“GRAVITATIONAL TIME DILATION EXPLAINS HOW
RELATIVISTICALLY ACCUMULATED MASS–ENERGY- IS DARK MATTER”
by Jose Gregorio Baquero
In a similar way of how relativistic length contraction explain electromagnetism, relativistic effects explain the observations evidencing the existence of Dark Matter around galaxies and galaxy clusters. When energy passes closer to galaxies it “slows down” due to gravitational time dilations. This decrease in speed compared to the speed that energy has when travelling farther away from galaxies cannot be perceived by observers within the frames of reference closer to the center of galaxies. Ultimately the accumulative effect is a substantial increase in energy density when seen from a frame of reference far from the center of galaxies and that increased energy density is imperceptible for an observer within the galaxy’s inner frames of reference. The higher concentration of travelling energy around galaxies in effect has an equivalent relativistic mass that is creating extra gravitational distortion in space-time. This extra gravity is at least a substantial part of the unaccounted gravity attributed to Dark Matter.
From the beginning of existence energy’s own movement created and still is creating new fabric of space-time. Every unit of volume of space in the universe is being crossed from all possible spatial directions by traveling energy in the form of neutrinos, electromagnetic waves, gravitational waves, cosmic rays, stellar winds, emitted vacuum energy and all other kinds of waves and particles yet to be discovered. The lines traced by each quantum of energy (world lines) can be seen as fibers that intersect themselves in a way that they weave space-time fabric. The “fibers” of this Energy Felt get denser around galaxies and galaxy clusters. Any field studied by quantum mechanics is a manifestation of the Energy Felt in a particular range of frequencies of the energy spectrum.
Using a thought experiment this document proves that traveling energy (neutrinos, electromagnetic waves, gravitational waves, cosmic rays, stellar winds, emitted vacuum energy and all other kinds of energy and particles yet to be discovered) is getting relativistically accumulated around galaxies and galaxy clusters and must be considered as an important factor on the calculation for Dark Matter’s mass.
Explaining “Observed” Dark Matter using General Relativity
Let’s imagine two identical laser pointers, Pa and Pb. Pa is going to be pointing in that way that its emitted photons (A) will pass parallel to Andromeda Galaxy galactic plane, at a minimum distance of one radius of Andromeda (Ra). Similarly Pb is pointing perfectly parallel to Pa in a way that its emitted photons (B) will pass at a minimum distance of 10 times Andromeda Radius (10Ra), for simplicity let’s omit gravitational lensing distortion. We set two light detectors, Da and Db at the other side of the galaxy in order to catch the photons emitted by Pa and Pb respectively.
General Relativity tells us that there is a different gravitational time dilation on the two different frames of reference; the frame of reference at Ra distance to the galactic plane and the frame of reference at 10Ra. For a rough calculation let’s use Andromeda’s mass including Dark Matter and calculate time for the two frames of reference at the moment when each photon gets to its closest point to the galaxy’s center.
There is a gravitational time dilation ratio (Ta/Tb) of 100.002% at the instant when the photons are closest to the center of the galaxy.
If time dilation for photons emitted by Pa were to remain constant along the 220.000 light years across the galaxy the photons hitting Da would take 161 days more than the photons hitting Db as measured by a clock on Db , but this is not the case. The gravitational field varies through the photons voyage.For all photons traveling on a plane parallel to the galaxy disk at Ra distance the ratio (Ta/Tb) decreases when the photons are farther away from the center of the galaxy depending on the actual mass-energy distribution of the galaxy. This is better visualized by a tridimensional distribution curved surface.Generalizing the time dilation ratio for two different radii (r, Ȓ) we can have:
Average Time of Relativistic Accumulation for a Static Frame of Reference at R
To calculate an average of time dilation ratio as measured on a static frame of reference we will integrate spherical surfaces (onion layers) multiplied by time dilation ratio while the radius varies and our product will be divided by the volume for Ȓ. Our approximation will use mass including Dark Matter as if it were inside an hypothetical black hole our integral will be calculated starting at Schwarzschild radius (rs) all the way to a non orbital (static) radius (Ȓ ).
The average time of relativistic accumulation for a sphere with r=Ȓ as measured on a static frame of reference:
This means that on average a traveling photon (boson or any relativistic particle) will stay at least 27 days longer inside the sphere with the radius of the galaxy as measured on a static frame of reference at R
Average Time of Relativistic Accumulation for an Orbiting Frame of Reference
If we calculate the average of time dilation ratio as measured from a orbiting star (frame of reference orbiting the galaxy) we will have to use the time dilation for a circular orbit for a given orbital radius (R ), a constant, as our denominator and for our numerator we will have the integral from the Schwarzschild radius to orbital radius ( R) of all spherical surfaces (onion layers) multiplied by time dilation ratio (non rotational) while the radius varies. The result will be divided by the volume for R. This approximation also will use mass including Dark Matter as if it were inside an hypothetical black hole.
This means that on average a traveling photon (boson or any relativistic particle) will stay about 1.76 seconds longer inside the sphere with the radius of the galaxy as measured on a orbiting frame of reference at R
Energy Felt Density Needed to Account for Dark Matter
Let’s imagine the Volume of Accumulation (U) by multiplying the surface of the sphere with radius (R ) times the distance a photon travels in the time of accumulation. The before mentioned Volume will need to have an Energy equivalent mass equal to the “observed” Dark Matter mass calculated for that particular orbital radius (R ) using the orbital velocity (V). Therefore the Energy Felt Density ( ) will be calculated by dividing the energy equivalent for Dark Matter Mass by the Volume of Accumulation.
Energy Felt Density calculated for a static frame of reference:
Energy Felt Density calculated for an orbiting frame of reference:
Model Test with Rotation Velocity Curves
The rotation velocity curve for an static frame of reference using the Energy Felt density calculated in that case did not follow curves as observed; in fact it followed, with smaller gradient variation, the expected”Keplerian” curve.
A completely different picture is plotted with the rotation velocity curve for the orbiting frame of reference using the calculated Energy Felt density (1.40*10^7 joules/cubic meter).Graph 1 shows how the calculated values for the rotation velocity on orbiting frames of reference follow the observed curves that evidenced the Dark Matter extra gravity.
Calculated Rotation Velocity Curve for an Orbiting Frame of Reference
Ω=1.40*10^7 joules/cubic meter
x Axis= Radius in Andromeda Radius, y Axis =Velocity in 100,000 m/s
Gravitational Lensing and Rotation Curve Discrepancies
When an observer on Earth calculates Dark Matter mass by measuring the bending of light caused by Gravitational Lensing there is a discrepancy with the calculated Dark Matter mass calculated from rotation velocities. While the rotation velocities calculation uses the orbiting probe at radius R (stars, dwarf galaxies, etc) frame of reference time dilation; the calculation from gravitational lensing uses the observer’s orbing frame of reference time dilation. This is needed since the observer becomes a part of the system (e.g. moving observer-rainbow system).
Discussion and Conclusion
The Dark Matter phenomenon explained as the Relativistic Accumulation of Mass-Energy allows us to mathematically calculate an approximation to Energy Felt density for a first time. No other phenomenon had given us the most remote chance to observe a quantifiable manifestation of this energy. Current Vacuum Energy Density (10^-9 joules/cubic meter ), calculated from universe expansion observations, differs greatly with predictions from quantum electrodynamics (QED) and stochastic electrodynamics (SED) where it is required a value 120 orders of magnitude larger (10^113 joules/cubic meter). Our calculated Energy Felt Density values indicate that there is a much higher energy density across the universe and that the observed values correspond only to the resultant of positive and negative (e.g. potential, gravitational) energy density interactions.
Cosmologists agree on an inflationary period in the early universe where similar energy (in nature but not in magnitude) to that of observed Vacuum Energy rapidly inflated the universe exponentially in a very short period of time. It is currently believed that most of the inflaton field energy got transformed into other types of radiation and matter particles which density were diluted by further expansion of the universe. It is not completely ruled out that the remnant of that energy is still been created across the universe in ways that are very difficult to measure experimentally although Casimir Effect experiments have demonstrated the existence of such energy. Some cosmologists believe that a small oscillation on the current Vacuum Energy value could be responsible for the creation of Dark Matter.
As an analogy, we could think of the universe as the proverbial inflating balloon which surface represents our spatial dimensions. Our balloon now is a special one; it is made out of latex fibers that when stretched won’t break. Rather, the fiber will separate slightly allowing the air to escape through the miniscule holes. With an initial colossal air pressure the balloon started inflating at an exponentially accelerated rate while all the air was contained (when energy density was as high that energy acted effectively onto itself). During the inflation period there was very little cooling in the universe since all new space created had new intrinsic energy created with it. There was matter formation; that is there was creation of elementary particles but they rapidly decayed back to energy because of the high temperature and continuous energy bombardment. There is the point when the balloon latex fibers got stretched and very little air started to escape. The universe started to cool down at a greater rate allowing for a big part of the remnant energy to transform into elementary particles like neutrinos, electrons and quarks to form and after it photons were able to travel through space. The balloon rapidly slowed its inflation since not only the air pressure diminished but at the same time the remnant pressure was less effective at inflating it. This analogy illustrates how nowadays the universe can have a higher vacuum energy density than that calculated from current universe expansion rates.
Also, it is currently accepted by the scientific community that neutrinos, specially primordial neutrinos, make out at least 10% of Dark Matter (Hot Dark Matter). This estimate is not taking into account the mechanism described on this paper. A bigger contribution has been ruled out because neutrinos are traveling at relativistic speeds; that is higher speed than any escape velocity (at any possible orbit). The phenomenon here described- Relativistically Accumulated Mass-Energy- can easily explain a significant accumulation of Hot Dark Matter not previously accounted by any calculation previously performed.
A star orbiting at Ȓ distance from the center of the galaxy will “feel” the combined effect of the gravity generated by the baryonic mass and the relativistically accumulated mass-energy inside that sphere. All the mass (baryonic and relativistically accumulated mass-energy) located outside that sphere is also “felt” but its gravitational net effect cancels out. This is similar to calculations for an object falling through a hole towards the center of a planet where its acceleration only depends on the mass of a sphere with a radius equal to the object distance to the center of that planet. Furthermore, we do not have observations of the gravitational effects of Dark Matter taken outside of the Milky Way and time in our frame of reference is dilated differently to time at a point far away from the Milky Way. Our observations therefore should show differences with those of an observer outside our galaxy.
In the space between galaxies energy density gets higher than that of the surrounding space because of both galaxies’ gravity compounded effect. These structures connect galaxies and galaxy clusters together forming the Cosmic Web. Dark Matter concentration ratio is different for different galaxies and galaxy clusters, not only because differences in mass distribution but also because galaxies have different masses. In addition, younger galaxies do have less Dark Matter in proportion to regular matter since the mass-energy relativistic accumulation has a compounding effect over time: The higher the energy density ratio; the more mass-energy relativistic accumulation, the more mass-energy relativistic accumulation; the more gravity, the more gravity; the more mass-energy relativistic accumulation. Even though in the early universe energy density was higher, that density was more uniformly distributed. That means that the further away we look to younger galaxies, the less Dark Matter we should find. What causes the mass-energy relativistic accumulation is the difference in energy density close to the galaxies in comparison with that of the galaxies’ surrounding space.
Traveling Energy particles and waves that come from outside the galaxy are not rotating with the galaxy. They will pass by if unobstructed. The time dilation on the frames of reference when close to the galaxy is greater specially if compared with orbiting frames of reference at a greater distance . Similarly to how relativity (length contraction) explains electromagnetism, relativistic particle and waves density is higher in the frames of reference closer to the galaxy than those far from the galaxy. In a way, there is a reservoir of traveling energy around galaxies that is constantly renewed with new energy.
It is this paper conclusion that there is no need for particularly special particles forming Dark Matter substance or even parallel universes’ escaped gravity to account for the extra gravity existing around cosmic structures. In a way, Dark Matter is just traveling energy. The purpose of this paper is to demonstrate the mechanism for mass-energy relativistic accumulation around galaxies and calculation for actual Dark Matter accumulated by this phenomenon will need to be performed using more precise models. Einstein was right and his findings continue to enlighten our understanding of the universe.
1 Einstein, A. “Relativity : the Special and General Theory by Albert Einstein”. Project Gutenberg. 2
2 Einstein, A. (1905), “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?”, Annalen der Physik, English Version