By Alfonso Leon Guillen Gomez
Bogota, Colombia, 2006
This work is published in Spanish in “Ciencia Abierta” and “La Flecha”
In General Relativity, a gravitational wave propagates in 5 dimensions. This is our discovery. The concept of gravitational wave in this theory is in contradiction with its equations. Gravitational waves consist on propagation of oscillations of spacetime itself, all dimensions must be taken into account. But, according to the equations, they consist on the propagation of the oscillations of X and Y dimensions of spacetime in the Z direction. This way, gravitational waves are really waves in space. This is actually the universally known and accepted definition. There is a big difference between both definitions. According to a mathematical law for N dimensional spaces, a variation in a N dimension space generates a N+1 dimension space. If spacetime has 4 dimensions, its oscillations must propagate in 5 dimensions. This concept also works for gravity within “The Everything Theory”, because if gravity has M dimensions, gravitational waves propagate in M+1 dimensions.
04.20.-q Classical general relativity (see also 02.40.-k Geometry, differential geometry, and topology)
04.20.Cv Fundamental problems and general formalism
04.30.-w Gravitational waves: theory
Particle movement, undulation movement and electro-magnetic wave propagation always take place within a frame: The 4 dimension spacetime (-ct, x, y, z).
Gravity is a phenomenon which occurs in the geometrical frame of physical phenomenon. Gravitational waves do not propagate in spacetime, because they consist on an oscillation of spacetime itself. (1). In this context: ¿Which is the gravitational wave frame? ¿How many dimensions does it have? ¿What does wave speed mean? ¿What does momentum mean?
General Relativity needs the formulation of a new physical theory in order to answer these last questions.
We will consider now the number of dimensions implied in gravitational wave propagation according to standard General Relativity where the expanding universe has no impact on the properties of gravitational waves, except the well known effect of redshift. Hence, we do not include Carmeli cosmology in which the expansion of the universe manifests as a fifth dimension.
Relativity, gravitational waves are oscillations of spacetime, undulations in spacetime, small distortions of spacetime geometry, or ripples of spacetime curvature
which propagate in the time through space as waves. They propagate in all
directions (1, 2, 3, 4, 5, 6,
7, 8). General
Relativity is broken by this Gravitational wave concept. It corresponds to
Newton idea of a universe of three dimensions modulated in the time, and not
universe of block of Einstein Relativity, in that the events are in four
dimensions. In Relativity does not exist time, space either since in Relativity
exist spacetime. Additionally, in the case of the gravitational waves the event
is the spacetime in itself.
By all means, the variation of an N dimensional space, ND; generates an N+1 dimensional space, (N+1)D (9). So, a cero dimension generates one dimension, one dimension generates two dimensions and so on.
Einstein’s gravitational wave requires a 5 dimension geometrical frame because spacetime consists of 4 dimensions.
In order to illustrate the geometrical problem of Einstein’s gravitational wave, we will consider, for example, the wave that is produced by earthquakes in the curvature of the Earth. Earthquakes occur in 3 dimensions; their waves occur in 4, and all changes which are produced in the curvature of the Earth occur in 3 dimensions, because all points of the curvature of the Earth have a latitude, a longitude and an altitude. Waves occurring in the curvature of tri-dimensional objects, like the Earth, occur in 4 dimensions.
According to a mathematical law for N-dimensional spaces, if ND gets out of its continuum N, in a direction k, which is contained in the (N+1)D, it generates an (N+1)D (9). Its variation within its continuum N actually corresponds to a new variation within the (N-1)D that generated it. For example, the variation in the line itself does not exist, but its extension does, due to the new variation of the point that produced it.
It is possible to create internal variations in (N>1)D; for instance, in a 2D, it is possible to vary only one dimension, X or Y. This way, alternative geometric configurations can be generated, starting from an existent configuration that keeps its original N dimensions. But, when the variation from a ND propagates, i.e., ND (t) always it generates an external variation (N+1)D.
ND can externally vary and generate an (N+1)D, or it can internally vary, generating alternative configurations NDa.
We have empirical evidence that the variation of electric and magnetic fields, which are components of the static electro-magnetic field (electric and magnetic fields uncoupled), produces an external variation of this field. The static electro-magnetic field is a geometrical 3D object. The electrical field exists in 2D, on an X, Y plane. The magnetic field exists in 2D, on an X, Z plane. Oscillation of these fields produces an electro-magnetic wave, which propagates in X, Y, Z, in the direction X. So, the electro-magnetic wave propagates in a 4D frame (-ct, x, y, z).
At the most fundamental level, in vacuum spacetime regions, the gravitational waves are studied in the perturbation theory of curved vacuum spacetimes. This theory is linearized gravity that supports any vacuum solution of the Einstein’s equations of the general form:
gμν = gμνν + εhμν + ε2jμν + O(ε3)
Here gμνν is the curved background metric; hμν is the linear order metric perturbation; jμν is a second order metric perturbation. ε is a formal expansion parameter [Page 23, 7].
Since gravitational waves propagate at infinitum they will reach to globally vacuum spacetimes in which Tμν = 0 everywhere, and which are asymptotically flat (hμν → 0 as r → ∞) [page 7, 7]. Thus, “in the absence of gravity, spacetime is flat and is characterized by the Minkowski’s metric, ημν” [Page 3, 10]. Equivalently gμνν is ημν and, the gravitational waves may be treated as deviating only slightly from the flat metric, η. In this case:
gμν = ημν + hμν, ||hμν|| ≪ 1 ..
“Here ημν is defined to be diag (−1, 1, 1, 1) and ||hμν|| means “the magnitude of a typical non-zero component of hμν”. Note that the condition ||hμν|| ≪ 1 requires both the gravitational field to be weak (Tμν → 0), and in addition constrains the coordinate system to be approximately Cartesian. We will refer to hμν as the metric perturbation; as we will see, it encapsulates gravitational waves, but contains additional, non-radiative degrees of freedom as well. In linearized gravity, the smallness of the perturbation means that we only keep terms which are linear in hμν — higher order terms are discarded [Page 4, 7].
In Minkowski’s metric, ημν, there is a tensor (0.2) for every ND. This function takes an input consisting on two tangent vectors, U, V, at a point P, of the ND, and produces an output consisting on a number that lineally depends on each U, V entry. This is the metric tensor (11), which allows to measure angles and distances.
In the Relativity, spacetime consists on the “collection of all the events in the Universe” (Page 385, 12), in 4D.
Therefore, metric tensors allow us to calculate the length S between two events in the spacetime. According to:
dS2 = -c2dt2 + dx2 + dy2 + dz2
The value S is an
invariant. However, it might vary, only due to oscillations that occur in the
squeeze of space → ∆s /s = h(t) [Page 3, 13].
In a simple form sometimes, the gravitational wave stretches all vertical distances between particles and, at the same time, squeezes all horizontal distances. At other times, all horizontal distances are stretched while all vertical distances are squeezed [Page 1, 14].
S can be expressed as the internal product of the base vector x in the direction xμ multiplied by the base vector x in the direction x ν according to:
dS2 = ημνdxμdxν
ημν is Minkowsky’s metric tensor:
-1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1
S is the value of the length between two events in a flat spacetime, in the absence of gravity, in vacuum. This flat spacetime is characterized by Minkowsky’s metric tensor, ημν.
The gravitational linear approximation is used to modify flat spacetime metrics, at the time in which a gravitational wave is passing.
At this essay, We use this great simplification of the gravitational wave; indeed gravitational wave is oscillating of metric curve; but, oscillating of Minkowski’s metric approximation is a very good simple picture. Thus
dS2 = gμνdxμdxν
, where gμν = (ημν
+ hμν) and hμν | << 1
hμν is the disturbance of the gravity in the vacuum metrics (8, 10).
Hence, the equation for gravitational waves is: (Page 1, 15)
(▼2 - ∂2) hμν = 0
perturbation of the metric tensor satisfies the wave equation in each component
(Page 386, 12).
The gravitational wave is traveling in direction z, and the metric tensor changes in coordinates x and y (Page 387, 12). This is the effect of the internal variation, of the length of the spacetime interval, caused by the gravitational wave.
Stretch and squeeze are transverse to direction of propagation [Page 4, 13].
waves have two polarizations plus, +, and cross, x. Each polarization has its
own gravitational-wave field [Pages 4-5, 13].
Those illustrations are taken of Thorne Kip S. Ph237 - Gravitational Waves, Caltech, 7 & 9. Cambridge. England. Pages 4-5.
Gravitational waves have a + polarity, if they are perfectly lined up with x or y axis (8). For a + polarity, hμν is:
0 0 0 0
0 ٤ 0 0
0 0 -٤ 0
0 0 0 0
hμν shifts the x axis in the (x, x + ε) span, and the y axis in the (y, y - ε) span.
-hμν shifts the x axis in the (x, x - ε) span, and the y axis in the (y, y + ε) span.
± hμν makes the x axis oscillate in the (x - ε, x + ε) span, and the y axis in the
(y - ε, y + ε)
Thus, a gravitational wave consists on the oscillation of the two orthogonal axes, and y axis, of flat spacetime, which propagates in the direction z, in 4D (-ct, x, y, z), similarly to electro-magnetic waves, but true gravitational waves consist on oscillations of geometrical curved dimensions. Since a wave is more than merely "gravitational oscillations" - periodic changes in distance for particles in one particular plane-, it is an oscillatory pattern that propagates in the time through space [Page 1, 14].
evolutions h+(t) is the waveforms [Page 5, 13].
This illustration is taken of Thorne Kip S. Ph237 - Gravitational Waves, Caltech, 7 & 9. Cambridge. England. Page 5.
A binary star system loses its energy from orbital binding due to radiation that is transported by the gravitational wave. This energy loss is the reason that makes their orbital separation decrease, together with the orbital period, as well. This fact causes star coalescence and the extinction of energy in the orbital binding field (Page 9, 16).
This way, in the binary pulsar PSR 1913+16 (17, 18, 19), orbital period decreases at a rate of -2.40247 × 10-12 second/second.
Gravitational radiation is only the loss of orbital binding energy in exchanges between potential energy and kinetic energy, without the intervention of binary pulsar internal energy-mass.
Binary pulsars involve a high degree of relativism because of their orbital speed, which is 0.1c for the PSR1913+16. Besides, pulses of binary systems are like the tic-tac of a watch, with a variable lapse. Changes in this lapse are predicted by Relativity, and pulsars are useful in measuring change in time dimension. Hulse and Taylor have measured the tic-tac lapse changes during the orbital passing of stars from periastron to apastron.
During the orbital period, intensity of gravity field and orbital speed from apastron to periastron increases, while it decreases from periastron to apastron. Its effect, in the pulsar-watch is that time, tic-tac lapse, contracts from apastron to periastron, and expands from periastron to apastron (17, 19) at the very same magnitudes that Einstein predicted.
Graph 1 shows the evolution
of time contraction-expansion (17).
This graph is taken of Astro 201 FAQ. Binary Pulsar PSR 1913+16. USA
(Based on Weisberg, 1981)
In Graph 1 we see according to our coordinates system of reference the evolution of the pulse, ticks, of the binary pulsar between periastron-apastron. “When they are closer together, near periastron, the gravitational field is stronger, so that the passage of time is slowed down -- the time between pulses (ticks) lengthens just as Einstein predicted. The pulsar clock is slowed down when it is travelling fastest and in the strongest part of the gravitational field; it regains time when it is travelling more slowly and in the weakest part of the field.” (17).
During the orbital period, potential energy of gravity field, in function of distance, increases from periastron to apastron and decreases from apastron to periastron. However, kinetic energy, in function of speed, increases from apastron to periastron and decreases from periastron to apastron. So, relation between potential and kinetic energy is inverse one; they experience an exchange during the orbital period; a decrease in potential energy corresponds to an increase in kinetic energy and vice versa. Nevertheless, total energy of a binary pulsar orbital binding would remain constant, according to the energy conservation law, but it decreases due to the gravitational radiation.
Gravitational energy is irradiated during the whole orbital period, in all points of local spacetime, in which the field of orbital binding energy exists.
Instant change of orbital binding energy, dE/dt,
depends on the punctual gravitational radiation (8, 16). Orbital period change index, TSij = (dEi/dt )/(dEj/dt),
behaves in an oscillatory way, TSij<1 between apastron - periastron, and TSij>1 between periastron – apastron. Where, i,j are consecutive two points.
TSij oscillates because it is directly related to changes in orbital binding
energy losses, between field points. dE/dt
increases between apastron-periastron due to the raising punctual gravitational
radiation, which increases during the orbital speed acceleration in a
progressively stronger gravitational field. But dE/dt
decreases between periastron-apastron due to the diminishing punctual
gravitational radiation because of the orbital speed des-acceleration in a
progressively weaker gravitational field. So, gravitational radiation is
directly related to principal momentum of inertia, I, according to
Einstein’s equation of quadrupole momentum.
dE / dt ≈ (G / 5c5 ) (d3 I / dt3)2 and I=r2m (Page 2, 20)
One solution for a binary system is:
Hence, in PRS1913+16, according to author and previous equation, TSap (where a is apastron and p is periastron ) can be measured like the change index between the value of dE / dt in apastron/periastron.
TSap = (dEa/dt)/(dEp/dt) → TSap = rp5/ra5
E is the orbital energy loss by energy radiation of gravitational wave
r is the separation of the stars
For PSR B1913+16 :
Hence, TSap= 0,0007
If gravitational radiation magnitude remains constant in all orbit points, the orbital binding energy losses is the same in apastron and periastron ( dEa/dt = dEp/dt ), then TSap = 1.
But the orbital binding energy losses in apastron is just ,07% of the orbital binding energy losses in periastron. These results actually verify the growing change in orbital binding energy losses between apastron-periastron and vice versa. So, punctual orbital binding energy changes, between orbital periods according to a sine function type ( ≈ sine (ωt), where ω=2π/T ).
Orbital binding energy is a component of the total energy density of a binary system, and regarding gravity radiation, it is very important because orbital binding energy is the sole source of gravitational waves.
Total energy density, of the orbital energy field, decreases in the direction of time, and oscillation of change in magnitude of orbital binding energy losses during the orbital period will cause an oscillation in energy density in the direction of time, with an oscillation period, equal to orbital period, of only 7.7522 hr ( T=27908 ± 7 sec ). During the orbital period the radiated punctual energy go on periastron 8,04325 1025 watts, to apastron 5,98189 1022 watts; newly, on periastron 8,04325 1025 watts to apastron 5,98189 1022 watts. This oscillation is reproduced in each orbital period and is transmitted to the metric tensor Gαβ, according to Einstein’s equation Gαβ = 8ЛG/c2Tαβ.
Oscillation of energy density, – T00 (8), a component of the tension-energy tensor, Tαβ, produces an oscillation of -ct coordinate, of metric tensor Gαβ. Gravitational radiation, -T00, is the orbital energy per punctual events in spacetime, in a similar way as the static electric field contributes to T00 in flat space (8). So, -ct coordinate oscillates within the lapse -(ct - ε, ct + ε).
Hulse and Taylor
discovered and measured time oscillations in pulsar PSR B 1913+16. Magnitude of
loss of orbital energy is directly related to time contraction-expansion, since
if TSij > 1, time expands, and if TSij < 1, time contracts.
Here oscillation of time means exactly the oscillation of the third dimension of the four dimensions of the spacetime. She oscillates together to the oscillation of the other two dimensions according to the discovery of Einstein. Since, in the spacetime the four dimensions mix for different observers (Clifford, 1999) and the time does not exist in itself.
Within a local region, with one source of gravitation, through a given event in spacetime, the diagonal components, of Tαβ, give the magnitudes of energy density “σ”, and the Px, Py, Pz pressures, which flow from “α” to “β” direction, where “σ”, “β” = -ct, x, y, z (22).
Since the gravitational radiation carries energy of the sources of the gravitational field (here the source is the pulsar PSR B 1913+16) to the asymptotical regions (Page 387, 12) and according to author find, in the linearized gravity model, gμν = (ημν + hμν) hμν | << 1, in the flat spacetime of an asymptotic region, in the global vacuum spacetimes (Page 386, 12), for + polarity, hμν is:
٤ 0 0 0
0 ٤ 0 0
0 0 -٤ 0
0 0 0 0
hμν shifts the –ct axis in the span –(ct, ct + ε), the x axis in the span (x, x + ε), and the y axis in the span (y, y + ε).
-hμν shifts the –ct axis in the span –(ct, ct - ε), the x axis in the span (x, x- ε), and the y axis in the span (y, y - ε).
± hμν causes oscillations of –ct axis in the span –(ct - ε, ct + ε), of x axis in the span (x - ε, x + ε), and of y axis in the span (y - ε E, y + ε)
Hence, linearized gravitational waves are oscillations of the three orthogonal axes -ct, x, y, of spacetime, which propagate in all directions z, in 5D (-ct, x, y, z, u).
Therefore, the equation of gravitational waves in 5D is:
(▼2 - 1 ∂2 ) hϊϋ = 0 and ϊ = (1,5), ϋ = (1,5)
In 1916, Einstein utilized his equations on gravity fields in order to establish the theoretical frame of gravitational waves. He used mathematics, not for measurement purposes, but as a tool, a method, to produce theory and knowledge. As a legacy, he left the foundations that allowed physicists to reach the mathematical concept of gravitational waves. He totally discarded his previous belief, that “thought and ideas, not formulas, are the bases for all physical theories.” (23).
General Relativity actually
considers as a four dimension event the geometrical effects of gravitational
waves in five dimensions.
waves cannot be experimentally discovered and measured, since it is physically
impossible to reach the five dimensions, starting from four dimensions. However, the partial reconfigurations of spacetime will propagate in 4D, for example, when the dimension of the time, according to a observer, does not intervene. These partial spacetime waves it can detect.
In physics, the main problem concerning a change from four to five dimensions consists on its philosophical consequences and implications.
In the Theory of Everything, which tries to unify all forces and particles in nature, points of the Standard Model are replaced by strings and gravity is explained under Planck’s scale, as having 5 or more dimensions, being only 4 detectable. In particular, in the 5 dimensional anti De-Sitter spacetime, brane world RSII model, “the fields of the standard model are confined to a brane, while only the gravitational field propagates in all the dimensions called the bulk”, “our four dimensional spacetime is embedded in the five dimensional anti De-Sitter(AdS5) spacetime with curvature scale l, and the four-dimensional gravity is effectively recovered in larger scales than l.˜ , where l ≤ 1 mm . In general, if gravity has M dimensions, these extra dimensions really fade, due to their smallness, into the 4 four dimensions. So, gravitational waves propagate in M+1 dimensions, and because of the Principle of Correspondence in the Macro cosmos, gravity has four dimensions and gravitational waves propagate in 5D.
A proper methodological order should consist on starting from elaborating a new conception of the world and, then, a new physical conception of the world. “Thought and ideas are the bases for all theories.”
Professor Gregorio Portilla. Astronomer. Universidad Nacional de Colombia.
Translation into English: Arry Constantin Casas.
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