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The Alternative Universe AltU: The Alternative Universe (Draft as of 2 February 2021) Ronald Ian Miller Abstract The physics of the conventional description of the universe are founded on the equations of special and general relativity. In special relativity theory an observer perceives that moving clocks run slow, and that moving objects shrink in the direction of their motion. In general relativity theory clocks slow and space expands within gravitational fields, and in the empty regions of the universe space expands continuously while time remains stable. The equations that describe these phenomena work well, but they provide no insight into an underlying physical process or processes. This paper postulates an alternative version of the universe, and a single physical process that underlies relativistic phenomena. This version of the universe has unchanging spatial distances, but a variable “speed limit” that determines the local speed of light and the local temporal rate. The combined gravitational fields of all the masses in the universe that affect a spatial location determine the local speed limit there. Prior to the start of the modern universe there was only energy, in the form of massless particles. There was no mass and no gravitational fields, and the speed limit everywhere was effectively infinite. At the start time for the modern universe mass particles formed from the energy, and their gravitational fields have been growing, at the (gradually-slowing) speed of light ever since that event, some 14 billion years ago. As a result, the gravitational intensity everywhere has steadily increased and the speed limit has steadily decreased. The alternative universe’s occupants observe cosmic redshifts due to the gradually reducing speed of light, rather than due to spatial expansion. The redshifts that we predict for that universe closely match the results of astronomical observations, despite the fact that the alternative universe is spatially stable and contains no dark energy. Where there are gradients in the gravitational intensity, the corresponding gradients in the speed limit underlie the gravitational accelerations of both masses and massless particles. Clocks, which work by counting cyclic motions involving masses, respect their local speed limit. Unlike the conventional universe, in the alternative universe the speed limit never drops to zero. The alternative universe’s black holes have no event horizons, and rather than being black they are just (very) red-shifted. Notation, Symbols and Equations Subscripts: We often embellish specific symbols with subscripts to attempt to clarify their meaning. There are three relevant identifiers: what, where, and from who’s perspective. Some common subscripts are: ∎0 Refers to the current universe. ∎푐 or ∎_푐 Refers to a “cosmic” observer’s perspective: an observer unaffected by local gravitational fields. Note that AltU’s observers are gender-less, and an observer is referred to as “it” where appropriate. ∎푙표푐 Refers to an observed location. ∎표푏푠 Refers to a local observer’s perspective, an observer experiencing the temporal rate of the observed object’s location. A local observer is assumed to be comoving with the cosmic observer. ∎푖, ∎푗, ∎표푏푗 Refers to a specific object. Numeric notation: powers of ten are shown using the “e” notation, so 1.23 × 109 is shown as 1.23e9. Vectors and matrices are identified by a bold font and an over-bar. The norm of a vector uses the same letter, without the bold font and over-bar. A unit vector is identified by a caret hat, e.g. 풗̂푗. List of Symbols 푎; 풂̅; 풂̅푗_푐 an acceleration; an acceleration vector; the acceleration vector of object j as observed by a cosmic observer. 2 2 2 2 푑 풙̅풔 푑 풙̅풔 푑푡푙표푐 푑풙̅풔 푑 푡푙표푐 풂̅풄 = 2 = 2 ( ) + 2 a local object’s acceleration, seen by a cosmic observer. 푑푡푐 푑푡푙표푐 푑푡푐 푑푡푙표푐 푑푡푐 푐 the SI value of the speed of light, 2.99792458e8 m/s. 푐푐(푡푐); 푐푐0 the cosmic observer’s local speed of light; its current value. 푐푙표푐 the speed of light at a specific location, expressed using its local time coordinate. 푐푙표푐_푐 the speed of light at a specific location, expressed using a cosmic observer’s time coordinate. 푐표푏푠 the speed of light at the observer’s location. 푐푠푡푎푟푡 the speed of light at the beginning (when 푡푐 = 0). 푑푠 an incremental spacetime separation. 푑푡; 푑푡표푏푠 a time increment experienced by an observer. 푑푡푙표푐 a time increment experienced by a local observer. 푑휏 a time increment experienced by an observed object. 푑휏/푑푡푐 an object’s temporal rate (compared to the cosmic observer’s rate). 푑푥푠 a spatial distance increment. i Notation, Symbols and Equations 퐺 the gravitational constant. 퐻(푡푐); 퐻0 the Hubble coefficient; its current value. 푚; 푚푛푒푎푟푏푦 a mass; a mass affecting a specific location. 푟 a spatial distance or radius. 푟ℎ(푡푐) the radius of the cosmic horizon. 푡푐; 푡푐0 the time coordinate for a cosmic observer; its current value. 푡푙표푐 the time coordinate for a local observer (typically in a gravitational field). 휏; 푑휏/푑푡푐 the time coordinate for a moving object (typically in a gravitational field); the “temporal rate” of the object. 퐺 푚 푈 the total gravitational intensity (or potential) at a point (푈 = 훴 = 푡표푡 푡표푡 푐2 푟 퐺 휌 ∫ 푑푣). 푐2 푟 푈 the local intensity or potential: the difference between the total intensity at an observed point and the intensity at a cosmic observer’s location. 푣; 풗̅ a velocity; a velocity vector. 푑풙̅풔 푑풙̅풔 푑푡푙표푐 풗̅풄 ≡ = a local object’s spatial velocity, seen by a cosmic observer. 푑푡푐 푑푡푙표푐 푑푡푐 풗̅풍풐풄 ≡ 푑풙̅풔/푑푡푙표푐 an object’s spatial velocity, seen by a local observer. 풗̅ ≡ 푑풙̅풔/푑푡 an object’s spatial velocity, generically. 풙̅풔 an object’s spatial coordinate vector. 푧 the redshift of an observed light or radio wave. 휵̅, 휵̅∥, 휵̅⊥ the gradient operator Del, its component parallel to an object’s velocity, its component perpendicular to an object’s velocity. 2 2 2 2 훾 the Lorentz coefficient, √1 − 푣푙표푐/푐푙표푐 or √1 − 푣푐 /푐푙표푐_푐. It can be calculated based on velocities observed from any inertial frame of reference. 1 0 0 0 [ 0 −1 0 0] 휼̅ The Minkowski metric matrix, 0 0 −1 0 0 0 0 −1 휃 the angle in a polar coordinates system. 휌 mass density. 휌푙푡 the long-term average mass density of the local universe. 휑 the conventional gravitational potential difference between an observed location and an observer. It reflects only the gravitational fields of masses that affect the observed location but not the observer. Typically, the observer is viewed as being in interstellar or intergalactic space and 휑 has a negative value. ii Notation, Symbols and Equations Within the paper the only equations that are numbered are those that are referenced elsewhere in the text. Numbered Equations Equation Section No. 푑푡푙표푐 푐푙표푐 푐푙표푐_푐 휑 ≅ ≅ ≅ 1 + 2 4.1.1 (4.1) 푑푡푐 푐푐 푐푙표푐 푐 2 푑푡푙표푐 푐푙표푐_푐 = 푐푐 ( ) 4.1.1 (4.2) 푑푡푐 2 푑휏표푏푗 1 푣푙표푐 = = √1 − 2 4.2 (4.3) 푑푡푙표푐 훾 푐푙표푐 2 퐺 푚푖 푈푗 = −휑푗/푐 = 2 훴푖_푛푒푎푟푏푦 4.3.1 (4.4) 푐 푟푖푗 퐺 푚푖 퐺 휌 푈푡표푡_푗 = 2 훴푎푙푙 = 2 ∫ 푑푣 4.3.1 (4.5) 푐 푟푖푗 푐 푎푙푙 푟 푑푡 푙표푐 ≅ 1 − 푈 4.3.2 (4.6) 푑푡푐 푐푙표푐 푑푡푙표푐 −푈 = = 푒 4.3.2 (4.7a) 푐푐 푑푡푐 푐 (푡 ) 푑푡 푐 푐 푐 −푈푡표푡(푡푐) = = 푒 4.3.2 (4.7b) 푐푠푡푎푟푡 푑푡푠푡푎푟푡 −2U cloc_c = cce 4.4.1 (4.8) 2 푣2 −푈 푑휏 −푈 푣푙표푐 −2푈 표푏푗_푐 2푈 푒 = 푒 √1 − 2 = √푒 − 2 푒 = 4.4.2 (4.9) 푑푡푐 푐푙표푐 푐푐 훾 푑풙̅풔 푑풙̅풔 푑푡푙표푐 −푈 풗̅풄 ≡ = = 풗̅풍풐풄 푒 4.4.3 (4.10) 푑푡푐 푑푡푙표푐 푑푡푐 ퟐ ퟐ ퟐ ퟐ 풅 풙̅풔 풅 풙̅풔 풅풕풍풐풄 풅풙̅풔 풅 풕풍풐풄 −2푈 −푈 풅푈 풂̅풄 = ퟐ = ퟐ ( ) + ퟐ = 풂̅풍풐풄 푒 − 풗̅풍풐풄 푒 4.4.3 (4.11) 풅풕풄 풅풕풍풐풄 풅풕풄 풅풕풍풐풄 풅풕풄 풅풕풄 −푈푡표푡(푡푐) 푐푐(푡푐) = 푐푠푡푎푟푡푒 5.1 (5.1) ( ) 푡푐 ( ) 푟ℎ 푡푐 = ∫0 푐푐 푡 푑푡 5.1.4 (5.2) 푐 (푡 ) 푧 = 푐 푒푚푖푡 − 1 = 푒푈푡표푡(푡표푏푠)−푈푡표푡(푡푒푚푖푡) − 1 5.1.5.2 (5.3) 푐푐(푡표푏푠) 푐푐̇ (푡푐) 퐻(푡푐) = − = 푈̇푡표푡(푡푐) 5.1.5.2 (5.4) 푐푐(푡푐) 퐺 푡 푡 푈 (푡 ) = 4휋 휌 [푟 (푡 ) ∫ 푐 푐 (푡) 푑푡 − ∫ 푐 푟 (푡) 푐 (푡) 푑푡] 5.1.7 (5.5) 푡표푡 푐 푐2 ℎ 푐 0 푐 0 ℎ 푐 4휋 퐺 퐻(푡 ) = 휌 푐 (푡 ) 푟 (푡 ) 5.1.7 (5.6) 푐 푐2 푐 푐 ℎ 푐 2 푑푈 2 풂̅풄 = 2 [푐푙표푐_푐휵̅⊥푈 − (푐푙표푐_푐훻̅∥푈 + ) 풗̅풄] ≡ 2[푐푙표푐_푐휵̅⊥푈 − 푈̇ 풗̅풄] (for photons) (6.1) 푑푡푐 iii Notation, Symbols and Equations 2 푟̇ 풂̅ = 2 [푐 휵̅ 푈 + U 풗̅ ] (for photons) 6.3.1.3 (6.2) 푐 푙표푐_푐 ⊥ 푟 풄 2 2 2 −2푈 2 2푈 dxs 2 2 dxs dτ = 푒 dtc − 푒 2 , i.e. dτ = dtloc − 2 6.5.2 (6.3) 푐푐 푐lo푐 2 2 −4푈 2 ̅ ̅ 푣푐 푑푈 푑푈 풂̅풄 = (푐푐 푒 + 푣푐 )휵푈 + [−4(풗̅풄 ∙ 휵푈) − (2 − 2 −4푈) (퐻(tc) + ) − ] 풗̅풄 (6.4) 푐푐 푒 푑푡푐 푑푡푐 2 ̅ ̅ dU 퐚̅퐜 = 2cloc_c훁U − [4(퐯̅퐜 ∙ 훁U) + 2 ] 퐯̅퐜 (for photons) 6.5.3 (6.5) dtc 퐺 푚푖 퐺 푚푗 푈푗 = 2 훴푖 = 훴푖≠푗푈푗_푖 + 푈푗_푠푒푙푓 , where 푈푗_푠푒푙푓 = 1.20 2 6.5.4.1.1 (6.6) 푐 푟푖푗 푐 푟푗 −푈 −푈푗 퐺 푚푖푒 푖 퐺 푚푗푒 푈푗 = 2 훴푖≠푗 + 푈푗_푠푒푙푓 , where 푈푗_푠푒푙푓 = 1.20 2 6.5.4.2.1 (6.7) 푐 푟푖푗 푐 푟푗 푑푈푗_푖 풗̅풊_풄∙풓̅풊풋 푑푈푖 = −푈푗_푖 [ 2 + ] 6.5.4.2.1 (6.8) 푑푡푐 푟푖푗 푑푡푐 iv Table of Contents Contents 1 Prelude: A Sample of What Lies Ahead ................................................................................................
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