSR J0337+1715 is a pulsar orbited by two white dwarfs.

That system was mentioned in my OP's first link as an example of how much GR alternatives are suffering.

Gravity-lab discovered: a pulsar in a unique triple star system | ASTRON
 PSR J0337+1715
A millisecond pulsar in a stellar triple system | Nature, [1401.0535] A millisecond pulsar in a stellar triple system
[1602.05725] Testing the strong equivalence principle with the triple pulsar PSR J0337+1715

The pulsar has a mass of 1.44 solar masses and the inner white dwarf a mass of 0.20 solar masses. They orbit with a period of 1.6 (Earth) days with a separation of 0.032 AU or 16 light seconds or 12 times the Earth-Moon distance.

This system is orbited by the outer white dwarf. It has a mass of 0.41 solar masses. They orbit with a period of 327 days and a distance of 1.18 AU or 590 light seconds.

The authors of that last one did some numerical integration of the objects' equations of motion and found parameters that would produce a good fit. They distinguished between active gravitational mass, passive gravitational mass, and inertial mass. The Newtonian-limit equations get modified to

m(i,inertial)*(d^x(i)/dt^2) = G * ( sum over j != i m(i,passive)*m(j,active) * (x(j) - x(i))/|x(j) - x(i)|^3 ) + (post-Newtonian terms)

Gravitational mass: self is passive, others are active

From that triple-pulsar system, the active and passive gravitational masses of the pulsar and its inner companion depart from their inertial masses by only 10^(-7). This translates into (Nordtvedt parameter eta) being less than about 10^(-6) because a neutron star's gravitational self-energy is about 0.1 of its mass. Or more precisely its mass-energy, by E = m*c^2.


From inspiral of pulsars with white-dwarf companions, we get upper limits of how much gravitational-dipole radiation from self-energies, something predicated by some alternatives to GR. By comparison, GR predicts only gravitational-quadrupole radiation to lowest order, analogous to electromagnetism's lowest order being electric dipole.

Dipole:
+ -
Quadrupole:
+ -
- +

Gravitational and electric monopole modes: forbidden by conservation of mass and electric charge.
Gravitational electric-like and magnetic-like dipole modes in GR: forbidden by conservation of linear momentum and angular momentum.

 

[lpetrich]

Gravitational self-energy is also called gravitational binding energy, the energy needed to unbind an object. Here are some numbers relative to the objects' masses:

• A rock about 0.1 m across: ~ 10^(-27)
• The Moon: 2*10^(-11)
• The Earth: 4.6*10^(-10)
• Jupiter: 10^(-8)
• The Sun: 3.6*10^(-6)
• White dwarf: ~ 3*10^(-6) to 3*10^(-4)
• Neutron star: ~ 0.1

That is why neutron stars are a good test of the Strong Equivalence Principle. So far, the SEP has held up well, in agreement with GR and contrary to many proposed alternatives.


How fast do gravitational waves travel? According to GR and many of its alternatives, G-waves travel at c, the speed of light in a vacuum, at least in the weak-field limit. This speed is determined by the geometry of space-time, and in a lot of theoretical work, it is set equal to 1.

That seems difficult to test, but it has recently been possible to test that hypothesis. G-wave detectors LIGO and VIRGO have detected six G-wave events so far, all of them from massive compact objects that spiral into each other to make a single one. Five of them are attributed to massive black holes, some 7 to 35 times as massive as the Sun, and well above plausible upper limits for neutron-star masses. These events have the expected appearance from GR calculations: G-waves from the final bit of inspiraling, followed by ringing of the resulting single black hole as it settles done.

A sixth one, GW170817, was of two objects with masses 1.36 - 1.60 and 1.17 - 1.36 solar masses, most likely neutron stars. About 1.7 seconds after that event, two satellites observed gamma-ray bursts, and the aftermath of that event was eventually observed in radio waves, visible light, and X-rays. It was an expanding cloud of neutron-rich material, what one would expect from two merging neutron stars.

This event happened 40 megaparsecs / 130 million light years away. The time interval between the G-waves and the gamma rays was about 4*10^(-16) of the travel time, meaning that G-waves and EM waves travel at the same speed to less than one part per million billion. That causes trouble for alternatives to GR that predict different speeds.

 

[lpetrich]

The OP's Quanta article mentioned that merging pair of neutron stars, and it mentioned a further conundrum of gravity: dark matter and dark energy.

Dark matter was proposed to explain why stars in galaxies are moving much faster than what one would expect from those stars' masses. That was also discovered for galaxies in clusters in them. From how fast stars in galaxies move, one would expect galaxies to fall apart in a few hundred million years. But galaxies have been observed over nearly all of our Universe's existence and there is indirect evidence of their persistence: the heavy elements that their stars contain. These were made by long-gone massive stars that had spewed them out, seeding the interstellar medium with them.

One would expect a galaxy's velocity curve to rise from its center, then fall as one goes further out. But instead it flattens out. That's consistent with a gas of nearly-collisionless particles with the same average velocity everywhere in that galaxy.

Several candidates have been proposed for dark-matter particles, but none have been observed, though there is still plenty of way to go in observing them. That has led some physicists to propose modifying gravity.

The simplest sort is MOND, modified Newtonian dynamics. A simple version of that is to make gravitational acceleration decline more slowly when it is small:
a = sqrt(a0*agrav)

That can duplicate galaxies' light curves, but it is theoretically ugly. One can create a theory with a Lagrangian, one called AQUAL, that gives this acceleration law. A Lagrangian is a sort of summary of equations of motion, and many of the more fundamental sorts of theories have Lagrangians. But AQUAL is Newtonian, and to make it GR-friendly, one needs a theory like TeVeS -- Tensor-Vector-Scalar gravity. It has two extra fields, and these fields' equations of motion are much more complicated than GR's. It can duplicate MOND in the Newtonian limit, and it can do gravitational lensing, but it has trouble with G-wave propagation.

 

[lpetrich]

Newtonian gravity can be written in Lagrangian form:
\( L = \frac{1}{8\pi G} (\nabla V)^2 + V \rho \)

for gravitational potential V, mass density ρ, and gravitational constant G. The acceleration of gravity
\( a = - \nabla V \)

This Lagrangian is integrated over all four space-time dimensions, and one readily finds Poisson's equation from it:
\( \nabla^2 V = 4\pi G\rho \)

In the spherical case, that gives us
\( a = - \frac{GM}{r^2} {\hat r} \)

In  AQUAL, developed by Jacob Bekenstein and Mordehai Milgrom in 1984, one uses this expression:
\( L = \frac{1}{8\pi G} F((\nabla V)^2) + V \rho \)

Poisson's equation becomes in it
\( \nabla ( F'(\nabla V)^2) \nabla V ) = 4\pi G \rho \)

reducing to
\( a F'(a^2) = - \frac{GM}{r^2} {\hat r} \)

To duplicate the velocity curves of galaxies, it is necessary to have F'(x) ~ sqrt(x), giving F(x) ~ x^3/2. This is theoretically ugly, because it has a cusp at x = 0.

 

[lpetrich]

If you think that that is horrible math, then you should look at TeVeS. That's much worse.

Now for the remaining puzzle of gravity. Dark energy. This was posited to make the Universe's expansion speed up over its most recent 5 - 10 billion years. It involves a kind of matter with a bizarre equation of state:

(pressure) = - (mass density) * c²

The minus sign is a feature, not a bug. An important consequence of this equation is that the mass density and the pressure both stay constant as the Universe expands. For flat 3-space, the Universe expands exponentially: a ~ exp(H*t) for Hubble parameter H.

The cosmic-inflation hypothesis involves some similar exponential expansion early in the Universe's history, and it is likely from some different sort of dark energy.


The dark-energy hypothesis goes back to Albert Einstein, who introduced it as Λ, his cosmological constant. He introduced it as a way of keeping the Universe at constant size. But it turned out to be unstable, and Edwin Hubble soon showed that the Universe did not have a constant size -- it was expanding.

I've seen theories that dark energy varies over cosmological time, theories of "quintessence" ("fifth stuff"). This is relative to the traditional Western four elements, with the celestial element, aether, being a fifth one. Here is a table:
[table="class: grid"]
[tr]
[td]Earth[/td]
[td]Baryonic Matter -- high density, the greatest solidity[/td]
[/tr]
[tr]
[td]Water[/td]
[td]Dark Matter -- high density, flows through everything[/td]
[/tr]
[tr]
[td]Air[/td]
[td]Cosmic Neutrinos -- low density, flows through everything[/td]
[/tr]
[tr]
[td]Fire[/td]
[td]Cosmic Microwave Background -- low density, reactive[/td]
[/tr]
[tr]
[td]Aether[/td]
[td]Dark Energy[/td]
[/tr]
[/table]