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Relativity, Cosmology, and Time


Gravitational Wave

In the weak field limit where the space-time metric tensors gik deviate only a small amount from flat space-time, the gravitational field equation (12a) is reduced to the form:


where the hki is small correction to gki, Tki is the energy-momentum tensor, and is the d'Alembertian operator in four-
EW and GW dimensional space-time. This equation looks similar to the electromagnetic wave equation except that it is now a second rank tensor field (with 10 components) instead of the more familiar vector field. It is responsible for many different characteristics in these two kinds of field. Figure 10v shows the differences in polarization and radiation pattern. There are two polarization states in gravitational wave. They alternatively squeeze and stretch the interacting particles shown as white circle in the diagrams (with direction of propagation perpendicular to the viewing page). Table 04 compares the properties of these two kinds of wave.

Figure 10v EW and GW
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See "Discovery of Gravitational Wave" in September 2015.

Property Electromagnetic Wave Gravitational Wave
Field Vector Second Rank Tensor
Wave Transversal Transversal
Polarization One State Two States
Radiation Pattern Dipole Quadrupole
Source Accelerating Charge Accelerating Mass-Energy
Interaction With Charges With Mass-Energy
Quantum Particle Spin 1 Photon Spin 2 Graviton
Rest Mass Massless Massless

Table 04 Electromagnetic and Gravitational Waves

GW Detector Gravitational wave have never been observed because of low radiation power and weak interaction strength. A rod about 1 meter long spun at the verge of breaking would radiate perhaps 10-30 erg/sec. The cross section for the interaction between gravitational wave of ~ 104 cycles/sec and an ammonia molecule is roughly 10-60 cm2. Figure 10w is the schematics of a gravitational wave bar detector. The impinging gravitational wave excites the fundamental longitudinal resonance (at ~ 1000 Hz) of the bar, kept at low temperatures. The induced vibration of the bar end face is amplified mechanically by the resonant transducer, which also converts the signal into

Figure 10w GW Detector
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an electromagnetic one. The signal is then amplified and acquired (see Figure 10w). It is suggested that large-scale astronomical motions of matter could generate appreciable gravitational energy flux.
GW from Binary Pulsars The binary pulsar PSR1913+16 was discovered in 1975. This system consists of two compact neutron stars orbiting each other with a maximum separation of only one solar radius. The rapid motion means that the orbital period of this system should decrease on a much shorter time scale because of the emission of a strong gravitational wave. The change predicted by general relativity is in excellent agreement with observations as shown in Figure 10x. Thus, the observation indirectly confirms the

Figure 10x GW from Binary Pulsars
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phenomena of gravitational radiation.

Black Hole Merger Figure 10y illustrates the merger of two orbiting black holes. Initially, the gravitational signals from such an event would show oscillation with increasing amplitude and decreasing wavelength as the black holes spiral toward each other. A chaotic pattern of gravitational waves may be given off at the moment of merger. Finally, the resultant single black hole is expected to "ring", creating waves with diminished amplitude. This event will emit no x-ray burst, not even a flash of light.

Figure 10y Black Hole Merger
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Gravitational waves may be viewed as coherent states of many gravitons, much like the electromagnetic waves are coherent states of photons. Since gravitational wave has evaded detection for over 50 years, it seems even harder to find the individual gravitons. However, it is suggested that in high-energy colliders such as the LHC, it is possible to produce gravitons, which can then disappear into the extra dimensions. This would lead to a "missing energy" signature, with unbalanced events. Such signatures are routinely used in particle experiments to detect the production of neutrinos (difficult to detect). The exchange of gravitons in the extra dimensions would also affect the dynamics of other scattering processes.

A leading cosmological model, known as inflation, predicts that our universe is just one part of a greater multiverse and that our Big Bang may have been one of many. In this model, our universe expanded extremely rapidly during the period of 10-35-10-32 second after the Big Bang.
GW and BB Another model, rooted in string theory, envisions a scenario in which the Big Bang occurred as a result of the collision between two parallel universes floating in higher dimensional space. Each of these models predicts a specific pattern of gravitational waves emitted from the Big Bang. NASA and ESA plan to launch the Laser Interferometer Space Antenna (LISA) to detect gravitational wave by 2015. It consists of three satellites orbiting the sun (Figure 10za). They will be linked by three laser beams, forming a triangle of light. They are designed to detect a change in their spacing as small as 1/10 the diameter of an atom. With such sensitivity LISA might be able to detect gravitational waves created immediately after the birth of the cosmos. It offers a chance to select between the contesting cosmological models, and also provides an opportunity to test the string theory.

Figure 10za LISA
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Meanwhile there are two gravitational wave observatories running in collaboration to measure stochastic (noisy) background signal (SGWB) from the earliest epochs in the evolution of the Universe. LIGO has built three multi-kilometer interferometers, two at Hanford, Washington, and
LIGO SGWB one at Livingston, Louisiana (Figure 10zb). Virgo is a 3-km interferometer at Italy. They have not yet detected the elusive gravitational wave, but managed to place an upper bound on the SGWB in the frequency band df around the frequency f ~ 100 Hz. The SGWB is defined by the formula:
GW = (f / c) (dGW / df)

Figure 10zb LIGO [view large image]

Figure 10zc SGWB
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where GW is the energy density of gravitational radiation contained in the frequency range df, and c is the critical energy density of the Universe.
Figure 10zc shows the different SGWB measurements designated as LIGO S4, LIGO S5, and the projected Advanced LIGO (AdvLIGO). Predictions by various models are also shown in different colored curves. The previous upper bound via BBN (Big Bang Nucleosynthesis) and CMB (Cosmic Microwave Background) at 100 Hz is about 10-5. LIGO and Virgo obtained a new upper bound of GW < 6.9 x 10-6. The new data rule out models of early Universe evolution with relatively large equation of state parameter, as well as cosmic (super)string models with relatively small string tension. Improved measurements will constrain other cosmological models such as the pre-Big-Bang model, which makes testable predictions of the gravitational wave spectrum as shown by the green curve in Figure 10zc (see a novel explanation for the noise from another observation).

Pulsar Beacons Gwave Sources and Detectors A novel method to detect gravitational wave is to measure the number of pulses per unit time from a pulsar. Since the gravitational wave stretches and compressed space, the arrival time of the pulses will be later and sooner correspondingly (Figure 10zd). In practice, though, the present technology is not sensitive enough to detect the minuscule changes. A cunning workaround is to map out millisecond pulsars in the sky and time their pulses for long enough to find out the average time it

Figure 10zd Pulsar Beacons [view large image]

Figure 10ze Gwave Sources and Detectors

takes them to reach Earth, any deviation in that time would indicate interference from gravitational waves. A positive map will show a pattern of variation in many pulses, all fitting the
expected stretch-squeeze template. It is estimated that a definitive detection can be achieved before 2015. A negative result will be more interesting as it will indicate that the present framework for general relativity has to be overhauled. Figure 10ze shows some sources (in green) of gravitational wave and the various detectors (in other colors) that are trying to detect them.

Gwave Detection Atom Interferometer On April 8, 2011, NASA announced that it would likely be unable to continue its LISA partnership with the European Space Agency, due to funding limitations. ESA began a full revision of the mission's concept and renamed it as the New (or Next) Gravitational-wave Observatory (NGO). A cheaper detector called "Atom Interferometer" has been proposed as the replacement. It uses atomic clouds (with the associated matter waves) instead of laser as the interfering medium, whereas the laser interferometer requires two or three long base lines (in the order of few kilometers) for gravitational wave detection via the change of those arms, the atom interferometer detects the changing space in between the clouds, which is typically at a separation about 5000 times shorter than the

Figure 10zf Gravity Wave Detection [view large image]

Figure 10zg Atom Interferometer
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LISA design, and thus reduces the cost of construction. Actually, the researchers estimate such a mission would cost between $100
million and 1 billion - not a bargain, but still significantly cheaper. An artist's rendition of the gravitational wave detectors in space is depicted in Figure 10zf. Two satellites are required to make sure the gravitational signal is real - by producing two different patterns on top of the laser noise (which is the byproduct from manipulating the atomic clouds as shown in Figure 10zg and explained briefly below).

    Here's the lowdown on the atom interferometer as shown in Figure 10zg :

  1. A cloud of ultra-cold atoms (e.g., the rubidium) at a temperature less than 50 nanokelvin and a size just 200 micrometers in diameter is launched by a laser pulse (not shown) from the bottom of the 10-meter-tall vacuum enclosure (a).
  2. A laser pulse (1) splits the cloud into two half with different velocity. The spread of velocity within each cloud is equivalent to the spread in wave length in optical interferometer according to the de Broglie matter-wave relation : = h/mv, where is the wavelength, v the velocity of the particle with mass m, and h the Planck constant.
  3. A second laser pulse (2) is applied to the clouds at the top of the enclosure (a) acting like a mirror in optical interferometer.
  4. A third pulse (3) recombines the clouds when they reach the bottom (b), producing interference pattern as shown in (c) and is recorded on the CCD cameras (a digital imaging device).
  5. Sensitivity of the atom interferometer for gravitational wave detection is estimated to be about 6.7x10-12g, where g = 980 cm/s2 is the acceleration of gravity at or near the Earth's surface. The device will be able to detect gravitational wave down to such magnitude if such wave happens to pass through between the two atomic clouds.
See a technical paper on "Atom Interferometry for Detection of Gravitational Waves".

On September 14 2015, the newly upgraded Advanced LIGO detected a transients (short duration event ~ 10-3 - 10 sec.) that stands above the background (detector noise) as shown in Figure 10zh (in red square, c ~ 8). Significance of the GW150914 event is about 5 (same as the
GW150914 Dectection GW150914 BHs Merger discovery of the Higgs). Further analyse of the data reveals that the gravitational wave pattern closely matches the one from the predicted merging of two black holes (Figure 10zi, notice the increase in frequency as they merged). The same figure also show the separation (in unit of Schwarzschild radius Rs=2GM/c2, and relative velocity v/c=(GMf)/c3, where f is the gravitational-wave frequency, M the total mass). The source is about 1.3 billion light years from Earth with initial masses 36Msun, 29Msun respectively, and final mass 62Msun - less than 3Msun of mass had been converted to gravitational wave.

Figure 10zh GW150914 Detection
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Figure 10zi GW150914 BHs Merger [view large image]


The insert in Figure 10zh shows the estimated position of GW150914. The colored contours denote the degree of confidence from 10% to 90% at the outer-most.

The discovery opens up a new widow to view the Universe. See detail in the LIGO home page.

LIGO has discovered another black holes merger on December 26, 2015. It seems to be a common occurrence once the gravitational wave detector becomes sensitive enough for the task.

However, the detections only discover the "classical" gravitational wave predicted by General Relativity. The existence of graviton in quantum theory has not been confirmed by such observations. Theoretically, unlike the photon in QED originated from Maxwell's Equations, there is no successful quantization scheme for gravitation and hence the existence for graviton is not clear even on paper. Experimentally, unlike the photon in photoelectric-effect, ... etc. providing evidences to show the particle nature of electromagnetic wave, the signal from the "quantum" gravitational wave (= graviton) is very difficult to detect because it would be very weak. BWT, the so-called quantum wave is associated with the wave-like component of the Fourier transform in the quantization procedure.

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