Spacetime and Gravitation Based on Logic, INSTEAD OF Einstein Assumptions
The Assumption of Inertial Frames in Special Relativity
Lorentz spacetime is defined for uniformly moving frames to measure light at the same speed regardless of frame velocity. The problem is that
“uniformly moving” frames are pre-selected before Lorentz spacetime scales are defined to justify its uniformity. Likewise, light and faster-
than-light (FTL) mono matter waves also cannot be distinguished until Lorentz spacetime scales are defined. The remedy: the strict spacetime
without presumption would be based on the principle that all light and FTL (according to Lorentz scales) mono waves be measured at the same
speed, and include both inertial and non-inertial frames, with “uniformly” moving frames defined “as being measured by the spacetime scales
being defined”.
Elimination of the assumption: the generalized 4+1 spacteime
Not being pre-occupied with Lorentz spacetime (thus, no wave packets, no earth, no stars, etc., only mono waves recognized), just imagine
what would be the spacetime definition which accommodates “all” mono matter waves most elegantly. It is a 4+1 spacetime, as mono waves
have but one more degree of freedom, namely, its speed. Adding one extra dimension to accommodate this degree of freedom naturally levels
all mono waves to the same (light) speed. The 4+1 spacetime is the “inevitable” conclusion of special relativity if the definition of Lorentz
spacetime is executed rigorously. The 4+1 spacetime is the most natural and symmetric spacetime for both massless and massive particles
as all of their associated waves share the same massless wave equation in the 4+1 spacetime
[ ∂2/(∂x0)2 - ∂2/(∂x1)2 - ∂2/(∂x2)2 - ∂2/(∂x3)2 - ∂2/(∂xm)2 ] φ = 0 (1)
But when observed from the Lorentz subspace, the same waves appear as
[ ∂2/(∂xL0)2 - ∂2/(∂xL1)2 - ∂2/(∂xL2)2 - ∂2/(∂xL3)2 - m2 ] φL = 0 (2)
where mass is gained automatically due to change of scales. (Subscript L means under Lorentz measurements, while no subscript is under the
4+1 scales). Superscript m is for the extra dimension, xm. The transformation between them, for “each” mono wave, is
dxL1 = dx1 (3a)
dxL2 = dx2 (3b)
dxL3 = dx3 (3c)
dxL0
= dx0 · [(p1)2 + (p2)2 + (p3)2 ]½ / [(p1)2 + (p2)2 + (p3)2 + m2]½
≤ dx0 (3d)
dxLm = 0 (3e)
Since Lorentz time scale is shorter than the 4+1 time scale, eq. (3.d), it measures “the same” mono wave as FTL. Just like SR uniting the
absolute Newtonian time and space to form a 3+1 space-time continuum, the 4+1 spacetime unites mass with Lorentz spacetime to form a 4+1
space-time-mass continuum, as demonstrated in eq. (1). Eq. (1) is also more beautiful than (2) as xm is treated on the same footing as all other
spatial dimensions. The 4+1 spacetime doesn’t contradict the verified Lorentz spacetime, but only expands it. It makes no sense to rule out the
FTL mono waves at the beginning as they are “not” faster than light in the more natural (inertial-frame-assumption-free) 4+1 spacetime.
Instead, what should be removed is actually the assumption of inertial frames. (Note: since in the real world we feel only one time flow, the
various Lorentz times for different waves are synchronized and spatial dimensions are expanded.)
The philosophy behind
SR and GR have reached the deepest quest for the nature of mass. We have to define inertial mass, gravitational mass and the equivalence
between them through the definition of space and time. Correct understanding will not emerge without correct definition, which usually is the
one which renders physics in the simplest and most elegant form. Before all these quantities are defined, we should just let physics define them
without imposing any assumptions. However, presuming knowledge of uniform frame velocity, we inadvertently closed the door of
uncovering the real nature of mass (thus “forced” mass to be “intrinsic”). Put differently, based on the energy-momentum-mass relation of SR,
(E)2 – (p1)2 - (p2)2 - (p3)2 - m2 = 0, space (inverse of energy), time (inverse of momentum) and mass can be defined. But there are 5, not 4,
quantities to be defined. All the 5 should be defined simultaneously, none should take precedence over the other. But Lorentz spacetime is
defined prior to mass by electromagnetism, and hence the “real nature of mass (i.e. nature of inertia and gravity) cannot be revealed”.
(Lorentz spacetime is defined based on a special case (m=0) of equation (2), while the 4+1 spacetime is from the general case, equation (1).)
Even though GR attempted to incorporate gravity later, it is still Lorentz spacetime oriented and the real nature of gravitation is already lost.
To have mass involved in equal status, both light and FTL (massive) waves must be used in its definition, as suggested by Lorentz and agreed
by Weyl. This leads unambiguously to the elegant 4+1 space-time-mass continuum, which has “built-in equivalence of inertial and
gravitational mass (i.e. the real essence of gravitation)”, and hence, if interpreted appropriately, the simplest and most elegant gravitation
theory should come out of it.
Mono waves (and hence the 4+1 spacetime) are the more fundamental part of matter than packet waves. How can one hope to uncover the real
Nature by ignoring the more fundamental part of it? Since packet waves are superpositional composite of mono waves, physical theories based
on packet waves and the 3+1 Lorentz spacetime (e.g. the standard model or General Relativity) are like mechanical laws extracted from
biological observations, which can hardly be simple and obvious. It is believed that only theories (whether it’s particle physics or gravitation)
based on mono waves and the 4+1 spacetime can touch the real fundamental essence of nature and hence able to reveal it in a simple and
obvious way.
While this may be a profound change not easy to adjust over night, we must reiterate before proceeding further that the most natural and
symmetric spacetime for objects with (inertial or gravitational) mass is the 4+1 spacetime, not the Lorentz spacetime, which is only a
subjectively selected subspace of the 4+1 spacetime. Hence, the most elegant and obvious particle theory should be based on the 4+1
spacetime, rather than on Higgs Mechanism and the most elegant and obvious gravitation theory should also be based on the 4+1 spacetime,
rather than General Relativity. It should be stressed that this is not because new concepts are imposed, but because the same spacetime
definition for Lorentz spacetime is carried out rigorously with subjectively imposed assumptions removed. That is, if special relativity were
introduced after quantum mechanics and mono matter waves could not be excluded, the same spacetime definition would result in the 4+1
spacetime instead of 3+1 Lorentz spacetime. This is the only logical conclusion which yields just perfect answer to quantum gravity and
unification with quantum mechanics as to be shown later.
4+1 universe and parity violation
This immediately leads to a 4+1 universe (the extra dimension, xm, is external, just like other spatial dimensions) with the universe we live in being a curved 3+1 manifold, most likely the curved 3-surface of a 4-sphere, in the flat 4+1 spacetime. If this were true, the same quasars (or whatever objects sitting at the other end of the spherical universe) would likely be observed from both opposite directions in each of the 3 dimensions of the universe (just like two persons traveling from north pole in opposite directions at equal speeds will eventually meet at south pole.) Actually, the numerous double quasars may already serve as partial evidence, where the whole spherical universe serves as the lens. The scarcity of galaxies immediately before the quasar region might also reveal the fact that the 3-volume near the quasar region is actually quite small, as conjectured by this model. In addition, the extension of electromagnetism to the 4+1 spacetime naturally covers parity violation and weak interaction.
The Assumption Of Consistent Proper Time In General Relativity; Gravity From The 4+1 Spacetime And Unification With Quantum
Mechanics
Curvature from stress-energy of uniform mass and uniform momentum
In defining spacetime scales over a curved manifold, general relativity lets proper time act as the standard yardstick and assumes it be consistent throughout the universe. The problem can be seen as follows. Einstein equation is written as
G = 8πGT (4)
Consider the curvature generated by the stress-energy of a single type of particles with uniform mass m and uniform momentum p (and
E = (p2 + m2 )½ ). As stress-energy tensor of such a curvature can be decomposed as a product of two 3+1 vectors,
T(E,p) = (E,p)×(E,p)/[V(E2 – p2 )½ ] (5)
Einstein curvature tensor can also be decomposed in the same way,
G(T,X) = (T,X)×(T,X)/[V (T2 – X2 )½ ] (6)
Hence Einstein equation can be written as
(T,X)×(T,X)/[V(T2 – X2 )½ ]
= 8πG (E,p)×8πG(E,p)/[V 8πG (E2 – p2 )½ ] (7)
Thus we obtain a 3+1 “vector” equation,
(T,X) = 8πG (E,p) (8)
While there are complicated mathematics with some parameters to be set between the metric gμν (distance and time units) and the vector (T,X), it is
reasonable to assume these parameters should be set to make local time unit dξ0 and distance unit dξ be proportional to nothing but T and X of the vector
(T, X). Let’s call this the “Assumption of Local Space and Time Units”. Let
dξ0 = kT (9a)
dξ = kX (9b)
Hence,
(dξ0 , dξ ) = (kT,kX) = 8πkG (E,p) (10)
From (10), the local proper time unit can be derived as
dτ = [(dξ0)2 - |dξ|2 ] ½ = k [T2 – X2 ]½
= 8πkG [E2 – p2 ] ½ = 8πkG m (11)
which clearly depends on the mass behind local stress-energy. In other words, Einstein equation itself dictates proper time variation according
to underlying masses (at least in this case). If a neighboring curvature is generated by a uniform-mass-uniform-momentum stress-energy of mass
twice as big, then its proper time unit should also be twice as big. Assuming the same proper time units would inadvertently reduce all 4-
vector and mass values by half for the neighboring locality. Thus, Einstein equation as currently understood is self-contradictory. This is
why 4-vectors and masses cannot remain constant after parallel transport. Even if the “Assumption of Local Space and Time Units”, eq.(9),
were wrong, it wouldn’t help; it only hides the problem, as 4-vectors and masses still cannot remain constant after parallel transport. The
“Assumption of Local Space and Time Units” only manifests the hidden problem. This also hints at the cause of dark matter and flat universe
mysteries.
Adjustment
The correct parameterization would be obtained by admitting dependence of local proper time on underlying masses and adjusting equation
(7) according to eq. (11),
[(T,X)×(T,X)/(dτ)2 ]/ [V(T2 – X2 )½ /(dτ)]
=[8πG(E,p)×8πG(E,p)/(8πkGm)2]/[V8πG(E2–p2)½/(8πkGm)] (12)
No more mass and 4-vector distortion under parallel transport. Without the adjustment, (7) is applicable only locally, which is the only
situation GR can be proved correct (as was done in the solar system). For a swarm of particles, each term must be adjusted separately before
being summed up,
∫dX Σ [(T,X)×(T,X)/(dτ)2 ]/[V(T2 – X2 )½ /(dτ)]
T
=
∫dp Σ [8πG(E,p)×8πG(E,p)/(8πkGm)2]/[V8πG(E2–p2)½/(8πkGm)] (13)
E
Notice that the summation over mass is now shifted to that over energy. Because mass is not an independent variable of Einstein tensor, thus
must be consistent, otherwise both energy, momentum and mass become independent variables, which are more than actually exist. On the
other hand, Einstein equation without adjustments,
∫dX Σdτ (T,X)×(T,X)/[V(T2 – X2 )½ ]
= ∫dp Σm 8πG(E,p)×(E,p)/[V(E2 – p2 )½ ] (14)
is a summation of oranges and apples, i.e. over terms of different masses and space/time scales (i.e. proper times, dτ’s), which is a real mess.
One can see the problem in Einstein tensor as currently understood is real serious, not only between different localities, but also is a mess
within itself at one single locality. Claiming Einstein tensor as currently understood being non-linear and non-decomposable, or claiming the
“Assumption of Local Space and Time Units”, eq.(9), being wrong cannot be an excuse and is irresponsible, as these claims are independent
of the mess in Einstein tensor and an arbitrary Einstein tensor is still not based on a consistent proper time (or mass) no matter what, and mass
and 4-vectors still distorts after parallel transport. They only hide the problem, which eventually shows up in the persistent failure of quantum
gravity, unfound graviton, gravitational waves and dark matter, etc. It is highly doubtful that gravity will be quantized under string theory or
other approaches, as they miss the point. It is said Einstein equation is beautiful. But what good does it do if mass and 4-vectors don’t even
conserve after parallel transport? Since Einstein equation has been verified in the solar system, we may claim that the validity of Einstein
equation is limited to one single locality for one single mass without summation.
Curvature vector equation
Even though the adjusted equation (13) solved the problem of scale distortion and messy summation of oranges and apples, it still suffers from
the problem of nonlinearity. Since none of the experimental tests of general relativity verifies that gravitation must be nonlinear, we are tempted
to cautiously introduce a fairly close linear equation which is valid for non-local scales and with multiple mass summations but approaches
Einstein equation locally in the solar system. It is Einstein’s highly intellectual insight to envision curvature in gravitation, but at non-local
scales it should probably be expressed in the language of 3+1 curvature embedded in a “flat” 4+1 spacetime, rather than in terms of Riemann
geometry. As mentioned earlier, the 4+1 spacetime has built-in equivalence of inertial and gravitational mass, hence the simplest and most
elegant gravitation theory, the following is one possible attempt based on the 4+1 spacetime. This linear theory from the 4+1 spacetime should
produce the same result in the solar system, but produce a different and much better result at large scales, like in galaxies and clusters of
galaxies, thus solving the dark matter and flat universe problems (in addition to quantum gravity and unification with quantum mechanics).
Instead of eq. (13), we adopt the closest linear approach to it, i.e. its basic ingredient, the vector equation
(T,X) = 8πG (E,p) (15)
Since inertia is manifested in each mono wave by the amount it’s faster than light, gravity should also be manifested in each mono
wave. Parallel to the superpositioning concept of quantum mechanics, each mono wave in the 4+1 theory also contributes its part
to the spacetime definition of the manifold. Thus, corresponding to a wave packet
Ψ = ∫dp Σ f(E,p) exp[-iπ(x0p0-x∙p)] (16)
E
there is the superpositioned “total curvature vector” (as opposed to curvature tensor)
∫dX Σ f(E,p)[(T,X)/(dτ)] / [V(T2 – X2 )½ /(dτ)]
T
= ∫dp Σ8πGf(E,p) [(E,p)/(8πkGm)] / [V8πG(E2–p2)½/(8πkGm)] (17)
E
Notice that left and right sides have their proper time and mass adjusted, i.e. summed up on consistent mass and proper time to avoid summation
over apples and oranges, with
dτ = 8πkG m (18)
This gives a genuine unification of gravitation with quantum mechanics. The mathematics in this theory is extremely simple, but, just like
electromagnetism, that is what the right theory for such a fundamental force should be.
4+1 Quantum Gravity, Gravitational Waves And Gravitons
The most fundamental form of gravity, eq. (15), can be combined with (18)
(T,X,dτ/k) = 8πG (E,p,m) (19)
One sees the proper time, dτ/k, in eqs. (18) and (19) are just the component of the extra dimension xm, because
(∂ξ0)2 - (∂ξ1)2 - (∂ξ2)2 - (∂ξ3)2 - (∂ξm)2 = 0 (20)
and
dτ ≡ [(dξ0)2 - |dξ|2 ] ½ = (∂ξm) (21)
Thus, we have the most elegant vector equation of gravitation in the 4+1 spacetime,
(T,X,dτ/k) = (T,X,xm) = 8πG (E,p,m) (22)
The linear equation (22) will generate gravitational waves in the 4+1 spacetime
Ψ = exp[-iπ(x0p0- x1p1- x2p2- x3p3- xmpm)] (23)
which are observed faster-than-light in the Lorentz spacetime as
ΨL
= exp[-iπ(xL0pL0 - xL1pL1 - xL2pL2 - xL3pL3 )]
= exp[-iπ(xL0pL0 - xL∙pL)] (24)
In other words, gravitational waves are but the FTL mono matter waves being denied all the time, and gravitons are just all elementary particles
formed from the mono waves, which are always observed but never recognized as gravitons. It is conjectured that cosmic rays could just be
gravitons emitted as gravitation radiations by astronomical bodies.
Dark Matter and Flat Universe
Unlike in Einstein theory where only local Lorentz scales is available for measurement, the 4+1 theory uses one consistent 4+1 scales throughout