Einstein postulated two principles of the general theory of relativity:

In the special theory of relativity, the inertial frames were used to describe the laws of physics. However, according to E. Mach’s principle [27], inertial properties depend on the presence of other bodies in the universe. For example, to determine an inertial frame in the laboratory frame on Earth, we have to take into account that Earth is a body that rotates around itself and the Sun.

According to the principle of equivalence, the gravitational mass for each object can be considered to be chosen equal to its inertial mass. Based on the Newtonian mechanics, mass has two distinguished properties. First is the gravitational property as an attraction for other masses through the gravitation’s Newton law: where $k_G$ is the Newton’s gravitation constant ($k_G=6.673\times {10}^{-11}\mathrm{N}·{\mathrm{m}}^2/{\mathrm{kg}}^2$), $m_g$ is the mass of any object above the surface of the Earth (with mass M) at a distance r (close to the surface) from the centre of the Earth. (Note that $r=R+h$, where R is the distance from the surface point to the centre of Earth and h is the height of the mass from the surface of Earth. Here, it is assumed that $h\ll R$ such that $r\approx R$). In Eq. (1), g is the gravitation acceleration: where R is the distance from the centre of Earth to its surface. Eq. (1) indicates that bigger the mass $m_g$ stronger the attractive force $F_g$; however, Eq. (2) indicates that the acceleration of that mass $m_g$ does not influence its acceleration g.

The second property is the inertial property of the mass, expressed through the second law of Newton in the form: where $m_i$ is the inertial mass of the same object moving under the resultant force ${\sum }_k{F}_k$ with acceleration a. As that the property, the inertial mass $m_i$ indicates the resistance of the object to acceleration. In the Newtonian mechanics, $k_G$ is chosen such that That is, $m_i$ and $m_g$ are considered numerically equal. However, experimentally, it is found that $m_i$ and $m_g$ are proportional to each other, and experimentally that proportionality constant is found within an accuracy of ${10}^{-12}$. Theoretically, the answer to that concerns is given by Einstein in 1916 with his general theory of relativity, who postulated that $m_i$ and $m_g$ are proportional to each other as a basic connection between the two unique concepts. Suppose a person in an elevator at rest. Then, that person is in a uniform gravitational field $\mathbf{g}$, and if that person drops an object, then the acceleration of the object relative to the floor of the elevator is g.

Now, suppose that the elevator is accelerated upwards by an external field with an acceleration $\mathbf{a}=-\mathbf{g}$. The person (and hence the object in his hands) is in a region where gravity does not exist. If that person drops the same object, then again that object undergoes the same acceleration relative to the floor of the elevator.

That “thought experiment” suggests that we can not distinguish between the two cases, and is an evidence of the two postulated principles by Einstein. That is, the physics is the same for observers in an inertial frame and an accelerated frame. Furthermore, it indicates that the gravitational mass and the inertial mass are equivalent.

Besides, consider an experiment in an elevator with laser light by an observer. If the observer sends a light pulse towards the wall of the elevator, when the elevator is accelerated upwards by an external force, then as the elevator moves up, the pulse strikes a spot which is not the same as it would have been when the elevator is at rest. That is, the trajectory of the light pulse bends downwards as the elevator accelerates upward. Based on the first postulate of the Einstein, we can not distinguish this phenomenon with the light pulse in a reference frame of the elevator accelerating with another non-accelerating reference frame in the gravitational field; therefore, the light beam bends downward by a gravitational field. Note that the experiments have already shown that a small bending exists.

Moreover, the time scales are changed in the presence of gravity. In particular, a clock runs slower under the influence of gravity than a clock located in the places where the gravity is negligible. For example, it has been detected that the frequencies of the radiation by the atoms in the presence of the strong gravity field are red-shifted to lower frequencies compared with the same frequencies in the absence or weak gravitational field. That is observed in the spectra of the atoms in massive stars and Earth by measuring the frequencies of the gamma rays emitted from nuclei that were separated vertically by a certain distance, so the gravitation field strength was varying.

It is a consequence of the general theory of the relativity that if we consider two identical clocks, one in the first floor of a high building and the other on the upper floors, for instance on the 10th floor, the one that is on the first floor run slower than the clock on the 10th floor because the gravity is more influential on the first floor compared to the 10th floor.

Another evidence of the general theory of relativity is the application of the so-called Global Positioning System (GPS) unit, which employs the relativistic correction of the time calculations when it analyses the signals received from the orbiting satellites. Note the clocks on the surface of the Earth run slower than the identical clocks in the Earth-orbiting satellites because they experience a stronger gravitation field of the Earth’s surface [27].

A consequence of the second postulate of Einstein is that the gravitation field may be made negligible at any point by an appropriate choice of the reference frame, such as a freely falling object as a reference frame. Also, Einstein introduced a concept that is known as curvature of space-time in which the gravitational strength can be specified in every point [1]. In fact, according to Einstein, the Newtonian concept of gravitation force does not exist, but rather every mass causes a curvature of space-time in its neighbourhood, which is followed by all freely moving objects in that space-time. Therefore, the two masses, which are at a distance r from each other, cause the curving of the space-time in between, and hence the two masses move towards each other.

As an evidence of the curved space-time of the general theory of relativity would be the deflection of the light ray passing near the Sun due to the curvature of the space-time caused by the Sun’s mass. Indeed, this is confirmed by the detection of the starlight bending near the Sun during one of the total solar eclipse.

Another consequence of the curvature of the space-time, in extreme cases, is the creation of the so-called black holes. That occurs when the mass concentration becomes very large, for example, when a massive star collapses to a small volume. Within a certain distance from the centre of the black hole, the curvature of the space-time is so extreme that it will not allow any matter, information, or even the light to escape. Note that recently there are reports that the creation of the black holes have been observed [28].

Also, the general theory of relativity predicts the existence of the so-called gravitational waves from the moving masses. More importantly, very recently, these waves are directly experimentally observed.

The general theory of relativity is mathematically described under the framework of non-Euclidean geometry. A common term used is the geodesic, giving the paths of the shortest distance. In general theory, the objects move along the geodesics of space-time. Assume that a set of geodesics starts parallel to each other. If the gravitational effects in the region extended by the geodesics are uniform, then the geodesics will remain parallel. However, if these gravitational effects are nonuniform within that region, the geodesics will deviate and start either approaching or receding each other. According to the general theory, the geodesic deviation is used as a measure of the gravitational field.

Note the analogy with the first law of Newton, which states that if there are no external forces, then the object continues moving along a straight line without acceleration.

As an example, we consider two objects separated by a distance of $\eta$ along the horizontal direction. Assume that the objects drop simultaneously from the same height above the surface of the Earth. Based on the principle of equivalence, locally, very close to each object and neglecting the gravitational mass of the objects, we can treat the region as an inertial frame. That is, in a local region of either object, the gravity can be made to vanish by the choice of the inertial reference frame. That system corresponds to a local to either object free fall frame, which will be used as an observer inertial system. As the objects fall toward the surface of Earth, their separation $\eta$ decreases. The change in their separation is a local measure of the gravitational effect of Earth since the choice of the frame can not neglect it. Fall itself of each object is not considered a gravitational effect because the selection of the frame can eliminate it, such as a frame that is accelerated by external forces with acceleration $\mathbf{g}$ upward.

In Figure 1, we show two geodesics, where two 4-vector fields are defined at each point, namely the 4-velocity field $\mathbf{u}$ of motion along the geodesic, and the separation field vector $\mathit{\eta }$ between two geodesics along the horizontal direction.

Let $\tau$ be a time when in each geodesic, there were observed test particles at the head and tail of the vector $\mathit{\eta }$. The proper time will be used at the tail of the deviation vector, and the head is at the point where the other test particle is at the same time. Note that as the motion of test particles progress along the geodesics, the proper times of the two test particles are not the same. In the Newtonian approximation, the equation of motion for the separation between the test particles is given by [22] where $ds$ is the length along the geodesic.

In Eq. (5), R is a constant that depends on the distance to the centre of Earth. Eq. (5) indicates that the acceleration of the distance between the two geodesics is proportional to their separation, $\eta$. Solution of Eq. (5) is given as (assuming an initial phase equal to zero): For the case of a two-dimensional example of the geodesics on the surface of a sphere, we assume that initially the two geodesics are parallel. Then, they meet after they have traveled $1/4$th of the circumference of the sphere, that is: where a is the radius of the sphere. Therefore, Solving it for R, we obtain Thus, R represents the gaussian curvature of the surface of the sphere, a two-dimensional space, which is product of two gaussian curvatures, namely $1/a$ and $1/a$ along the two separate one-dimensional directions on the sphere’s surface.

The invariant expression of the square of a linear element is written as where $d{x}_{\nu }=g_{\mu \nu }d{x}^{\mu }$. In Eq. (10), $g_{\mu \nu }$ is a covariant second rank tensor, which is symmetric: $g_{\mu \nu }=g_{\nu \mu }$. That tensor is also called by Einstein as fundamental tensor because of its role in physics basis of understanding the gravitation. Based on Einstein, the contravariant second rank tensor, $g^{\mu \nu }$, is defined such that where and in Eq. (11), based on Einstein’s rule, the repeating indices up-down indicate summation.

Therefore, based on Einstein, the contravariant tensor is defined by calculating the determinant of covariant tensor $g=\mid g_{\mu \nu }\mid$, then for each element of $g_{\mu \nu }$ we can consider the co-factor and divide it by g to determine $g^{\mu \nu }$. That is, in terms of matrix notations, $g^{\mu \nu }$ is the inverse of $g_{\mu \nu }$. Therefore, or in matrix form $\mathbf{g}{\mathbf{g}}^{-1}=\mathbf{1}$, where $\mathbf{1}$ is a diagonal matrix with all diagonal elements equal to 1. Eq. (10) can also be written as we can write Combining Eq. (14) and Eq. (15), we write that Eq. (16) indicates that $g^{\mu \nu }$ is the contravariant form of the same second rank tensor $\mathbf{g}$.

The tensor $g_{\mu \nu }$, which is a covariant second rank tensor obeys the Lorentz transformation (a physical quantity can be defined in terms of a contravariant vector in four-dimensional space $A^{\mu }$ or $A^{\mu \prime }$, respectively, in the inertial frame S and $S^{\prime }$, if between them there exists a relation as follows: That is, $L_{\nu }^{\mu }$ is given as ); that is: Taking the determinant of both sides in Eq. (19), and using the properties of the multiplication of determinants, we can write: where $\mathbf{L}$ is the matrix given by: which the so-called Lorentz transformation tensor. It can be seen that $\mathbf{L}$ is a symmetric matrix.

Denoting by $g^{\prime }=\mid g_{\mu \nu }^{\prime }\mid$ and $g=\mid g_{\mu \nu }\mid$, we can write

We can now apply the rule of transformation of the volume in the four-dimensional space for an element of volume Using Jacobi’s theorem, in the new coordinates Combining Eq. (22) and Eq. (24), we obtain that We can denote by $\mid g\mid$ the absolute value of g to have a positive number, then Eq. (25) reduces to Eq. (26) implies the invariance of $\sqrt{\mid g\mid }v$, $\sqrt{\mid g\mid }$ characterises the hyperbolic nature of the space-time continuum, as stated by Einstein. Moreover, $\sqrt{\mid g\mid }v$ gives the magnitude of the four-dimensional element of the volume in a local reference frame in the same sense as in the spacial theory of relativity of a rod-clock measurement. Therefore, based on the time form and the space-like volume form, the scaled space-time volume form is where g is the determinant of the covariant form $g_{\mu \nu }$. Furthermore, $g_{\mu \nu }$ is dependent on the point in the four-dimensional space-time space, $x^{\mu }$, according to the general theory of relativity. That is in contrast to special theory of relativity, where there exists a finite region in four-dimensional space such that $g_{\mu \nu }$ are assumed to be constant.

In the general theory of relativity, the elements $g_{\mu \nu }$ physically describe the gravitational field concerning the chosen reference frame, and they are no longer constant but functions of the four-dimensional space point $x^{\mu }$; that is, they depend on both space and time. Furthermore, if we describe the motion of a free particle, then its movement in the new coordinates will be a curved nonuniform motion. The law of this motion will be independent of the nature of the moving particle, and, based on Einstein, it can be interpreted as a motion under the influence of the gravitation field, which occurs because of the dependence of $g_{\mu \nu }$ on $x^{\mu }$.

In contrast, based on the special theory of relativity that may apply to a particular four-dimensional region with some properly chosen coordinate reference frame, then $g_{\mu \nu }$ has components constant, the free particle moves relative to this frame with uniform motion in a straight line.

Therefore, if we are no longer able to chose a coordinate system, where the special theory of relativity applies to a finite region of the space-time, then we should consider that $g_{\mu \nu }$ elements describe the gravitation field. They are a function of the four-dimensional space point $x^{\mu }$.

We now describe the motion of a point-like particle in the presence of the gravitation field. In particular, we will derive the equation of the geodesic line. For that, we will use the formalism for describing the motion of a free particle from the special theory of relativity viewpoint.

We call again the principle of the least action. The action integral I is defined as a time integral of Lagrangian function L along a possible path of the particle. The path that we would like to know the equation. The action integral is written as: where the integral is evaluated in the proper time interval between ${\tau }_1$ and ${\tau }_2$ measured in the local coordinate system of particle. In Eq. (28), s represents a parameter in the sense that it is monotonically increasing function of $\tau$, that is, $ds=cd\tau$. Furthermore, $ds$ represents the magnitude of the element length along the geodesic line given by Eq. (10), which is Lorentz invariant.

First, we write Eq. (28) in the following form: where is some 4-vector velocity related to the particle, which is a function of $x^{\mu }$. We can define an effective Lagrangian as: which is a function of particle position in four-dimensional space $x^{\mu }$ and its 4-vector velocity ${\overline{u}}^{\mu }$. Eq. (29) reduces to We can now use the principle of least action $\delta I=0$ to determine the equation of geodesic line, which corresponds to the shortest path. However, in contrast to special theory of relativity case, $g_{\mu \nu }$ is also a function of $x^{\mu }$. We can obtain the solution in the following covariance form, known as Lagrangian equations: where $L(x^{\mu },{\overline{u}}^{\mu })$ is given by Eq. (31). We can evaluate the second term, and obtained: and for the first term, we got We can write Eq. (36) in a more convenient form as follows: Now, we can assume that the element length $ds$ is along the geodesic line, that is, $ds\ne 0$; furthermore, we take it such that $\sqrt{g_{\mu \alpha }{\overline{u}}^{\mu }{\overline{u}}^{\alpha }}=1/c$. Then, Eq. (35) and Eq. (37), reduce to and We can take the derivative of the expression in Eq. (39) with respect to s, and we obtain Combining Eq. (38) and Eq. (40) in Eq. (34), we obtain for $\alpha =0,1,2,3$. For convenience the rescaled space-time metric can be defined [29]: In addition, we can replace $ds=cd\tau$, and obtain Eq. (41) as follows for $\alpha =0,1,2,3$: or, where the following Christoffel’s tensor is introduced Combining Eq. (44) and Eq. (45), we write We can multiply both sides of Eq. (46) by the contravariant metric tensor $g^{\alpha \sigma }$ and use $g_{\mu \alpha }{g}^{\alpha \sigma }={\delta }_{\mu }^{\sigma }$, to get where Eq. (47) and Eq. (48) represent the geodesic line equation of a particle in gravitational field.

Moreover, the following third rank tensor is introduced: and Eq. (47) is written as follows:

It is important to emphasise that Eq. (50) is general; that is, it is valid to describe the geodesic line equation independently of the coordinate reference frame. If we consider two reference frames, the coordinate reference system $S_0$ in which the particle is free and not subject to any external forces and the coordinate reference system $S_1$ in which there exists a gravitation field described by the dependence of metric tensor $g_{\mu \nu }$ on the positions of the four-dimensional point $x^{\mu }$. Then, in $S_0$ particle moves, according to the special theory of relativity, in a straight geodesic line uniformly. Eq. (50) applies for the motion of the free particle by taking $g_{\mu \nu }$ constant, and hence ${\Gamma }_{\mu \nu }^{\sigma }=0$. On the other hand, if we consider the motion of a particle in the coordinate system $S_1$ in which $g_{\mu \nu }$ are not constant, then the particle moves in a gravitation field, and hence ${\Gamma }_{\mu \nu }^{\sigma }\ne 0$. Therefore, Eq. (50) describes the motion in any arbitrary coordinate reference frame, either $S_0$ or $S_1$. The term on the right-hand side can also be interpreted as the 4-gradient of a gravitational potential. That is, it is a contravariant field vector.

Consider a vector $\mathbf{A}$ undergoing a parallel translation between infinitesimal nearby points $x^{\mu }$ and $x^{\mu }+d{x}^{\mu }$. The changes on the covariant components of $\mathbf{A}$ are $\delta {\mathbf{A}}_{\mu }$ that are linearly dependent on the values of the covariant components $A_{\mu }$. To describe that, we can consider that the scalar product is invariant under the parallel translation, that is where the summation is assumed over the repeating indices. Thus, the dot product between two vectors represents a physical quantity that has a magnitude but no direction, and hence, it does not change from point to point. Therefore, or We can assume a linear dependence between $\delta B^{\mu }$ and $B^{\mu }$ via Christoffel’s symbols: Furthermore, using Eq. (53), we write If we change the summation index on the left of Eq. (55), we obtain After rearranging, it gives Since $B^{\alpha }$ is an arbitrary chosen contravariant vector, then Eq. (57) implies that which indicates a linear dependence between the covariant components $A_{\mu }$ and the change $\delta A_{\alpha }$ in the covariant components of vector $\mathbf{A}$ undergoing a parallel translation between infinitesimally close points $x^{\mu }$ and $x^{\mu }+d{x}^{\mu }$.

The difference between the evaluated vector $A_{\alpha }\left(x^{\alpha }\right)$ and $A_{\alpha }(x^{\alpha }+d{x}^{\alpha })$, and the change of vector $\mathbf{A}$ upon the parallel translation from points $x^{\alpha }$ and $x^{\alpha }+d{x}^{\alpha }$ is defined as the absolute covariant differential $D{A}_{\alpha }$ in terms of the covariant components $A_{\alpha }$: Combining Eq. (58) and Eq. (59) yields or, Note that the repeated indices represent summation.

It is important to note that if the surface is flat, then in a Cartesian coordinate system, also called Galilean, the volume is rectangular and ${\Gamma }_{\alpha \beta }^{\mu }=0$. Therefore, for a flat surface indicating that both contravariant and covariant absolute differential components reduce to the respective ordinary differential components, and they are identical.

In contrast, Eq. (61) represents the covariant absolute differential of the component $A_{\alpha }$ in a curvilinear coordinate system for curved surfaces.

Similarly, we can calculate the absolute differential of a contravariant vector $A^{\alpha }$: Using the linearity $\delta A^{\alpha }=-{\Gamma }_{\mu \nu }^{\alpha }{A}^{\mu }d{x}^{\nu }$, and substituting it in Eq. (63), it gives

Furthermore, the covariant derivative can be introduced as the following:

Consider now the case when the vectors are expressed in terms of the contravariant components. The contravariant tensor is expressed as a product of their contravariant components: For a parallel translation, the differential of $C^{\mu \nu }$ is given as Using the linearity expressions for both $\delta A^{\mu }$ and $\delta B^{\nu }$ in terms of Christoffel’s symbols, we obtain Eq. (68) suggests that The absolute differential $D{C}^{\mu \nu }$ of the contravariant tensor is Using Eq. (69) and Eq. (70), it can be found that Thus, the covariant derivative of the contravariant tensor is as the following:

The tidal forces have also been described in the framework of Newtonian mechanics. These represent forces that exist because of the gravitational waves. It is interesting to describe these forces in the context of the general theory of relativity. It has been shown that Riemann tensor is used to express the second derivatives of the gravitational potential in the equation of motion for the separation $\eta$ of two nearby particles moving in the gravitation field, where each can be described by Eq. (50).

For that, we consider two nearby geodesic lines, each described by Eq. (50), that represent the motion of two freely falling objects. Let $x_1^{\sigma }\left(\tau \right)$ represents the four-dimensional position of the first object, and $x_2^{\sigma }\left(\tau \right)$ that of the second falling object. The trajectories of two falling objects are described by the following equations: for $\sigma =0,1,2,3$. The same proper time $\tau$ is used to describe the motion along each geodesic. It is important to note that the geodesic lines are very close to each other and are parallel to each other at $\tau =0$. We would like to describe the time evolution of the difference: ${\eta }^{\sigma }\left(\tau \right)=x_2^{\sigma }\left(\tau \right)-x_1^{\sigma }\left(\tau \right)$.

First, a Taylor expansion of Christoffel’s symbol about the position of the first object, $x_1^{\sigma }$, is obtained:

Then, the equation of the second object (see Eq. (73)) becomes (for $\sigma =0,1,2,3$) or, we obtain the approximate equation by ignoring the terms second order on $\eta$: Using the geodesic line equation for the first object from Eq. (73), we obtain:

Employing Eq. (64), we obtain absolute derivative as Taking the second absolute derivative, we get where $u^{\mu }=d{x}_1^{\mu }/d\tau$ is the 4-velocity of the first object. We can use the following identity: where summation over repeated indices is considered. Then, Eq. (80) becomes Combining Eq. (79) and Eq. (82), we get

Using Eq. (73), we can write Substituting Eq. (84) into Eq. (83), we obtain We can relabel some of the indices in Eq. (85) and obtain where is the famous Riemann tensor. Eq. (86) is written in more convenient form: Eq. (88) is first order in small 4-vector difference ${\eta }^{\sigma }\left(\tau \right)$; that is, it ignores higher order terms $O\left({\mathit{\eta }}^2\right)$. Eq. (88) describes the relative acceleration of the two objects initially $u^{\mu }=d{x}^{\mu }/d\tau$ and $u^{\nu }=d{x}^{\nu }/d\tau$ being their initial motions that describe the flow along the geodesic. On the other hand, ${\eta }^{\mu }=d{x}^{\mu }/ds$ (where $ds$ is some parameter) describes the motion from one geodesic to the other. Another interpretation of Riemann tensor, as description of the curvature of the surface, is given in Figure 2. The two vectors $u^{\mu }$ and $u^{\nu }$ describe the flow along each geodesic form a parallelogram. Then, a vector $A^{\sigma }$ can be translated parallel along that parallelogram. The deviation of the vector $A^{\sigma }$ would be proportional to Riemann tensor: where the minus sign indicates that the vector can either increase or decrease its length. Note that Eq. (88) can be generalised to a family of the geodesics parameterised as $x^{\mu }(s,\tau )$, where s is a parameter describing the family of geodesics and $\tau$ is the proper time describing the geodesic flow. Let ${\eta }^{\mu }\left(\tau \right)=d{x}^{\mu }/ds$ be the 4-vector characterising the motion from one geodesic to another nearby geodesic within the same family, and $u^{\mu }\left(s\right)=d{x}^{\mu }/d\tau$ representing the flow along a geodesic. Then, we would find that if the 4-vector ${\eta }^{\mu }$ is transported parallel along the geodesic flow, its second covariant derivative is related to Riemann tensor as in Eq. (88).

Time solution of Eq. (88) is determined by the curvature of space-time, which is represented by Riemann tensor. It has been applied to strong gravity to test the limits of the general theory of relativity. Contraction of the Riemann tensor produces the so-called Ricci tensor with components defined as By definition, the curvature scalar R is defined as Note that those equations become Newtonian gravitational field theory equations for weak gravitational fields and when speed is much lower than that of the light. Besides, they converge to special theory results, if there is no gravitational field. It is important to mention that these equations predict the existence of the gravitational waves from moving masses, as predicted by Einstein [30], which are very recently detected experimentally [31].

First, we will discuss the gravitation field equations in the absence of matter. According to Einstein, with matter should be understood everything that is not gravitation field; that is, including the electromagnetic field; however, in some future work, we will discuss the possibility of joining the gravitation and electromagnetic fields in a unified field. The main idea is to generate gravitation field equations that are Lorentz invariant. From Eq. (50), if there is no gravitation field, then ${\Gamma }_{\mu \nu }^{\sigma }=0$. Therefore, ${\Gamma }_{\mu \nu }^{\sigma }$ are the components of the gravitation field. In addition, from Eq. (87), if ${\Gamma }_{\mu \nu }^{\sigma }=0$, then Riemann tensor components vanish, $R_{\mu \alpha \nu }^{\sigma }=0$. However, Riemann tensor elements are not the components of the gravitation field. That is, if ${\Gamma }_{\mu \nu }^{\sigma }=0$, then Riemann tensor elements are necessary zero. Based on the formalism of the general theory of relativity, the gravitation field causes curvature of the four-dimensional space, that is the space-time. However, the gravitational field is described by the metric tensor $g_{\mu \nu }$, and precisely on the dependence of $g_{\mu \nu }$ on the 4-vector positions $x^{\mu }$, and hence ${\Gamma }_{\mu \nu }^{\sigma }$ based on its definition contains the components of the gravitation field. On the other hand, Riemann tensor is a mathematical object to describe the curvature of the space-time (in general, of any multi-dimensional space), and therefore, it is necessary for it to vanish if ${\Gamma }_{\mu \nu }^{\sigma }=0$. Mathematically, it is also sufficient, but based on the general theory of relativity the components of the gravitation field are described by ${\Gamma }_{\mu \nu }^{\sigma }$. Based on this introduction, we expect that gravitation field equations in the absence of matter to be expressed in terms of the components of ${\Gamma }_{\mu \nu }^{\sigma }$.

The least action principle can be used with a Lagrangian given by: and $\sqrt{-g}=1$, where g is the determinant of $g_{\mu \nu }$. Then, the action integral is where $\tau$ is a proper time indicating the flow along the geodesic. Using the principle of the least action, we have where where in the third line the indices $\mu$ and $\nu$ are swapped and the symmetric property of metric tensor $g^{\mu \nu }=g^{\nu \mu }$ are use.

We can use the following identity: to obtain Using Eq. (45) and Eq. (48), we can write: Therefore, Therefore, Eq. (97) can be written as We can now write Lagrangian equation by considering it (as also indicated from Eq. (100)) as a function of $g^{\mu \nu }$ and its derivative (so-called velocity) as follows: where the second equation is a constraint. Eq. (102) give Lagrangian equations of gravitation in a Lorentz invariant form.

Using Eq. (100), we have Substituting Eq. (103) into Eq. (102), we obtain Eq. (104) are know as Einstein’s equations of gravitation field in the absence of matter.

To write Eq. (104) in terms of physical quantities, such as energy and momentum, Einstein introduced some other important quantities. For that, first we multiply both sides of the first expression in Eq. (102) by $g_{\sigma }^{\mu \nu }$: Using the following identity, we have Combining Eq. (107) and Eq. (108) yields Introducing the tensor where Then, The symmetric tensor $T_{\mu \nu }$ is called energy-momentum density tensor. The expressions in Eq. (111) imply the conservation law of energy-momentum. Furthermore, $T_{\sigma }^{\alpha }$ describes the energy component. $\kappa$ is related to the universal gravitation constant as defined by Newton’s mechanics. To Eq. (111), we have also to add the constraint $\sqrt{-g}=1$.