1. Introduction

It is important to begin this paper remembering that solving the Einstein's equations for a static distribution of mass [^{1} – ^{4}] we can obtain the unperturbed metric tensor g
^{1} – ^{6}] theories predict the existence of gravitational waves (GW) and that they are expected to have **extremely small amplitudes** with h
^{5},^{6}], in which the fundamental equations of the Einstein gravitation theory are constructed, experimental tests [^{5} – ^{7}] are shown, and there are deductions of the basic equations predicting the emission of gravitational waves (GW) [^{8}]. These equations have also been used to estimate the intensities, wave amplitudes (or strain) h
^{9},^{10}] like, for instance, binary stars, neutron star pulsations, the precession of deformed neutron stars with oscillating quadrupoles, rotating bars and cataclysmic processes that give rise to supernovae. Many different detection techniques have been proposed to detect GW [^{3}, ^{11},^{12}]: laser interferometric techniques, resonant solids, fluctuations in the distance between Earth and Moon, crust oscillations of the Earth, normal modes of vibrations in solids in the form of rectangles, forks and rings, spinning rods and spinning tubes with fluids inside. In **Section 2** we present the basic principles of the *Laser Interferometry*
^{9}, ^{10} – ^{12}]. Detection of GW by another systems would be fortuitous. This occurred only after 20 years of intense researches with different techniques in USA, Japan, Italy, Australia, Holland,… Interferometric detectors were first suggested in the early 1960s and the 1970s [^{8} – ^{12}]. In **Section 3** we analyzed the two recent GW observations performed by the two interferometric detectors (**LIGO)** installed in the United States of America. These GW have been created in the merge of a binary black hole (BBH) system. As it will be seen in **Section 2** the entire merge evolution of the BBH can be divided into three stages: ”inspiral”, ”merger”(or ”plunge”) and ”ringdown”. To calculate exactly the GW emitted in the complete merge process it is necessary to solve the full equations of general relativity theory (GRT). This can be done only in numerical relativity simulations. Some simplified algebraic models can be used, for instance, in the ”inspiral”stage. In **Section 4**, using a simple relativistic approach, we explain approximately the amplitude (”strain”) of the GW signals detected by the LIGO in the ”inspiral”stage. Of course, these estimations are non rigorous, only instructive.

2. Laser interferometric technique

The laser interferometric detector **(LID)** is basically a laser large Michelson [^{11} – ^{13}] interferometer with perpendicular arms where there are three mirrors M, M1 and M2 as shown in Figure 1. M1 and M2 mirrors are attached to suspended blocks that can swing freely like pendulums. With the aid of a *highly potent*
*monochromatic laser* the interferometer measures the relative displacement of the mirrors which would be generated by GW.

Since 1990 several interferometric detectors were being built and designed: **GEO600** (cooperation between Germany and England), **LIGO** (USA), **VIRGO** (collaboration between Italy and France), **TAMA 300** (Japan), **AIGO** (Australia), **LISA** (spatial design, NASA, ESA) and **LISC** (LISA- international space project). As there is a huge amount of technical information involved in such major projects, we suggest to readers to search at Google using the words ”Interferometric Detectors”. The LIGO, VIRGO and GEO600 are very similar in concept. All projects have L-shaped facilities with multi-kilometer-long arms (4 km for LIGO, 3 km for VIRGO and 600 m for GEO600) with evacuated tubes that contain laser beams monitoring the positions of precision mirrors using interferometry. There is an intense collaboration between these international groups. According to Einstein's theory, the relative distance of the mirrors along the two arms changes very slightly when a GW passes by. The interferometers are set up in such a way that change in the lengths of the arms as small as

Let us see the principle of operation of these detectors. Thus (see Fig. 1) consider the GW incident on the z-axis direction (perpendicular to the plane of the figure) and the polarizing axes (+) along the x-axis (passing through the laser, M and M2) and y (passing by M and M1). Following our article [6c], let us see how this GW changes the distance between two particles located along the x axis (y = z = 0) at a point x = -dx/2, and the other at x = dx/2 . Supposing that the GW have a single polarization (+) with an amplitude or strain h the distance d

where k=

where

showing that the amplitude

Under analogous conditions for the two points along the y axis,
^{5}] along the x axis or y axis, as it is expected. The distance changes

According to (4) the maximum

Due to the extremely small intensities of GW [^{5},^{6}], the interferometer must be optically perfect and extremely well isolated from the rest of the world (seismic isolation, cosmic rays, electromagnetic fields, etc.). Lasers must be very powerful, extremely monochromatic and stable, the mirrors must have high reflectivity, the light path must be done over extreme high vacuum tubes, etc.

The measurement accuracy is mainly limited by fluctuations in the interference fringes, the number of detected photons as they mimic the effect of changes in the optical path. Suppose at a point P (in the mirror M) the intensity of the light from one arm is A and the intensity from the other arm is exp (i2
*effective paths* of the two light beams along the arms 1 and 2, respectively, and

The distribution of the number N of photons in the interference fringes is then given by [^{13}]

Due to the incident GW the distances originally undisturbed L

As the statistical error of N is N

where we see that d(

The coefficient N
^{7},^{9}],

where

Using (4) we have seen that the maximum amount of displacement due to the GW is achieved when the path length was equal to

According to (4) we conclude that the minimum strain h
*effective length* must obey the relationship h

Using (13) and P = 100 W,
**h**

As in (13) we supposed that f = 1 kHz and the optical path L =

Note that the interferometers can in principle detect GW with any frequency. Eq.(13) is a simple estimation of the minimum strain h
*sensitivity curve* h
**VIRGO** interferometer [^{10},^{11}] that was projected [^{14}, ^{15}] in 1990 and today is installed at Cascina (Pisa-Italy); each arm of the interferometer has 3 km.

As it will be shown in Section 3, the current Advanced LIGO and VIRGO Interferometers are able to detect signals with h
^{6} – ^{12}].

3. Observations of Gravitational Waves by LIGO

As we have mentioned above, interferometric detectors were first suggested in the early 1960s and the 1970s. Long-baseline broadband laser interferometers began to be proposed in 1989 - 2000. Combinations of these detectors made joint observations from 2002 through 2011, setting upper limits on a variety of gravitational-wave while **LIGO** and **VIRGO** became the first significantly **more sensitive network** of advanced detectors to begin observations [^{16},^{17}].

In Figure 2 it is shown a simplified diagram of the Advanced LIGO detector (see details in reference ^{12}), a modified Michelson interferometer: each arm is formed by two mirrors connected to test masses, separated by L
^{16}]). In this figure is also shown the two identical LIGO detectors that have been constructed in USA : one at Hanford (H1) and another at Livingstone (L1), distant by 10 *ms light
* with arms in different orientations.

A passing gravitational wave (Figure 3) effectively alters the arm lengths such that the measured difference is

Figure 4 shows for the **(Advanced) LIGO** the strain sensitivity h
**LIGO** detector is able to detect signals with h

The current strain sensitivity curves h
**VIRGO** and **LIGO** are very similar [^{18}].

3.1. Binary Black Hole (BBH)

For many years, proving the existence of BBHs was quite difficult [^{19}] because of nature of the BHs themselves, and the limited means of detection available. However, in one event that a pair of BHs were to merge, an immense amount of energy should be given off as GW, with distinct waveforms that can be calculated using the GR. Therefore, during the late 20th and early 21st centuries, BBHs became of great interest scientifically as a potential source of such waves, and a means by which gravitational waves could be proven to exist. BBH mergers would be one of the strongest known sources of gravitational waves in the Universe, and thus offer a good chance of directly detecting such waves [^{19}]. These sources have been finally observed in the events known as GW150914 [^{16}] and GW151226 [^{20}].

The entire merge evolution of the BBH can be divided into three stages: ”inspiral”, ”merger”(or ”plunge”) and ”ringdown”. The ”inspiral”is the first stage of the BBH life which resembles a gradually shrinking orbit and takes a longer time, as the emitted GWs are very weak when the BHs are distant from each other. This phase contributes only with a small fraction of the GW energy emitted in the entire process. As the BH orbit shrinks, the speed increases, and the GW emission increases. When the BHs are close the GW, they cause the orbit to shrink rapidly. In the final fraction of a second the BH can reach extremely high velocity. This is followed by a plunging orbit and the BH will ”merge”once they are close enough. At this time the GW amplitude reaches its peak (it is only clearly observed in the GW151226 event [^{20}]). Once merged, the single hole settles down to a stable form, via a stage called “ringdown”, where any distortion in the shape is dissipated as more gravitational waves. To calculate exactly the GW emitted in the complete merge process it is necessary to solve the full equations of general relativity. This can be done only in numerical relativity simulations [^{16}, ^{17}, ^{19}, ^{20}]. Some simplified algebraic models can be used, for instance, in the ”inspiral”stage. In **Section 4** using the general relativity theory we estimate approximately the amplitude (”strain”) of the GW signals detected by the LIGO in the ”inspiral”stage.

3.2. Event GW150914 observation

The existence of stellar-mass BBHs (and gravitational waves themselves) was finally confirmed on September 14, 2015 at 09:50:45 UTC when the two LIGO detectors H1 and L1 have detected simultaneously a wave transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain h = 1.0 X 10
^{12}]. The signals h(t) can be divided into three phases : ”inspiral”, ”merger”(or ”plunge”) and ”ringdown”, that will be clearly shown in Figure 7. The merging BBH with a pair of BH with masses
^{19}, ^{20}].

In **Figure 6** we compare the strain h(t) measured by H1 and L1 . We verify that they are very similar, in good agreement within the experimental errors.

Observing the evolution of the gravitational-wave signal h(t) one could deduce [^{16}, ^{8}] that the most plausible explanation for this is that a binary system formed by black holes (BH) with masses m

3.3. Event GW151226 observation

On December 26, 2015 at 03:38:53 UTC the twin LIGO detectors H1 and L1 have detected simultaneously [^{20}] a second wave transient gravitational-wave signal GW151226 shown in Figure 8. The signal persisted in the LIGO frequency band for approximately 1 s, from t = −1.0 up t = 0, increasing in frequency and amplitude over about 55 cycles from 35 to 450 Hz, and reached a peak gravitational strain
*third row* of the Fig. 8 [Signal-to-noise (SNR) time series]. For

Comparing Figs.5 – 6 and 8 we see that they are very similar. That is, observing the evolution of the gravitational-wave signal h(t) one could also deduce that the most plausible explanation is that a BBH with masses m

From Figs.4–7 and Figs. 8,9 we observe a good agreement within the experimental errors between theory and measurements. Comparing orbital with observed GW frequencies one can see [^{16}, ^{20}] that the emitting objects of the binary systems can only be very compact BH. These results confirm that GW have finally been detected and that they are generated by merging BBH system. In next **Section** using the GTR we estimate approximately the GW strain h(t) in the ”inspiral”stage of a BBH system.

4. Strain estimation for a binary-star system

We remark that our calculations are not rigorous; they have only a didactical objective. As we have said before, to describe exactly the detected signals of the GW in the complete merge BBH process of the BBH (”inspiral”, ”plunge”and ”ringdown”) it is necessary to use a completely relativistic model that can be treated only with numerical simulations [^{16}. ^{17}, ^{19}, ^{20}]. Our approach describes only approximately the ”inspiral”phase of the observed BBH.

Let us begin calculating the GW strain h(t) emitted by a *generic* binary-star system. We take R as the distance between the detector and the GW source that has a dimension D. According to preceding papers [^{4}, ^{8}], the emitted GW in the radiation zone (R » D) is represented by the second-rank tensor

where R is the radius and Q

For a binary system (Figure 10) composed by stars with masses m
^{9}] that

where

Under these conditions, and using (14) and (16), one sees that the GW would be given by

showing that the GW frequency is

4.1. Strain estimation

To obtain the *effective*
*strain* h of the GW that arrive at the detector it is necessary to take into account the contributions of all h
**L**
**L**
**A** = dA **n** = R
**n** = d
**n** is given by

where

where the brackets

As the energy of the GW in the radiation zone is transported by a **plane wave** with amplitude h and orbital frequency

Thus, with (19) and (20) let us define the following protocol to obtain h: **first** calculate L
**second** calculate h from L

For a binary-star we can show [6b] that

From (21) and (22) we have h

As for a binary
**orbital angular frequency
**

It is important to note that the relation

4.2. Strain h(

To estimate the strain h(
^{16}]: distance R = 410 Mpc = 1.3
**BH** mass M

According to Fig. 8 for initial orbital frequencies (t in the range
**Figs. 5**–**7**: up to t

4.3. Strain h(

In this event we have [^{20}] R
**BH** masses, m

During signal detection the GW frequencies f
*second row* of Fig. 9), that is, v/c

4.4. Strain h as a function of time

Let us give a rough description of the strain as a function of the time h = h(t) as it is seen in Figs. 7 and 8 in the ”inspiral”stage. According to (17) in the initial or ”inspiral”phase we have h(t)

remembering that
^{3}, ^{9}, ^{11}, ^{12}], the mechanical energy of the binary decreases with time and the distance r(t) decreases according to the law r(t) = r
^{3}],
*third row* of Fig. 8).