This section is aimed at specialists in General Relativity. It exposed from an original point of view how it is possible to travel in time, specifically to the past, without resorting to overly complex models. Used specifically a simple model of perfect fluid without pressure and rotation.
First it is assumed that the Gravity field created by the fluid itself is negligible, which implies a flat space-time. The novelty is that in this space-time background, no temporary loops are created, nor huge amounts of matter bend space-time, nor resort to exotic matter or negative energy to travel to the past is needed. Instead of all that are solved analytically the equations of fluid dynamics which facilitates the discussion of results.
This is possible because equality between gravity and inertia postulated by Einstein’s equivalence principle of general relativity. As you can intuit much easier to handle inertia appearing with the accelerated movement gravity , although the effects in both cases have to be equivalent.
Finally we consider the case where the Gravity field of fluid is important and therefore the gravity is present. Along the way we find the dark matter and dark energy.
To that currents or argue the ideas presented, the results, the hypothesis, the deductive method or any other aspect from a technical point of view offers on the FEEDBACK TECHNICAL section.
Of course the results obtained are entirely theoretical . Should submit to experimental test. For this I encourage readers to make contributions in this regard in the section related items. Also I suggest the generalization for the case of a perfect fluid pressure and fluid accelerated charged particles in an electromagnetic field.
In these sections you can upload text files to facilitate communication.
Nothing, I invite you to sit back and dispongas you to make what could be the most amazing journey of your life. I hope you surprised. Thank you very much for your attention.
The Physics of Time Travel
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THE PHYSICS OF TRAVELS IN TIME
Summary
The principle of General Covariance allows us to understand the dynamics of accelerated reference systems using the formalism of General Relativity. Using this principle, the metric, the related connection and the equations of the dynamics in an accelerated reference system, in the absence of gravity, are easily deduced. The equations obtained explain: the geometry of a space-time without gravity, the differences between gravity and inertia and what it is and how it can travel in time. These results apply to the dynamics of a simple system: a perfect fluid without pressure and in circular motion. In this case there are some indications about the possible origin of dark matter and energy. Finally, the external force to be applied to this rotating fluid is calculated to go back in time. In the Appendix the formalism is generalized for the case in which gravity is present.
INTRODUCTION
The invariance of the equations of Physics to Lorentz transformations constitutes one of the pillars of Physics. This principle determines the shape of the equations of physics in inertial reference systems.
It makes sense that there is some more general invariance principle to express the equations of physics in any system of reference. This is the principle of invariance of the equations of Physics before any transformation of coordinates implicit in the principle of General Covariance.
In general, the invariance to Lorentz transformations is satisfactory when in the equations of the field the mass does not intervene, as in electromagnetism. If mass is present, as in the case of dynamics or gravity, then general invariance is more appropriate.
The general Covariance principle of the equations of physics is suitable for finding the equations of dynamics in accelerated reference systems, metrics, and related connection in a space-time without gravity. Its main advantage is that in the equations of dynamics the transformations of coordinates between observers in movement appear explicit. This makes that starting from a known transformations of coordinates can determine the dynamics of the movement. Another important advantage comes from the equality between gravity and inertia that Einstein postulated in his Principle of Equivalence of General Relativity. As you can intuit, it is much easier to deal with the inertia that appears with accelerated motion than gravity, although the effects in both cases have to be equivalent.
This article shows the treatment of reference systems in motion from a non – quantum and relativistic point of view and using the formalism of General Relativity.
First, sections I to IV consider movement in a space-time without gravity. In section I study the reference systems in motion and the field. In Section II, the dynamics in a rotating reference system is analyzed. In section III, the obtained results are applied to establish the dynamics of the rotation of a perfect fluid without pressure. In section IV, it is determined what should be the force to be applied on this model of fluid in rotation to go back in time. Finally in the Appendix connects with the equations of the field and includes gravity.
- REFERENCE SYSTEMS IN MOVEMENT
Dynamics equations
When studying relativistic dynamics, it is convenient to define two observers linked to two reference systems: one that is considered fixed and the other in motion in relation to it. These will be referred to hereafter as the reference system at rest S ‘and reference frame in motion S.
Let S be an inertial reference system in which the laws of Special Relativity are valid globally. In this reference the metric system is the Minkowski metric (1)
| (1.1)
|
An observer located in S ‘defines Cartesian coordinates to measure events.
Let S be a reference system in motion with respect to S ‘in which other coordinates are used to measure the events.
The equations of motion in S ‘of a fluid moving with respect to S’ subjected to the action of an external force are
| (1.2) |
where
| (1.3) |
Is the energy-momentum tensor of the fluid considered perfect in S ‘, is the external force density,
| (1.4) |
Is the inverse of (1.1), that is,
| (1.5) |
Is the generalized velocity of a fluid particle,
| (1.6) |
Being the proper time and y are the pressure and mass density of the fluid.
The dynamics equations must be invariant to any coordinate transformation. If the transformations between the coordinates of S ‘and S are continuous functions of the form
| (1.7) |
Global and covering all space-time or at least extend to the entire volume of the fluid and inverse transformations are functions of the form
| (1.8) |
Then, by effecting a coordinate change such as (1.7) in the equations of motion (1.2) of the fluid in S ‘, we can find the equations of the fluid dynamics in any reference system, for example the S
| (1.9) |
where
| (1.10) |
Is the external force density acting on a fluid particle in S,
| (1.11) |
Is the energy-momentary tensor of the perfect fluid in S and is the generalized velocity of a fluid particle measured in S.
Also, from (1.6) it follows
| (1.12) |
where
| (1.13) |
Is the metric in S.
The related connection is defined as
| (1.14) |
It is also the inverse tensor of (1.13), that is
| (1.15) |
Equations (1.9) are invariant to any coordinate transformation. They explicitly show the coordinate transformations (1.7) through the affine connection (1.14).
Degrees of freedom
Equations (1.9) together with condition (1.12) form a system of five equations with 14 variables or unknowns: the four components of generalized velocity, density, pressure, four components of applied external force density, and The four functions that relate the coordinates in one and another reference system, which leave nine degrees of freedom. The state equation
| (1.16) |
Provides another equation that reduces the degrees of freedom to eight.
If the reference system S is chosen so that the fluid remains at rest with respect to it, it must be
| (1.17) |
And three more equations are obtained.
Equations (1.17) do not fix the Gauge alone, that is, the reference system. Needless to add another condition: the pure time component of the metric is to be invariant (1)
| (1.18) |
Where use has been made of (1.1).
The three equations (1.17) plus equation (1.18) simplify the degrees of freedom to four.
In addition, the equation of continuity of the fluid must be verified, that when the mass-energy in S ‘is conserved it takes the form
| (1.19) |
And in this way the degrees of freedom are reduced to three, as corresponds to a perfect fluid in motion.
The same applies to S ‘: the four equations (1.2), together with condition (1.6), the equation of state (1.16) and the equation of continuity (1.19) constitute a system of seven equations with ten variables or unknowns, Which allows only three degrees of freedom. These three degrees of freedom correspond to the three spatial components of the applied external force density that can be chosen arbitrarily. The rest of dynamic variables will be expressed as functions of them.
However, it is possible to choose any three functions that intervene in the dynamics as independent variables, for example, the functions that relate the spatial coordinates in one and another reference system and the rest will be functions of them.
Meaning of the time
As is well known, the three spatial dimensions are characterized by freedom of movement. The following is demonstrated why this is not so in the case of time.
Introducing (1.13) into (1.18) gives
| (1.20) |
from where
| (1.21) |
On the other hand, the components of the velocity of S with respect to S ‘are
| (1.22) |
Where it has been used
| (1.23) |
Since S is at rest in its reference system.
In addition, the velocity module can be calculated with (1.22) and (1.21):
| (1.24) |
This equation indicates, as expected, that it can never be greater than unity, that is, that no reference system S can be moved at a speed greater than that of light, with respect to S ‘.
From here, you get
| (1.25) |
This equation explains the phenomenon of temporal dilatation that occurs with increasing velocity. It indicates that, in the absence of gravity, a clock at rest relative to another will march at the same pace, while one that moves relative to another will mark its own time, different from the other.
It shows that movement in the temporal dimension depends on movement in space. In particular, it indicates that the velocity of movement in time of S with respect to S ‘depends on its speed of displacement in space with respect to S’.
Consequently, it can be affirmed that time is a dimension in which there is no freedom of movement because the movement in the temporal dimension is not independent of movement in space, but depends on it in the manner indicated by (1.25).
This dependence, which is explicit in the equations of the fluid dynamics (1.9) in S through the affine connection (1.14), makes it possible to determine at what speed a fluid must move in space to go back in time. The answer will be found in the following sections.
Inertial Field
The affine connection can be calculated using equations (1.14) or with the metric
| (1.26) |
The curvature tensor in S can be defined as a function of the related connection as
| (1.27) |
Since the tensor of curvature in S ‘is clearly zero, it follows from the laws of transformation of the tensors which is also zero in any other reference system
| (1.28) |
This result implies that the space-time in S is flat instead of curved and that in the moving reference system S Gravity does not appear.
The tensor of Ricci is obtained by contraction of the tensor of curvature and also is zero according to (1.28)
| (1.29) |
This equation indicates that the field in S has no sources but is originated only by the movement of S with respect to S ‘.
In short, the field in S has no sources that curves space-time.
- ROTARY REFERENCE SYSTEM
Coordinate transformation
A reference system S in rotation is an accelerated reference system with an interesting particularity: it neither approaches nor distances from the reference system at rest S ‘.
In this case it is convenient to define in S cylindrical coordinates for practical reasons. If the motion is circular, we must add two more equations to (1.9) and (1.12):
| (2.1) |
Which reduces to one the degrees of freedom for a perfect fluid at rest with respect to S.
The transformations between the coordinates of the reference system in rotation S and the coordinates of the fixed reference system S ‘(1.7) compatible with (2.1) are functions of the form:
| (2.2) |
The metric tensor
The metric in S can be calculated with (1.13), (1.1) and (2.2):
| (2.3) |
Where it has been used (1.18) in the first of the equations. The determinant of the metric tensor is
| (2.4) |
The first of equations (2.3) can be written as
| (2.5) |
This equation supports two solutions. The positive solution implies that it increases with, which is known as the phenomenon of temporal dilation. In section IV it will be seen under what circumstances the negative solution is admissible, which ensures a decrease of as it increases what would allow trips to the past.
From now on, in a general way, both signs will be shown except when one of them is justified.
Inverse of the metric tensor
The inverse of the metric can be calculated with (1.15) and equations (2.3):
| (2.6) |
The related connection
Deriving (2.5) is obtained
| (2.7) |
The related connection can be calculated with (1.26), the derivatives of the metric (2.3), the inverse of the metric tensor (2.6) and with (2.7):
| (2.8) |
The rest of the components are zero.
III. DYNAMICS OF THE ROTATION OF A PERFECT FLUID WITHOUT PRESSURE
Equations of dynamics in the reference system in rotation
In this section we apply the above results for the simplest model of fluid, that is, a perfect fluid without pressure. In this case equation (1.16) is replaced by
| (3.1) |
And the energy-momentary tensor of the perfect fluid in S is obtained by introducing (3.1) into (1.11)
| (3.2) |
This model represents a fluid without interaction between its particles, that is to say, without distance forces or collisions between them so that it resembles rather a model of dust.
If the fluid is in rotation and a reference system S is chosen which is at rest with respect to the fluid, then the equations (1.17), which in cylindrical coordinates take the form
| (3.3) |
These equations together with (1.12) and (1.18) imply that
| (3.4) |
In this reference system the tensor-energy moment is obtained with (3.2), (3.3) and (3.4)
| (3.5) |
The components of the external force density applied to the fluid in S are calculated with (1.9), (2.8) and (3.5)
| (3.6) |
| (3.7) |
| (3.8) |
| (3.9) |
The components of the external force density in S ‘can be calculated by inverting (1.10)
| (3.10) |
In particular, the temporal componte of (3.10) can be calculated with the first one of (2.2) (3.6) and (3.7)
| (3.11) |
The principle of conservation of mass-energy in S ‘requires that (3.19)
| (3.12) |
Integrating this last equation between y can know the mass density of the fluid
| (3.13) |
Deriving (3.13)
| (3.14) |
Finally, the equations of the dynamics (3.6) to (3.9) with (3.13) and (3.14) are
| (3.15) |
| (3.16) |
| (3.17) |
| (3.18) |
As seen at the beginning of Section II, a perfect fluid in rotation has only one degree of freedom. By inspecting Eqs. (3.15) to (3.18) we see that this degree of freedom corresponds to the angular velocity of the rotation of S ‘with respect to S, which can be chosen as an independent variable. The rest of variables will be a function of it. However, although there is only one degree of freedom, the applied external force density has two components that are responsible for the rotational movement: the radial component and the tangential component calculated in (3.16) and (3.17). In order for the circular movement to occur, both components must be applied. These two components will now only be referred to.
Equations of the dynamics in the reference system at rest
It is also desirable to determine the dynamics of fluid rotation in the reference system S ‘. This can be done, in Cartesian coordinates through equations (3.10), (2.2) and (3.15) to (3.18). Now from a practical point of view it is convenient to use cylindrical coordinates also in S ‘. The relation between the cylindrical coordinates in S ‘and S is obtained with (2.2)
| (3.19) |
This relationship allows the external force density to be obtained directly in S ‘in cylindrical coordinates using (1.19), (3.10) and (3.15) to (3.18)
| (3.20) |
| (3.21) |
| (3.22) |
| (3.23) |
The same result would have been obtained if it was first calculated in S ‘in Cartesian coordinates and then a transformation to the cylindrical coordinates in S’ is made through (3.10), to finally obtain in cylindrical coordinates. Also for the same reasons given for S, hereinafter only the radial and tangential components of the external force density will be referred to.
Classic Limit
The classical limit of the equations of the dynamics (3.21) and (3.22) is obtained for speeds much smaller than that of light. Introducing (2.5) into (1.24)
| (3.24) |
It follows that if
| (3.25) |
Taking into account (3.25) and (3.16) the equations (3.21) and (3.22) in classical limit are
| (3.26) |
| (3.27) |
Where it has been assumed by simplicity that the fluid starts from rest at the initial instant of S in which the density is. It is convenient to express and as a function of time in S ‘. For this purpose, (2.5) is solved with the approximation (3.25)
| (3.28) |
Finally, equations (3.26) and (3.27) with (3.28) provide
| (3.29) |
| (3.30) |
Which coincide with the classical result for a fluid without pressure and in circular motion.
Dark matter
If the angular velocity of rotation is constant
| (3.31) |
Then if the fluid starts from rest at the initial instant of S in which the density is the relativistic centripetal force (3.21) with (3.16),
| (3.32) |
And the classical centripetal force (3.26)
| (3.33) |
Do not depend on time and what implies
| (3.34) |
This means that in the classical case at constant angular velocity, it will be necessary to assume a higher mass density than in the relativistic one. The interpretation is obvious: to explain the dynamics of the rotation of this fluid it is necessary to resort to dark matter in the classical case while it is not necessary with relativistic formalism.
| Fig. 1. Graphical representation of
opposite to angular velocity constant. |
Fig. 2. Graphical representation of
against a constant angular velocity. |
The discrepancy between one outcome and another will be more evident the greater. Introducing (3.31) into (3.24)
| (3.35) |
Is observed to increase with. This means that for points near the center of rotation it is non-relativistic, whereas for distances sufficiently far from the center of rotation it becomes relativistic, as it corresponds to a rotation movement. Specifically for small angular velocities will approximate the unit for large values of. On the contrary, if it is large it will be relativistic even for small values of.
The greatest negative value occurs when
| (3.36) |
Which with (3.32) occurs when
| (3.37) |
Dark energy
The first term of (3.17)
| (3.38) |
Is a term of dark energy. It is a relativistic consequence due to the dependence of con. When
| (3.39) |
During the movement, then if the angular acceleration and the angular velocity of the rotation have the same sign, which indicates positive or negative acceleration, the dark energy decreases the force of inertia. If on the contrary they have different sign, a positive or negative deceleration occurs and the dark energy increases the force of inertia.
For small angular rotational speeds this effect will be important for high angular velocities. If the angular velocity of the rotation is large, the effect will be noticeable for small values of.
Observe that according to (3.22) the dark energy does not appear in the reference system at rest S ‘but is exclusive to the reference system in rotation S.
The presence of dark energy is responsible for the fact that S is not conserved mass-energy as can be deduced from (3.15).
- TRAVELS IN THE TIME
In this section we apply the results obtained in the previous sections to determine how the dynamics of the rotation of a perfect fluid without pressure to be back in time should be. This involves finding the external force to be applied to said fluid which causes it to rotate in the space associated with a temporary recoil. For this purpose the only degree of freedom possessed by a perfect fluid in rotation is fixed. It is convenient to choose, among all the variables involved in the dynamics, the angular velocity of the rotation that appears in the equations of motion (3.16) and (3.17) as independent variable. This variable is chosen so that the associated time shift obtained from (2.5) allows the temporary recoil of the fluid. The trip will take five stages.
Temporal Acceleration
Before the movement begins, the observers located in S ‘and S are in an inertial reference system in which the metric (1.1) is valid. This allows both systems clocks are synchronized according to the criterion of the light signals proposed by Einstein [1] .
The first stage of the temporal journey takes place during the interval for an observer located at S. If the rotation starts from rest at the initial instant the simplest function is of the form
| (4.1) |
The angular velocity is
| (4.2) |
Which, as part of the rest at the initial time, is zero as follows from (4.2)
| (4.3) |
The angular acceleration is
| (4.4) |
Where it is a constant that can be chosen so that
| (4.5) |
Introducing (4.2) into (2.5)
| (4.6) |
And integrating the function
| (4.7) |
Deriving (4.7)
| (4.8) |
Introducing (4.2) to (4.4) in (3.16) and (3.17) is obtained for this fluid in the reference system S during the first stage of the temporal journey
| (4.9) |
| (4.10) |
Where it is given by (4.8).
The first stage of the journey passes for an observer located in S ‘during the interval. The dynamics of the rotation of this fluid in this reference system during this stage is obtained by introducing (4.2) and (4.4) with (3.28) into (3.29) and (3.30)
| (4.11) |
| (4.12) |
Expressions that are valid if it is verified (3.25) that in this case, taking into account (4.2) it takes the form
| (4.13) |
Temporary investment
The second stage of the temporal journey takes place during the interval for an observer located in S. The simplest function, which gives rise to a temporal inversion, is of the form
| (4.14) |
Where, and are constants checking
| ; ; | (4.15) |
The angular velocity is
| (4.16) |
The angular acceleration is
| (4.17) |
The functions and must be continuous in
| (4.18) |
Which with (4.1) and (4.14) provides
| (4.19) |
and
| (4.20) |
This last equation with (4.2) and (4.16) leads to
| (4.21) |
Introducing (4.16) into (2.5)
| (4.22) |
It is observed that it changes sign in as it was discussed in (2.5). Integrating (4.22) yields
| (4.23) |
Continuity implies
| (4.24) |
That with (4.23) gives
| (4.25) |
Introducing (4.7) into (4.25)
| (4.26) |
Inserting (4.26) into (4.23)
| (4.27) |
In order to avoid divergences in (4.27) it
| (4.28) |
It is the form of the function (4.27) that reaches a maximum in which it allows the temporary recoil. At the instant of S the temporal inversion occurs. Of course this process is only possible if the external force density (3.16) and (3.17) applied on this fluid in S and the external force density (3.21) and (3.22) applied according to S’ are finite.
Fig. 3. Plotting opposite to fixed during the second phase of time travel
Deriving (4.23), making use of (4.25) and (4.8) in
| (4.29) |
The mass density at the instant is obtained from (3.13) and (4.3)
| (4.30) |
Introducing (4.16), (4.17) and (4.30) in (3.16) and (3.17) and making use of (4.20) is obtained for this fluid in the reference system S during the second stage of the temporal journey
| (4.31) |
| (4.32) |
Where it is given by (4.29). It is observed that as much as they are finite throughout the interval.
| Fig. 4. Plotting opposite to fixed during the second phase of time travel. | |||
| Fig. 5. Plotting front for fixed during the second phase of time travel. | |||
The second stage of the trip takes place for an observer located in S ‘during the interval. The dynamics of the rotation of this fluid in this reference system during this stage is obtained with (3.21) and (4.31), introducing (4.16), (4.17) and (4.30) into (3.22) and taking into account (4.3) And (4.20)
| (4.33) |
| (4.34) |
Fig. 6. Graphing
address for fixed during
the second stage of time travel.
It is convenient to express the results (4.33) and (4.34) as a function of the time measured by S ‘. For this it is necessary to clear as a function of (4.23)
| (4.35) |
| (4.36) |
| (4.37) |
Which are the same equation, given by (4.26).
By introducing (4.35) and (4.36) into (4.33) and (4.34) we arrive at
| (4.38) |
| (4.39) |
Where it is given by (4.26). It is observed that as much as they are finite throughout the interval.
During this stage of the temporal journey the fluid reaches the speed of light in S ‘. This can be checked by inserting (4.16) into (3.24)
| (4.40) |
so that
| (4.41) |
At the instant of S or at the instant of S ‘where
| (4.42) |
Is obtained by doing in (4.23) and is determined by (4.26).
It is striking that a fluid as described can reach the speed of light in S ‘. This is unthinkable in the case of a material particle. The difference is in the equation of continuity (1.19) which is only verified in the case of the fluid. This equation has the consequence that the mass density of the fluid obtained from (3.13) with (4.30), (4.3), (4.20) and (4.16) during the second stage of the trip
| (4.43) |
Can be annulled at the instant of S in which the velocity of light at S ‘is reached and the time inversion occurs. This happens gradually in S ‘at the instant given by (4.42) because it depends on. As mass-energy is conserved in S ‘by checking the continuity equation (1.19) this means that the mass of the fluid is progressively transformed into energy. This can happen instantaneously in S ‘if the fluid is confined to a ring of infinitesimal width. It is thanks to this phenomenon that the fluid can reach the speed of light in the reference system S ‘.
Temporary regression
The third stage of the temporal journey takes place during the interval for an observer located in S. It is convenient that during this stage the angular velocity of the rotation is constant. As the angular velocity of the rotation must be continuous this means that
| (4.44) |
And also that according to (4.44) and (4.16) the angular velocity is
| (4.45) |
Integrating (4.45)
| (4.46) |
Continuity of function implies
| (4.47) |
As can be deduced from (4.14) and (4.19).
Deriving (4.45) gives the angular acceleration
| (4.48) |
Introducing (4.45) into (2.5)
| (4.49) |
And integrating is obtained
| (4.50) |
Continuity implies
| (4.51) |
That with (4.50) gives
| (4.52) |
Introducing (4.23) into (4.52) and making use of (4.26)
| (4.53) |
Substituting (4.53) into (4.50)
| (4.54) |
Deriving (4.50) and making use of (4.52) and (4.29) in
| (4.55) |
It is necessary to determine the speed (4.49) of the temporary setback. As it depends on to determine this speed it is advisable to fix one so that to verify
| (4.56) |
Where is a constant greater than zero
| (4.57) |
Inserting (4.49) into (4.56) yields
| (4.58) |
| (4.59) |
| (4.60) |
Which are the same equation.
The mass density at the instant is obtained from (3.13), making use of (4.30) and (4.20)
| (4.61) |
Introducing (4.45), (4.48) and (4.61) in (3.16) and (3.17) and making use of (4.44) is obtained for this fluid in the reference system S during the third stage of the temporal journey
| (4.62) |
| (4.63) |
Where it is given by (4.55).
The third stage of the journey takes place for an observer located in S ‘during the interval. The dynamics of the rotation of this fluid in this reference system during this stage is obtained with (3.21) and (4.62) and introducing (4.48) into (3.22)
| (4.64) |
| (4.65) |
Temporary reinvestment
The fourth stage of the temporal journey passes during the interval for an observer located in S. The simplest function that gives rise to a new temporal inversion is of the form
| (4.66) |
Where, and are constants checking
| ; ; | (4.67) |
The angular velocity is
| (4.68) |
The angular acceleration is
| (4.69) |
The functions and must be continuous in
| (4.70) |
Which with (4.46), (4.47) and (4.66) provides
| (4.71) |
and
| (4.72) |
This last equation with (4.45) and (4.68) leads to
| (4.73) |
Introducing (4.68) into (2.5)
| (4.74) |
It is observed that it changes sign in as it was discussed in (2.5). Integrating (4.74) is obtained
| (4.75) |
Continuity implies
| (4.76) |
That with (4.75) gives
| (4.77) |
Introducing (4.50) into en (4.77) and making use of (4.53) gives
| (4.78) |
Introducing (4.78) into (4.75) gives
| (4.79) |
To avoid divergences in (4.79) it is
| (4.80) |
It is the form of the function (4.79) that reaches a minimum in which it allows to reverse the temporary setback. At the instant of S, temporary reinvestment occurs. Again, this process is only possible if the external force density (3.16) and (3.17) applied on this fluid in S and the external force density (3.21) and (3.22) applied according to S ‘are finite.
Deriving (4.75), making use of (4.77) and (4.55) in
| (4.81) |
The mass density at the instant is obtained from (3.13), (4.61) and (4.44)
| (4.82) |
Introducing (4.68), (4.69) and (4.82) in (3.16) and (3.17) and making use of (4.72) is obtained for this fluid in the reference system S during the fourth stage of the temporal journey
| (4.83) |
| (4.84) |
Where it is given by (4.81). It is observed that as much as they are finite throughout the interval.
The fourth stage of the journey takes place for an observer located in S ‘during the interval. The dynamics of the rotation of this fluid in this reference system during this stage is obtained with (3.21) and (4.83), introducing (4.68), (4.69) and (4.82) into (3.22) and taking into account (4.72)
| (4.85) |
| (4.86) |
It is convenient to express the results (4.85) and (4.86) as a function of the time measured by S ‘. For this it is necessary to clear as a function of (4.75)
| (4.87) |
| (4.88) |
| (4.89) |
Introducing (4.87) and (4.88) into (4.85) and (4.86)
| (4.90) |
| (4.91) |
Where it is given by (4.78). It is observed that as much as they are finite throughout the interval.
Again, and just as it happened in the instant, it can be verified that at time S, the fluid returns to reach the speed of light in S ‘. This happens in S ‘at the instant obtained by doing in (4.75)
| (4.92) |
Where it is given by (4.78).
Temporal deceleration
The fifth stage of the temporal journey passes during the interval for an observer located at S. If the rotation stops at the end of the trip, at the instant of S, the simplest function is of the form
| (4.93) |
The angular velocity is
| (4.94) |
Which when stopped at the final instant is zero
| (4.95) |
The angular acceleration is
| (4.96) |
Where is a constant such that
| (4.97) |
The functions and must be continuous in
| (4.97) |
Which with (4.93), (4.66) and (4.71) leads to
| (4.99) |
| (4.100) |
The latter equation with (4.94) and (4.68) gives
| (4.101) |
Inserting (4.94) into (2.5)
| (4.102) |
And integrating the function
| (4.103) |
Continuity implies
| (4.104) |
That with (4.103) gives
| (4105) |
Introducing (4.75) into (4.105) and making use of (4.78) gives
| (4.106) |
The above expression is simplified if the constants are chosen so that
| (4.107) |
Introducing (4,107) into (4,106)
| (4.108) |
Substituting (4.108) into (4.103) yields
| (4.109) |
It is fitting that the journey should end at a certain point in S’s past. As it is advisable to fix one in which this happens. This will be what
| (4.110) |
Which with (4.109) gives
| (4111) |
Expression that is simplified if the constants are chosen so that
| (4.112) |
Introducing (4.112) into (4.111)
| (4113) |
Which with (4.58) and (4.59) can be written in the form
| (4.114) |
It is convenient to introduce a new parameter defined by equation
| (4115) |
In summary, the boundary conditions (4.21) and (4.60) and equations (4.107), (4.112), (4.114) and (4.115) provide ten independent equations
| ; ; ; |
| ; ; ; ; ; |
| (4116) |
Because equations (4.73) and (4.101) are identically satisfied. In addition there are 13 unknowns which leaves three parameters free, for example, and. The unknowns will be functions of,,,, and that are known, in addition to the free parameters, and.
Solving these equations is obtained
| (4.117) |
| (4118) |
| (4.119) |
| (4.120) |
| (4.121) |
| (4.122) |
| (4.123) |
Deriving (4.103) and making use of (4.105), (4.81) in (4.107) and (4.112)
| (4124) |
The mass density at the instant is obtained from (3.13), (4.82) and (4.72)
| (4125) |
Introducing (4.94), (4.96) and (4.125) into (3.16) and (3.17) and making use of (4.100) is obtained for this fluid in the reference system S during the fifth stage of the temporal journey
| (4126) |
| (4.127) |
Where it is given by (4.124).
The fifth stage of the journey passes for an observer located in S ‘during the interval. The dynamics of the rotation of this fluid in this reference system during this stage is obtained by introducing (4.94) and (4.96) with (3.28) into (3.29) and (3.30)
| (4.128) |
| (4.129) |
Expressions that are valid if (3.25), which in this case is satisfied by (4.112) and (4.13)
| (4.130) |
And this concludes the journey into the past.
As it is easy to imagine the journey of the return to the present moment can be carried out in a similar way and in only three stages that do not entail as much difficulty as the trips to the past.
APPENDIX: EINSTEIN EQUATIONS WITH SCALAR FIELD
The above results are adequate in a flat space-time in the absence of gravity. In practice this happens when the gravitational field created by the fluid itself is so small that it barely curves space-time. Although it is not relevant in the exposed developments, it is convenient to generalize the formalism to the case in which the gravitational field generated by the fluid is important. The results in the absence of gravity are obtained as a particular case. To do this we have to resort to the Einstein equations of the gravitational field.
First, it should be noted that the energy-momentum tensor (1.11) is not conserved, as is deduced from (1.9), so that it does not fit Einstein’s equations
| (A.1) |
For which the energy-momentum tensor is conserved
| (A.2) |
This means that the energy-momentary tensor (1.11) is not adequate to calculate the gravitational field created by a perfect fluid subjected to external non-gravitational forces.
The simplest way to avoid this difficulty is to introduce a scalar field into Einstein’s equations (A.1).
This scalar field has to verify a new differential equation. The simplest differential equation that is usually covariant for a scalar field is
| (A.3) |
Where it is a coupling constant and is the energy-momentary tensor of matter.
The most general energy-moment symmetric tensor for this field must contain second derivatives and first derivative products of the field
| (A.4) |
Einstein’s equations (A.1) including this scalar field are
| (TO 5) |
Where it is the energy-momentary tensor of matter that for a perfect fluid is given by (1.11) and is the energy-momentary tensor of the scalar field (A.4).
In order to determine the functions, which appear in (A.4), the covariant derivative of equation (A.5)
| (A.6) |
Bianchi’s identities ensure that the first term in equation (A.6) is nullified
| (A.7) |
Which is equivalent (A.6) to
| (A.8) |
This last equation guarantees that the energy-momentary tensor defined as
| (A.9) |
Is preserved, as intended.
As reflected in equation (1.9) the first term of (A.8) is the external force density
| (A.10) |
That with (A.8) gives
| (A.11) |
The covariant derivative of (A.4) is
| (A.12) |
Where it has been used
| (A.13) |
Introducing (A.12) into (A.11) gives
| (A.14) |
These are the equations of the scalar field dynamics that determine the four functions,, and.
Reference systems
The ten equations (A.5) that determine the gravitational field are not independent but are related by the four identities of Bianchi (A.7) which reduces the equations (A.5) to six independent equations. To determine the metric univocally it takes four more equations to fix the Gauge, that is, the reference system.
In the presence of gravity, it is also desirable to define a reference system at rest S ‘as described in section I. Since equations (1.9) are also applicable in the case of gravity, then in the reference system S’ the equations of The dynamics of the fluid are
| (A.15) |
where
| (A.16) |
Is the energy-momentum tensor of the fluid as in (1.11) and
| (A.17) |
The affine connection, as a function of the metric in S ‘is obtained as in (1.26)
| (A.18) |
In this case the metric satisfies Einstein’s equations (A.5)
| (A.19) |
Clearly, in the presence of gravity, the metric is different from the Minkowski metric (1.1) and the curvature tensor (1.27) is no longer zero, so spacetime is curved and the field has fountains.
Now, in the presence of gravity and by similarity with (1.2), the reference system at rest S ‘can be defined as that in which the equations of the fluid dynamics acquire the form
| (A.20) |
This equates, in view of equations (A.15), which are valid in any reference system and in particular in S ‘, to choose a gauge in which
| (A.21) |
These four equations that determine the Gauge, that is, the reference system S ‘, are not covariant so they are not valid in any reference system, but only in S’ in particular.
The reference system in movement S is chosen, as in section I, so that the fluid, which originates the gravitational field, is at rest with respect to it. Similarly, the equations of the fluid dynamics in the reference system S are
| (A.22) |
where
| (A.23) |
Y
| (A.24) |
Is the invariance equation of the interval from which the relation between the metric in S and S ‘
| (A.25) |
Again, the affine connection as a function of the metric in S is obtained as in (1.26)
| (A.26) |
The metric in S also satisfies Einstein’s equations (A.5)
| (A.27) |
If the fluid is at rest with respect to S, the three equations (1.17)
| (A.28) |
In addition, in order to completely fix the reference system S, it is necessary to add another condition that can be obtained by similarity with (1.18) by imposing that the pure temporal component of the metric must be the same in any reference system
| (A.29) |
Where is the pure temporal component of the metric in the reference system S ‘at rest.
The three equations (A.28) together with equation (A.29) fix another Gauge, that is, the reference system S.
Degrees of freedom
In addition, the equation of continuity of the fluid must be verified, that when the mass-energy in S ‘is conserved it takes the form
| (A.30) |
This equation is also not covariant so it only applies in the reference system S ‘.
In general and in any reference system, Einstein’s six independent equations with scalar field, the four equations that fix the reference system, the four equations of the fluid dynamics, the four equations of the scalar field dynamics, the Equation of invariance of the interval, the equation of state, the equation that verifies the scalar field and the equation of continuity of the fluid form a system of 22 equations with 25 variables or unknowns: the ten components of the metric tensor, the four components of the density Of applied external force, the four components of the generalized velocity of the fluid, the four functions,, and, the pressure and density of the fluid and the scalar field. This leaves degrees of freedom as corresponds to a perfect moving fluid.
In practice, it may be agreed, as in Section I, that the coordinate transformations appear explicitly in equations (A.22) of the fluid dynamics in S. For this purpose the affine connection (A.26) Which appears in equations (A.22) as a function of the metric in S ‘through (A.25).
In this case the six Einstein independent equations (A.19), the four equations (A.21) that fix the reference system S ‘, the four equations of the dynamics (A.22), the four equations of the dynamics Of the scalar field (A.14), the three equations (A.28) together with the equation (A.29), the invariability equation of the interval (A.24), the equation of state (1.16), the equation A.3) that verifies the scalar field and the continuity equation (A.30) form a system of 26 equations with 29 variables or unknowns: the ten components of the metric tensor, the four components of the applied external force density, the Four components of the generalized velocity of the fluid, the four functions, and, the four functions that relate the coordinates in one and the other reference system, the pressure and the density of the fluid and the scalar field. This again leaves degrees of freedom.
These three degrees of freedom allow us to choose, for example, the functions that relate the spatial coordinates in one and another reference system as independent variables. The rest of dynamic variables will be expressed as functions of them.
Relevance of the scalar field
The introduction of the scalar field yields the Minkowski (1.1) metric as a solution of the Einstein equations (A.19) in the Gauge (A.21). This solution is adequate in the absence of gravity. In practice this happens when the gravitational field created by the fluid is so small that it barely curves space-time. In this situation, in any reference system
| (A.31) |
And the curvature scalar
| (A.32) |
With (A.31) and (A.32) and (A.5) it can be verified that for this fluid it is verified that
| (A.33) |
So the field is especially important.
In the void
| (A.34) |
And equation (A.3) will be written
| (A.35) |
While equation (A.10) gives
| (A.36) |
Introducing (A.35) and (A.36) into (A.14) gives
| (A.37) |
These equations have as solution
| (A.38) |
With (A.35) and (A.38) the energy-momentary tensor (A.4) of the scalar field in the vacuum is
| (A.39) |
Equations (A.34) and (A.39) imply that in the vacuum equations (A.1) and (A.5) have the same solutions, ie in this case the scalar field does not alter the solutions of The equations (A.1) that explain, among other phenomena, the planetary dynamics.
The same happens when the applied external force density is zero
| (A.40) |
Since in this situation equations (A.14) provide
| (A.41) |
Which with (A.4) lead again to (A.39) and the field ceases to have influence. In other cases, their significance will have to be determined.
Cosmological dark energy
Finally, it should be noted that in the absence of gravity or weak fields, the only way to produce acceleration over the fluid is, as has been seen, by the application of a non-gravitational external force density. However, when the fluid-generating gravity is intense, it alone is capable of producing acceleration over the fluid itself. This acceleration causes a force density as described, which produces the same effects, but which has a purely gravitational origin to appear. This is because the force density appears whenever inertial forces act on the fluid. In other words, when the gravitational field generated by the fluid is intense, the energy-momentum tensor (1.11) of the perfect fluid is not conserved. In this way the scalar field can be present even if only gravitational forces intervene in the movement of the fluid.
This has important cosmological implications: in Cosmology appears the scalar field and the dark cosmological energy can have its origin in the term that appears in the equations (A.5). In this case, three of the four functions, which appear in (A.4), are independent and can be adjusted to the observational data of accelerated expansion of the Universe.
CONCLUSIONS
The exposed developments demonstrate the theoretical possibility of traveling to the past, in the case of a perfect macroscopic fluid without pressure, in the absence of gravity and without violating the laws of Special Relativity. This is only possible if the fluid is accelerated to the speed of light. To demonstrate this, a relativistic treatment of rotation has been made using the principle of General Covariance, which has proved to be practical. Finally, the need to introduce a scalar field in Einstein’s equations to explain relativistic dynamics satisfactorily has been justified. These equations have made it possible to generalize the formalism to the case in which the fluid generates an appreciable Gravitational Field.
And to finish, just add this phrase: “time will give or take reasons.”
BIBLIOGRAPHY
- Weinberg: “Gravitation and Cosmology”, Wiley, New York, 1972, Cap. 2 to 7.
REFERENCES
(1) Henceforth a unit system is used in which c = 1 and the agreement sum over repeated indices.
(1) This condition is necessary for the observer S and the fluid to move jointly in time plus space.
[ 1] AP French, Special Relativity, Massachusetts Institute of Technology, 1968, p. 71
