Transverse Gravitational Redshift

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Transverse Gravitational Redshift: a previously-unmodeled relativistic gravitational phenomenon having a subtle measurable effect on time metrology and spacecraft telemetry Alexander F. Mayer • [email protected]

ABSTRACT The following formula rests on the Einstein equivalence principle; it is derived simply, in a few logical steps, guided exclusively by first principles. It predicts the existence of a symmetric redshift zδ between two points at identical coordinate radius r separated by a constant distance [δ ≪ r] in a static, spherically-symmetric gravitational field; the field absorbs a minute portion of a photon’s energy when the photon is transmitted transverse to the local gravitational gradient. Circa Q4 2015, textbook general relativity (GR), overlooks this clearly-implied relativistic phenomenon (see Fig. 1). The accuracy of the following formula is dependent on the accuracy of the approximation sin(δ/r) ≈ δ/r, which imposes the ‘small-distance’ restriction [δ ≪ r] on measurement.

2 g 2 δ µ δ µ zδ ≈ 3 2 = 2 [δ << r] • µ ≡ GM • g = 2 r c rc r This reveals an oversight in canonical GR: time is endowed with geometric properties that were not incorporated in the original presentation of the theory. When this omission is remediated, a symmetric time dilation is modeled between points at the same Newtonian potential. The following formula, which is derived from pure geometry arising from the principles of relativity, calculates an accurate value for the relativistic “transverse gravitational redshift” (TGR) effect between any two points in the weak ﬁeld, which ﬁeld assumes an idealized static point source. The symbol b is a conventional impact parameter. E(ϕ|m) represents an elliptic integral of the second kind; additional details are contained in this report. When evaluated between antipodes (i.e., b = 0), zt = 0.

⎡ ϕ2 ⎤ ⎧ ⎫ RS ⎛ ϕ ⎞ ⎤ ⎪ ⎡ π π ⎤ ⎪ 2µ zt = sec ⎢2 E ⎜ 2⎟ ⎥ ⎥ −1 ⎨ [b > 0] & − ≤ ϕ ≤ ⎬ • RS ≡ 2 ⎢ b ⎝ 2 ⎠ ⎥ ⎣⎢ 2 2 ⎦⎥ c ⎣ ⎦ϕ1 ⎦ ⎩⎪ ⎭⎪ Plugging in values for the realistic test case of a 1-km horizontal separation distance on the geoid (p. 21), these two radically different formulas agree to one part per billion (10−9):

Approximation: zδ(1 km) = 1.71503723208E−17 ……………… zδ(10 km) = 1.715037232E−15 Authoritative: zt(1 km) = 1.71503723031E−17 ……………… zt(10 km) = 1.715037056E−15 This feature of the mathematical physics provides a theoretical ‘proof’ that the predicted phenomenon is real. A quantitative survey of the literature reveals that it is routinely observed astrophysically; these observations were either unexplained or misidentified. Numerous university physics laboratories have the necessary off-the-shelf equipment and technical expertise required to run various possible experiments to empirically verify the predicted phenomenon conclusively at nominal cost. Unexplained anomalies in GPS can be attributed to the unmodeled TGR effect, which may be partially absorbed in estimated user equivalent range errors (UERE).

“We are not to tell Nature what she’s gotta be.” – Richard P. Feynman (1979)

Introduction Reliable first principles in physics, (e.g., the Einstein equivalence principle), are proven direct predictors of empirical phenomena, (e.g., the Einstein redshift). This report presents the prediction of a testable, previously unrecognized relativistic gravitational phenomenon implied by the Einstein equivalence principle. Two independent predictive formulas for the same phenomenon are presented: an approximation valid for ≲100 km horizontal distances on Earth, which rests on the Einstein equivalence principle, and a general formula applicable between any two points in the weak external field, which is based on the revolutionary concept of relativistic temporal geometry, which itself arises from first principles. The respective mathematical forms of the two equations are radically different and it is impossible to derive one equation from the other, yet they agree quantitatively to one part per billion in a test case that can be verified by dozens of university laboratories. This feature of the mathematical physics, alone, provides strong assurance that the predicted phenomenon is real. A quantitative survey of the professional literature reveals that it has been previously observed numerous times and in numerous circumstances dating back to the early 1900s when the center-to-limb variation of the solar wavelength was first reported.1 However, since the Einstein field equations and their practical physical interpretation as expressed by the parametrized post-Newtonian (PPN) formalism have been assumed to be an accurate gravitational model, and they failed to predict such a phenomenon, anomalous measurements that could not be explained within the context of conventional relativity have been typically attributed and quantitatively modeled by some ad hoc phenomenon (e.g., a ‘gravitational anomaly’ or ‘atmospheric effect’) that could be invoked without challenging long-accepted and widely-respected conventional thinking. Alternatively, the subset of such anomalous measurements reported in the literature were either summarily removed from the raw data as an inexplicable apparent nuisance, or they were simply ignored or ridiculed. A “gravitational equipotential surface” is a physical concept arising from pre-relativistic classical mechanics. According to the germane principles and mathematics, all points at rest on a spherical surface centered on an idealized gravitational point source are at the same energy potential. Moreover, according to Newton’s theory of universal gravitation, the gravitational field is conservative; for any closed path through the field, the net work (integrating force times distance) is zero. Similarly, modeling an ideal radar round trip of a photon in vacuum, there is no change in photon wavelength (i.e., photon energy) induced by propagation through gravitational fields. Originating with Newton, these two notions became axioms (i.e., propositions regarded to be self-evidently true) that were unchallenged with the advent of modern physics, which holds Newtonian physics to be a low-error practical approximation to more generally applicable relativistic mechanics.

1 Paul Marmet, “Redshift of Spectral lines in the Sun’s Chromosphere,” IEEE Tran. Plasma Sci. 17 (2), 238 (April 1989); < online > < online2 >.

© 2015 A. F. Mayer 2 ver. 15.11.05.Z17 The question arises as to whether these notions are in fact low-error approximations in the context of relativistic physics. Might a requisite but overlooked empirical relativistic effect exist that is subtle enough in the weak field to have escaped ready detection and distinction from various possible constituents of experimental error budgets? Even for experts in relativity theory, interpreting the cryptic mathematics of general relativity is notoriously difficult; a subtle error in the assumptions underlying such interpretation or an oversight of the relevant principles will result in an erroneous model. A conviction in the infallibility of the conventional textbook treatment of general relativity produced by a century of academic work in relativistic gravitational physics is judiciously subject to skepticism and reexamination; science is about challenging the status quo. In theoretical physics, simple qualitative models that rest on established first principles provide a “reality check” on the interpretation of quantitative models in the form of complex mathematical symbology. More importantly, such intelligible qualitative models serve to guide the derivation and evolution of quantitative mathematical models that accurately predict empirical phenomena. Such a quantitative predictive model must be empirically measurable with available technology, allowing the underlying model to be tested and either falsified or verified within evolving boundaries of experimental error. The Fig. 1 schematic depicts four ideal clocks (A – D) at rest in an idealized static, spherically-symmetric gravitational field that is isolated from all external influence and a fifth inertial clock (I ) free of gravitational influence at arbitrarily-large coordinate radius.

Scalar approximation† source of ‘tick’ pulses A clock at rest for the speed of light faster* relative to μ (over chords AB & AC r B * B:C relative A.tick arrival I dt # " & I rate per dtI v ! c%1+ 2 ( 0 r → ∞ $ c ' slo θ we − r* C Φ → 0 " AC < " AB < 0

) vAC < vAB (per the clock-I inertial-observer increasing µ ! = " reference: v ≡ c) N I μ ≡ GM potential e r w (Φ < 0) t o n ia n ‘e q † u Accurate expression i p π requires a tensor feld. ot en tia l su The speed of light between clocks rface’ Ideal clocks A – D are at rest in A & C is less than between A & B, an idealized static, spherically- which has relativistic implications. D symmetric gravitational feld. Fig. 1

© 2015 A. F. Mayer 3 ver. 15.11.05.Z17 The Schwarzschild metric is conventionally applied external to the source mass, where dτ represents local proper time (e.g., the time recorded by clock A) and dt represents coordinate time recorded by an inertial clock, here represented by clock I :

−1 2 2 ⎛ RS ⎞ 2 2 ⎛ RS ⎞ 2 2 2 2 2 c dτ = 1− c dt − 1− dr − r dθ + sin θdφ (1) ⎝⎜ r ⎠⎟ ⎝⎜ r ⎠⎟ ( ) For a static location (dr = dθ = dϕ = 0), the metric reduces to Eq. (2), which models the empirically-veriﬁed phenomenon of gravitational time dilation. The relative rate of ideal clocks at rest in a gravitational ﬁeld is related to the local gravitational potential, which is correlated to the coordinate radius r; clocks at lower potential (i.e., lower r) tick slower.

1 − 2 2 ⎛ RS ⎞ 2 dt ⎛ RS ⎞ 2µ dτ = ⎜1− ⎟ dt → = ⎜1− ⎟ • RS ≡ 2 (2) ⎝ r ⎠ dτ ⎝ r ⎠ c

According to this well-known predictive formula, ideal test clocks (A – D) all tick at the same reduced rate with respect to the ideal inertial reference clock I. This being the case, one might then jump to the conclusion that clocks A – D are all synchronous relative to one another. A simple example taken from special relativity suggests that such assumption of mutual synchronization of a set of ideal clocks, based on their individual synchronization with an ideal reference clock, is unwarranted and requires more careful consideration. In Fig. 2, each of the three ideal test clocks (A – C ) have the identical relative speed (|vα| = v) with respect to the ideal rest-frame reference clock. Accordingly, all three test clocks record local proper time at the same reduced rate with respect to clock I. However, since no two of the test clocks in are in the same frame of reference, it is clearly the case that no two of these clocks are synchronous with respect to one another, which is quantiﬁed by the non-zero Δv between them.

vA vB v v v v A A = B = C = B v dtI dtI dtI C ! = = C d! A d! B d! C

I Fig. 2 rest frame

Referring to Fig. 1, let the test clocks transmit the measured rate of local proper time by emitting a pulse of light per unit of local time (e.g., clocks emit 1 GHz pulses). From the point of view of an inertial observer referencing local clock I, the average speed of light integrated over the null geodesic AC is less than that over AB because the latter geodesic travels through a lower potential (i.e., space having higher average curvature). Ergo, in reference to clock I, the inertial observer records regular timing pulses sourced at clock A to arrive at clock C at a lower frequency than at clock B; within an arbitrary

© 2015 A. F. Mayer 4 ver. 15.11.05.Z17 interval of coordinate time Δt, more ‘clock ticks’ sourced at clock A arrive at clock B as compared to clock C. Because clocks B and C tick at the identical rate relative to clock I, it is also true that the local proper time interval between received clock-A timing pulses is less as measured at clock B than as measured at clock C. These unequivocal facts and the applicable symmetry imply a bilateral relativistic time dilation that is a function of the azimuthal angle (θ ) between ideal clocks at relative rest and at the same Newtonian gravitational potential. From the point of view of each of two such clocks, the remote ideal clock ticks at a lesser rate than the local clock. It is important to note that this relativistic time dilation effect is not a monotonically increasing function of the distance between the two clocks (i.e., the azimuthal angle θ between them). This is immediately evident from the symmetry between clocks A and D in Fig 1; considering the signal path, the magnitude of the gravitational blueshift to the center of mass is identically equal to the magnitude of the gravitational redshift from the center of mass to the other clock. Accordingly, antipodal ideal clocks (A, D) at the same coordinate radius in the Fig.1 schematic are synchronous with respect to one another. It is exclusively the component of a null geodesic that is transverse to the local gravitational gradient that produces the relativistic bilateral time dilation effect, “transverse gravitational redshift” (TGR). As the magnitude of this transverse signal-path component initially increases from zero as θ increases from zero and then returns to zero as θ increases to pi radians, the TGR effect between endpoints at the same coordinate radius is a function of the azimuthal angle between them, which function f (θ) is periodic over [0 ≤ θ ≤ π]: f (0) = f (π) = 0. After consideration of the forgoing argument, it is necessary to concede that a formal mathematical system that contradicts this prediction (i.e., historical conventional GR) must incorporate some error in expressing the constituent fundamental principles of the general theory of relativity. Moreover, prior empirical measurements and tests of GR that certainly measured the phenomenon, but failed to recognize it, employed procedures to obscure the anomaly so that reported results would be consistent with canonical theory.

Background It will be shown that the canonical model of relativistic time dating back about a century incorporates a very subtle modeling error (i.e., omission) for measurable phenomena. The prevailing conception of universal spacetime as a “3+1” manifold proves to be an incorrect general application of a geometric model that is valid only for a localized reference frame. The conception of spacetime as having one geometric time dimension is similarly false to the seemingly rational but primitive misconception of one geometric altitude dimension, parallel everywhere in three-dimensional space over a naïvely modeled ‘ﬂat’ Earth. Just as we experience the ‘altitude dimension’ (i.e., local vertical) of each distinct point on a sphere to be geometrically distinct, the time dimension of each quantitatively-distinct reference frame in relativistic physics must be similarly visualized, which is inherently in accord with the fundamental principles of relativity. By deﬁnition, ideal clocks that accurately record local time in such distinct reference frames are not synchronous due to relativistic effects. A key new idea presented here is the physical interpretation of this measurable phenomenon: although a differential clock rate is the familiar observable and it is clear that relativistic time is an empirical reality, such ideal

© 2015 A. F. Mayer 5 ver. 15.11.05.Z17 clocks having a relativistic relationship are not measuring local time at different rates per se. Rather, the more fundamental physical reality is that these clocks are measuring somewhat different orientations of time in a physically abstract four-dimensional manifold. “Physically abstract” means that no measurable quantity such as space or time can be attributed to any of the four arbitrarily-oriented manifold dimensions. There is no universal ‘direction’ (i.e., manifold dimension) of time; time is a strictly-local geometric distinction in “spacetime.” This seemingly abstract physical concept may not be intuitive upon initial consideration, but subsequent discussion will make it seem obvious. Similarly, the fact that the orientation of the local gravity vector is not globally parallel, which seems obvious today, is not naturally intuitive; for example, it remained obscure in China until the spherical Earth was introduced by European missionaries in the seventeenth century.2 In 1905 Einstein abandoned the false Newtonian concept of absolute universal time; the algebra expressed temporal relativity as a difference in the measurable rate of clocks. Some months prior to his untimely death in January 1909, Hermann Minkowski put forward a more robust expression of Einstein’s special relativity in the context of geometry (i.e., a metric).3, 4 What was overlooked then, and over the ensuing century, is that the fundamental physical interpretation of Minkowski’s mathematics concerns the relative orientation (i.e., geometry) of time for distinct reference frames in the four- dimensional spacetime manifold. The concept of distinct observers measuring time in an arbitrary number of directions in a four-dimensional manifold was as unintuitive in the twentieth century as was the concept of altitude having different directions in space for ancient Western cultures prior to the ﬁrst century of the Common Era. Although both ideas are actually simple and even empirically manifest, they are a radical departure from preceding ‘common sense’ thinking based on a naïve physical perspective. Accurate physical interpretation of the fundamental mathematics underlying relativity theory requires the respective time dimensions of distinct reference frames to be geometrically distinct. Minkowski’s mathematical formulation of Einstein’s revolutionary ideas concerning relativistic time implies that relativistic time involves relative geometry of geometrically-distinct time dimensions associated with distinct reference frames. Physically interpreted, the mathematics implies that these geometrically-distinct time dimensions are not collinear (i.e., parallel) in 4D spacetime. This idea of the progress of time having the similar property of orientation in a multi-dimensional manifold, as is true and familiar for arbitrary translation vectors in three-space, is highly non-intuitive because the common human experience of time is bidirectional; particularly in Western culture, we experience and model time as a linear phenomenon (i.e., a timeline), with time bifurcated into the past and the future by the elusive existence of the ‘present moment.’ The relativity of simultaneity informs us that this bifurcated ‘common sense’ linear model of time is a naïve perceptual illusion. A universally-linear model of time, which is practical for everyday human activity, is not generally valid in modern physics, which necessarily must deal with phenomena at vast and inﬁnitesimal extreme scales of time and space far beyond the narrow boundaries of common human physical experience.

2 Carlo Rovelli, “Science Is Not About Certainty,” New Republic (11 July 2014); < online >. 3 H. Minkowski, “Das Relativitätsprinzip,” Annalen der Physik 352 (15), 927 (1907/1915); < online >. 4 H. Minkowski, “Raum und Zeit,” Jahresber. Deutsch. Math.-Verein, 75 (1909); < online >.

© 2015 A. F. Mayer 6 ver. 15.11.05.Z17 Physics Let us first review the known phenomenon of Einstein redshift: Consider ideal clocks at rest in a gravitational field (Fig. 3). Due to gravitational relativistic effects, the measured rate of clock A as compared to an ideal inertial clock (I ) at radius r → ∞ from the isolated gravitational point source is correlated to the energy differential between the two clocks, which is quantified by the characteristic gravitational escape velocity (Eq. 3) at the location of clock A. The magnitude of the Einstein redshift (zE), which is correlated to the measurable differential clock rate, is accurately expressed in terms of this escape velocity (Eq. 4). Alternatively, a close approximation to the Einstein redshift (Eq. 5) in a weak (i.e., dg/dh → 0) gravitational field over short vertical distances (Δh ≲1 km) may be derived using energy conservation or the Einstein equivalence principle. Eq. (4) and Eq. (5) have distinct mathematical forms and it is impossible to derive one formula from the other.

I r → ∞ g → 0 inertial observer

dtI = f (vesc ) dtA $dg ' ! "h # 0 %&dh ()

A g h

gravity (v ) Fig. 3 esc

2µ v = • µ ≡ GM (3) esc r

dt 1 1 1 2µ z = I −1 = −1 = −1 = −1 • R ≡ E 2 S 2 dtA 2µ R c (4) ⎛ vesc ⎞ S 1− 1− 2 1− ⎝⎜ c ⎠⎟ rc r

g ⋅ Δh z ≈ (5) E c2 Using a purely mathematical process, the Eq. (4) prediction is typically derived from the Schwarzschild solution to the Einstein ﬁeld equations. In contrast, the Eq. (5) prediction is easily derived directly from the Einstein equivalence principle (EEP), as elegantly

© 2015 A. F. Mayer 7 ver. 15.11.05.Z17 described in the Feynman Lectures on Physics:5 For a vertical separation (Δh) of two ideal clocks inside a rocket accelerating at rate g in inertial space (Fig. 4), one considers the nominal inertial-frame photon propagation time between the two clocks (Δh/c), which value ignores the second-order component induced by a small acceleration. In this time the rocket has increased its velocity in the inertial frame by the following magnitude.

Δh Δv = g ⋅ Δt ≈ g ⋅ • g ∼ 10 m s−2 (6) c From the point of view of an unaccelerated free-falling inertial observer external to the rocket, photons emitted from a lower clock in the rocket’s tail ‘chase’ a higher clock in the rocket’s nose and are therefore redshifted upon arrival. The Eq. (7) approximation yields a measurably-accurate value of this Doppler redshift for sufﬁciently small Δv.

Δv zD ≈ [Δv << c] (7) c

Substituting the Eq. (6) expression for Δv into Eq. (7) yields the familiar Eq. (5) formula.

A subtle property of this inertial-frame equivalent to the clocks shown in Fig. 3 is the speed of the rocket (|v|) relative to an inertial observer (I ). I B g ≡ 0 Δh inertial observer v dt I = f ( v ) d! A A g

An equivalence that includes relationship I ⇒ A

requires |v| = vesc, where vesc is defined in Fig. 3. Fig. 4

Accelerated observers inside the rocket experience the clocks to be at rest within their reference frame, yet they are required to measure the same frequency shift between the clocks as is ‘measured’ by the inertial observer, based on the comoving motion of the clocks in the inertial frame over the photon propagation time. Accordingly, accelerated observers inside the rocket must attribute the observed photon frequency shift between clocks at relative rest exclusively to a relativistic time-dilation effect between the clocks. This shift is readily observed to be a function of the measured local acceleration (Eq. 5) when g is varied. Per EEP, Eq. (5) is equivalently applicable for gravity, yet a notable

5 Richard Feynman, Robert B. Leighton & Matthew Sands, The Feynman Lectures on Physics, Volume II (Addison-Wesley, Reading MA, 1964), §42-6, p. 42-9; < online >.

© 2015 A. F. Mayer 8 ver. 15.11.05.Z17 difference between an inertially-accelerated frame (e.g., a rocket in gravity-free space) and a real gravitational field is the absence of gravitational tidal forces in the rocket. In the context of relativity, there are two important measurable properties associated with a particular location in a gravitational field: the local gravitational acceleration and the energy relationship (i.e., Newtonian potential) between this location and a distant inertial observer in gravity-free space. In accurately modeling the inertial equivalent to such a location, including time dilation, both of these two properties must be considered. Accordingly, the accelerating rocket cannot have an arbitrary speed relative to a free- falling inertial observer; it must be considered exclusively at the ideal moment that its speed relative to this inertial observer is equal to the local gravitational escape velocity (vesc) at the location of the gravitationally-accelerated reference frame being modeled (Fig. 4). In this case, an accelerated ideal clock (A) within the rocket will tick measurably slower than an inertial ideal reference clock (I ), having the identical temporal relativistic relationship to that between the modeled gravitationally-accelerated ideal clock at rest with respect to a distant inertial reference clock (Fig. 3). At any other relative speed between an inertial observer and the accelerating rocket (Fig. 4), equivalence does not hold with respect to this fundamental relationship (relativistic time dilation), arising from a difference in gravitational potential energy (Fig. 3). This subtle detail of equivalence is not relevant to modeling the Einstein redshift in an inertially-accelerated frame, so it was easily overlooked in the past. However, it is critically relevant when lateral gravitational tidal force associated with real astrophysical gravitational fields is taken into consideration in the equivalence model. In such fields, gravitational acceleration is not parallel between points separated by any appreciable horizontal distance (Fig. 5). The model geoid in this schematic is idealized in that we consider |g| to be equal everywhere on the surface and the source body has no angular momentum, so that Sagnac and any relative velocity effects are eliminated. The small angle approximation (sin θ ≈ θ) must hold for the identical angle of separation between the two clocks in the Fig. 5 and Fig. 6. schematics (i.e., θδ ≪ 1). This essential restriction for the vector equation [dg/dθ · θδ → 0] in Fig. 5 is similar to the essential restriction for the scalar equation [dg/dh · Δh → 0] in Fig. 3. Accordingly, the distinct acceleration vectors (gA , gB), though not parallel, are very nearly so (i.e., the vector difference in acceleration between the two compared clocks is very small, but not zero). Clocks A and B are in different frames of reference due to the subtle vector distinction Δg ≠ 0.

A B g g A δ ≪ r B (v ) dg al geoid esc # 0& ide % !!" " ( [ω = 0] $d! ' r (radius) " ! = !1 " r Fig. 5

g g

rockets accelerating in gravity-free space dv/dt = g |vA| = |vB| = vesc

vesc and θδ reference the empirical values on the Fig. 5 geoid.

θδ Fig. 6

The two ideal clocks separated by the arc δ in Fig. 5 comprise a system. We now consider the inertially-accelerated equivalent of this system. Accordingly, there are two distinct accelerating rockets on divergent trajectories (Fig. 6), having the identical small angle between these trajectories as angle θδ in Fig. 5. Clearly, the divergence of these trajectories implies that respective observers in A and B make local measurements in distinct inertially-accelerated frames of reference. The relativistic relationships [I ⇒ A] and [I ⇒ B] are identical; the clocks in both accelerating rockets tick measurably slower relative to an inertial reference clock (I ) in exactly the same proportion because the only relevant factor is the identical speed of the rockets (vesc) relative to this clock, where this speed is the characteristic escape velocity of the model geoid in Fig. 5. The fact that the rockets are on different trajectories has no part to play in these two specific relationships. Symmetry considerations require that an inertial observer will measure both of the accelerated clocks A and B (moving at the identical speed vesc relative to clock I ) to each tick at the same rate relative to this inertial reference clock. However, this symmetry is broken in considering the relationship between A and B themselves, which clearly represent distinct frames of reference. The divergent velocity vectors (vA , vB) imply an energy differential between frames A and B quantiﬁed by the change in velocity between them, which is a vector quantity with a non-zero magnitude (Δv). Due to the restriction (θδ ≪ 1), the component of Δv (similarly Δg) that is collinear with the acceleration vectors

© 2015 A. F. Mayer 10 ver. 15.11.05.Z17 is vanishingly small. This energy differential represents an inherent symmetric energy cost associated with the propagation of an electromagnetic signal between the two reference frames that is transverse to the direction of acceleration. This symmetric energy cost implies a readily-quantiﬁable symmetric relativistic relationship between the respective ideal clocks in frames A and B: from the point of view of clock A, clock B appears to tick slower, and from the point of view of clock B, clock A appears to tick slower, both according to Eq. (8). dt dt 1 A = B = (8) d d 2 τ B τ A ⎛ Δv⎞ 1− ⎝⎜ c ⎠⎟

The Einstein equivalence principle implies that what is true for the energy relationship between the two rockets (i.e., inertially-accelerated clocks) in Fig. 6 is identically true for the two gravitationally-accelerated clocks in the Fig. 5 schematic. According to this principle, the 17th-century Newtonian concept of a “gravitational equipotential surface” must be abandoned. This assertion, which is justiﬁed in what follows, corroborates the same qualitative conclusion drawn in §Introduction, which references Fig. 1. Per the following discussion, it immediately follows from Eq. (8) that the predicted redshift (zδ) for a separation distance δ restricted to the condition [δ ≪ r] is:

2 g 2 δ µ δ µ (9) zδ ≈ 3 2 = 2 [δ << r] • g = 2 r c rc r

This restricted approximation formula is of a similar nature to Eq. (5), which yields an accurate approximation for the Einstein redshift with immeasurable error for sufficiently small vertical displacements (Δh) in the weak field. To be clear, Eq. (9) is derived directly from Eq. (8) using the following applicable approximations of very high accuracy under the stated respective restrictions. The first approximation (LHS of Eq. 10) is well known and arises from binomial expansion of relativistic gamma. The validity of the second approximation is dependent on the accuracy of the approximation sin(δ/r) ≈ δ/r.

1 Δv2 z = −1 ≈ Δv ≪ c 2 2 [ ] δ Δv 2c Δv ≈ vesc [δ ≪ r] (10) 1− r c2

It is natural to initially reject or even ridicule Eq. (9) based on preconceived notions from pedagogical learning and from some prior-reported empirical evidence, which together provide a false sense of certainty. Especially when an anomaly is quite small, expectancy theory suggests that observers will overlook such an anomaly and report measurements that are consistent with ‘correct’ measurement according to prevailing theory, which is universally assumed to be consistent with empirical reality. Evaluated over a 1-km separation distance on the surface of the Earth, Eq. (9) implies a symmetric redshift −17 measurement zδ ~ 10 as compared to a zero value consistent with conventional theory.

© 2015 A. F. Mayer 11 ver. 15.11.05.Z17 “That cannot possibly be right” or “everybody knows that Newton’s 17th-century idea of a gravitational equipotential surface is consistent with relativity” do not hold water in the context of the new idea of relativistic temporal geometry: In short, the reason that there is indeed a minute, non-zero energy cost associated with translation between the two points (A↔B) in Fig. 5 is that the ‘direction of time’ in spacetime is not identical for these two distinct frames of reference. This geometric distinction between the respective time coordinates, which is readily quantifiable in direct accord with first principles, implies the relativistic relationship. The concept of relativistic temporal geometry and its associated empirical phenomenon are not reflected in conventional general relativity, nor in its embodiment in the current PPN formalism. However, they are clearly and unequivocally reflected in the conceptually-simple Fig. 1 and the comparison of the Fig. 5 and Fig. 6 schematics when viewed in the context of the Einstein equivalence principle. The full development of relativistic temporal geometry, which constitutes an amendment to general relativity, is beyond the scope of this introductory report and its intended broad technical audience, inclusive of those who are inexpert in the mathematics of GR. However, the following predictive general equation (Eq. 11) arises from this amendment to that widely accepted theory. Derived from pure geometry motivated by first principles (see addendum to this report), it may be used to evaluate the magnitude of this previously-unrecognized relativistic phenomenon (TGR) that inherently exists between any two points in the weak field between which there exists a non-zero transverse (horizontal) gravitational tidal force. An idealized point-source mass producing a static, spherically-symmetric gravitational field is assumed. This ‘Schwarzschild simplification’ closely approximates the field of typical astrophysical bodies such as the Sun or the Earth, so the formula is accurate for any major solar system body. This formula predicts energy loss for an electromagnetic signal (i.e., a redshift, zt) incurred due to translation transverse to the local gravitational gradient, which energy loss is not now predicted by conventional general relativity. A corresponding symmetric difference in the measured frequency of ideal clocks exists. At first glance, Eq. (11) may seem cryptic; fundamentally, what this geometric formula calculates is the relativistic time dilation effect caused by the divergence of distinct proper time coordinates between two points in a gravitational field (i.e., two distinct frames of reference) due exclusively to lateral displacement in the field and excluding vertical displacement (i.e., Einstein redshift) and relative velocity (i.e., special relativity):

⎡ ϕ2 ⎤ ⎧ ⎫ RS ⎛ ϕ ⎞ ⎤ ⎪ ⎡ π π ⎤ ⎪ zt = sec ⎢2 E ⎜ 2⎟ ⎥ ⎥ −1 ⎨ [b > 0] & − ≤ ϕ ≤ ⎬ (11) ⎢ b ⎝ 2 ⎠ ⎥ ⎣⎢ 2 2 ⎦⎥ ⎣ ⎦ϕ1 ⎦ ⎩⎪ ⎭⎪

The function sec[u] is the reciprocal of the cosine. RS is the Schwarzschild radius of the source mass and b is a conventional “impact parameter,” or the distance of closest approach of the signal path to the gravitational point source. As shown in the Fig. 7 schematic, this distance may be virtual (i.e., the actual signal path between the points of emission and reception for which the value of zt is evaluated may not extend to the point at which this distance is measured). The term in parentheses prefaced by and inclusive of the symbol E represents an elliptic integral of the second kind. The physical meaning

© 2015 A. F. Mayer 12 ver. 15.11.05.Z17 of the angular term (φ) in Eq. (11) is illustrated in the Fig. 7 schematic. The parameter φ1 typically corresponds to the location of the signal source, while the similar parameter φ2 typically corresponds to the location of the receiver, but this convention is arbitrary, having no effect on the quantitative evaluation of Eq. (11). Note that Fig. 7 does not use this convention. Employing the standard convention, for a signal source that exists beyond the point of closest approach to the the gravitational source at which b is evaluated and φ1 is zero, the value of φ1 is negative. For a distant signal source that is is in opposition to and nearly occulted by an intervening astrophysical mass, for example interplanetary spacecraft telemetry in opposition to the Moon, limit values are φ1 → −π/2 and φ2 → π/2, being the maximum possible difference between these two parameters. The argument of the secant function in Eq. (11) is the result of the following integral:

R ϕ 2 u = S cosϕ dϕ (12) b ∫ϕ1

Accordingly, the single right bracket in Eq. (11) that expresses limits (φ1, φ2) means that the elliptic integral term is evaluated between φ1 and φ2 per Eq. (13).

ϕ2 ⎛ ϕ ⎞ ⎤ ⎛ ϕ2 ⎞ ⎛ ϕ1 ⎞ E 2 = E 2 − E 2 (13) ⎝⎜ 2 ⎠⎟ ⎥ ⎝⎜ 2 ⎠⎟ ⎝⎜ 2 ⎠⎟ ⎦ϕ1

zenith GPS SV

ground station a horizon ψ R a ≈ 26,560 km φ = ψ b 1 R ≈ 6,371 km φ2 φ φ = cos-1 b b = R cosψ 1 2 ( a ) Fig. 7

GPS SV: “GPS Space Vehicle” (i.e., “satellite”) zenith: located directly overhead (ψ = 90°) Note: Transit (the point of highest elevation angle ψ) is generally below zenith.

© 2015 A. F. Mayer 13 ver. 15.11.05.Z17 It should be clear that this predicted effect (Eq. 11) contributes to a total effect, which many include the influence of conventional gravitational time dilation due to relative vertical displacement or actual motion of the source relative to the point of reception. This new prediction corresponds to a radial pseudo-velocity (czt); although the frames may be at relative rest, they behave as if they were slowly moving away from each other. The respective time-dilation effects of the two velocities (real and pseudo), as well as any effect induced by difference in gravitational potential, are independently calculated and then summed to arrive at a total effect. Measured empirically, but not modeled, the TGR effect gives the illusion of recessional motion. However, it can be differentiated from such motion due to the correlated first-order time dilation: For radial motion, Doppler shift is of first order (v/c) and time dilation is of second order (relativistic γ); for TGR, both correlated relativistic effects are of first order (zt). The predicted effects of TGR have apparently been previously observed astrophysically dating back about a century and in various more-recent radio science experiments, although they were never recognized as an anomaly requiring an amendment to conventional GR and the PPN formalism. The argument of the sec[u] function in Eq. (11) represents the effective angle of divergence in spacetime between two distinct time coordinates, which respectively correspond to the two points in the field between which zt is being evaluated. The secant function should be recognized as the inverse dot product of normalized time-coordinate vectors. The nature of this geometric function is physically intuitive in the context of evaluating a relativistic time dilation between two distinct reference frames having strictly-local time coordinates (i.e., distinct linear time axes) that have different orientations in spacetime. The secant function evaluates the projection of the linear time coordinate of one reference frame (parallel transported) onto the geometrically distinct linear time coordinate of the other reference frame. Although this concept has not heretofore been incorporated in the pedagogy of relativity theory, the correct physical interpretation of relativity concerns the relative geometry of distinct local time coordinates; the measured relative rate of ideal clocks is a secondary effect produced by this more-fundamental physical reality. This is the only rational mathematical model of the symmetry inherent to special relativity (SR) whereby each observer measures the rate of the remote ‘moving’ ideal clock to be slower than the local ideal reference clock (i.e., each clock is falling behind the other clock). The algebra of the Lorentz transformations belies the fundamental underlying physically-real relative geometry of geometrically-distinct local space and time coordinates in spacetime. If two individuals starting from the same point each measure their progress in the plane in different directions (0 < Δθ < π/2) using odometers, then both odometers will display the same interval after each vehicle reaches the same arbitrary distance. The odometer readings are strictly-local measurements of progress, which is defined in a particular direction in space. Each odometer records the magnitude of the interval, but they do not incorporate any information about the corresponding direction vectors that are inherent to the measurement. Because each traveler defines progress in a different direction in space, each is entitled to state that relative to their strictly-local (i.e., “proper”) geometric definition of progress, the other individual has fallen behind in making such progress. Replacing the clocks shown in Fig. 8 with odometers and the time intervals with spatial

© 2015 A. F. Mayer 14 ver. 15.11.05.Z17 intervals, the symmetric geometric relationship between the two equal intervals (dtA, dtB) shown in the schematic illustrates the foregoing scenario of the two travelers. When one speaks of “proper time” in relativity, the origin of the term comes from the French as coined by Henri Poincaré (1854 – 1912) where the term “mon propre temps” in French literally means “my own time.” In accord with the fundamental principles and mathematical foundation (i.e., H. Minkowski) of relativity, the progress of an ideal clock’s “proper time” (i.e., it’s own time) is being recorded in a particular direction in spacetime. Recall from §Background that, rather than the conventional concept of a “3+1” manifold, spacetime is a physically-abstract four-dimensional manifold in which no dimension has a physical interpretation. Like an odometer, although a clock records the interval of time progression, it displays no information concerning the underlying geometry inherent to the measurement. However, when one observes the apparent rate of an ideal clock (B) in comparison to the proper time recorded by an ideal clock in the local frame (A) and ﬁnds that clock B is “falling behind” relative to the time recorded by clock A, one can state unequivocally that clock B is measuring an interval of time in a different direction in spacetime than clock A. This is patently obvious in the case of the symmetry inherent to special relativity, which is reiﬁed in the context of this intuitive geometric model (Fig. 8).

Symmetric relativistic temporal geometry in SR: geometrically, no ‘universal time coordinate’ exists in spacetime. A is “falling behind” B is “falling behind” the progress of B. the progress of A.

dtA dtB

" "

o os

c c B

v A t

sin! = ! " d dt

= = c B A

! !

1 d d sec! = ! # ζ 2 Euclidean 1" " spacetime Fig. 8

To reject this concept simply because it is not in the current textbooks reflecting prior thinking by respected authorities in the field is similar to a person of religious persuasion rejecting a new idea because it is not recorded in cannon law. Regardless, like Kepler’s three laws of planetary motion that effectively completed the Copernican revolution and laid the foundation for Newton’s comprehensive transformation of physics, this new idea leads to the prediction of subtle empirical observables that cannot be explained without it (i.e., observable effects of transverse gravitational redshift). In the idealized Euclidean (i.e., ‘flat’) spacetime of special relativity, the angle zeta (ζ ) is a simple mathematical function of the measured relative velocity (v) between two reference frames:

If the two distinct time coordinates diverge in spacetime by the angle zeta, the projection of the remote clock’s time coordinate on the local clock’s time coordinate corresponds to secant(zeta). Accordingly, the symmetric relativistic relationship between the respectively measurable frequencies of the two clocks (A, B) has a simple and intuitive geometric (i.e., physical ) interpretation, which is reflected in the definition of relativistic gamma. Note that the initial algebraic equality in Eq. (15), reflecting the inherent symmetry of special relativity, is sensible in the context of geometrically-distinct time coordinates in an idealized “flat” manifold (i.e., Fig. 8). This elegant mathematical symmetry cannot be achieved in the context of a spacetime defined such that all reference frames have universally collinear (i.e., parallel) time coordinates. Such a naïve temporal geometry (i.e., geometric model of time) having a single absolute time axis is inconsistent with abandoning the pre-relativistic Newtonian concept of absolute universal time.

dt dt 1 A = B = secζ = γ = ⎡dt = dt = dt = dt ⎤ 2 ⎣ A A B B ⎦ (15) dtB cosζ dtA cosζ 1− β

Special relativity is just as fundamentally rooted in geometry as is general relativity; the Lorentz transformation equations are merely a reflection of this underlying geometry in the four-dimensional spacetime manifold. Ironically, Einstein’s (i.e., Lorentz’s) algebra is really a misleading and somewhat superfluous mathematical formality as compared to Minkowski’s more fundamental, more informative and more physically-meaningful and accurate geometric expression of the theory.6 The concept of “curved spacetime” in GR arose from interpretation of its mathematical expression in the form of the Einstein field equations, rather than as an intuitive physical interpretation, but such an interpretation does indeed exist. The idealized condition of locally-Lorentzian space implies that the Minkowski metric (Eq. 16) applies locally within the neighborhood of any point, assuming a free-falling inertial reference frame. ds2 = −c2dt 2 + dr 2 + r2dθ 2 + r2 sin2 θ dφ 2 (16)

Accordingly, within this differential neighborhood, the locally-measurable time dimension (dt) is mutually orthogonal to the three locally-measurable space dimensions such that the strictly-local time dimension at each point may be globally deﬁned in the context of geometry as the “local vertical” to the three-dimensional spatial ‘surface’ at every point. Clearly, physical (i.e., measurable) three-dimensional space is embedded in the four physically-abstract (un-measurable) dimensions of spacetime. The presence of mass causes this 3-manifold (space) to literally “curve” in the 4-manifold as a function of the

6 It is reported that a querulous young Einstein initially ridiculed Minkowski’s vital mathematical contribution to relativity as “superﬂuous erudition” (and subsequently never properly understood it). This occurred after Minkowski’s premature sudden death from appendicitis in January 1909. Source: Abraham Pais, Subtle is the Lord… The Science and the Life of Albert Einstein, (Oxford University Press, Oxford, 1982), p. 152.

© 2015 A. F. Mayer 16 ver. 15.11.05.Z17 radial coordinate (r). It follows that the measurable locally-orthogonal time dimension rotates with this curvature, just as the local vertical to a sphere rotates with the curving surface of the sphere. This ‘rotation’ in spacetime is a physical conversion of time into space producing the canonical “excess radius” ( ρ − Δr) of a gravitational ﬁeld:

1 − R 2 R ⎛ S ⎞ ⎡ S ⎤ (17) ρ = ∫ ⎜1− ⎟ dr ⎢0 < ≪1⎥ ⎝ r ⎠ ⎣ r ⎦typical

The measured phenomenon of gravitational time dilation (Eq. 2), which is correlated to the Einstein redshift (Eq. 4), has been well-understood for about a century. The concept of relativistic temporal geometry (Fig. 9) correlates this empirical phenomenon with the rotation of the local time coordinate at some finite radial coordinate r (clock A) by a specific angle [η(r)] relative to that local proper-time coordinate corresponding to any inertial reference clock I at r→∞, where both clocks are collinear to the identical radial. This angle (~ 3.7E−5 rad on the geoid and ~ 2.1E−3 rad at the surface of the Sun) is defined in Eq. (18) and illustrated in the Fig. 9 schematic, which demonstrates that the angle between respective time coordinates for antipodal points at the same coordinate radius is zero by inverting the proper time vectors on the left hand side. Respective time coordinate arrows at the same coordinate radius on the left and right hand sides are parallel; two ideal clocks situated at antipodal points at the same coordinate radius in an idealized static, spherically-symmetric gravitational field (and, equivalently, on a spinning disk in the laboratory) are synchronous. The geometric relationships in 4-dimensional spacetime expressed by the general mathematics (e.g., Eq. 11) cannot be diagramed so that the corresponding arrows on both sides of Fig. 9 have the same orientation in the schematic, however the fundamental mathematical concept of relativistic temporal geometry in the context of the gravitational field can be intuitively illustrated as shown. Note the similarly between Eq. (14), which applies to special relativity, and Eq. (18), which applies to general relativity; the only difference between the two mathematical expressions is that the latter is defined in terms of a virtual (gravitational escape) velocity while the former is defined in terms of a real relative velocity between inertial reference frames. Both velocities represent a similar energy differential. v sinη = Β → cosη = 1− sin2 η = 1− Β2 • Β ≡ esc (18) c

According to the operative geometric principle (Fig. 9), ideal clock A, shown in blue, is “falling behind” ideal clock I, shown in red, which is serving as a reference. The absolute frequency ratio of the two ideal clocks is specified by Eq. (19), which is a function of the coordinate radius, per Β(r). A difference in coordinate radius r (i.e., vesc) between two points in a gravitational ﬁeld implies preferred locations that can be distinguished by a difference in the respective local measurements of gravitational acceleration; thus, the symmetry of special relativity is lost.

Plugging the deﬁnition of Β into Eq. (19) yields Eq. (20), duplicating Eq. (4). As required in this case, one gets the same results as canonical GR, but in a much simpler way within the context of an intuitive geometric model for “spacetime curvature,” which has a direct correlation with the similar projective geometry of special relativity. dt 1 1 1 z = I −1 = −1 = −1 = −1 (20) E 2 dτ A 2µ R ⎛ vesc ⎞ S 1− 1− 2 1− ⎝⎜ c ⎠⎟ rc r

local proper time “coordinate time”

dtI η dτA

“flat” space “curve r→∞ d” spac μ Δρ I e A η→0 hips b = 0 relations r

The inherent parallelism of antipodal time coordinates, corresponding to a null geodesic through the center of mass (i.e., b = 0) having ideal spherical symmetry , is shown by inverting the antipodal mirror image of the above time coordinates. Re Eq. (11): if b = 0 then z = 0. t Fig. 9

Especially for an expert steeped in the complex and cryptic mathematical formalism (e.g., “index gymnastics”) of general relativity, the descriptive and predictive power of the foregoing intuitive geometric expression of the theory may not be immediately apparent. Both the newly-introduced underlying principle of relativistic temporal geometry and its most basic mathematical expression (assuming an ideal gravitational point source, which is the same facilitating simplification that was adopted by Karl Schwarzschild) are so extraordinarily lucid in comparison to the conventional approach that it may be easy for some to initially believe that this is all mathematical “smoke and mirrors.” Quite to the contrary, this approach provides important new physical insights into the phenomenon of gravitation, in particular a testable prediction (Eq. 11) of a previously unsuspected empirical phenomenon. And when it is verified by numerous empirical observations, the underlying theory must overthrow existing physical interpretations of general relativity. According to this reification of general relativity, the respective time coordinates of every unique point at rest in a static, spherically-symmetric gravitational field are geometrically distinct, with the noted exception of antipodal points at identical coordinate radius. Barring this exception, and within the boundary (r < ∞), no two such strictly-local time

© 2015 A. F. Mayer 18 ver. 15.11.05.Z17 coordinates are parallel in spacetime. In the context of relativistic temporal geometry, this non-parallelism implies a relativistic time dilation between the distinct reference frames associated with these time coordinates. The Eq. (12) integration, which yields an expression for the effective TGR angle between the distinct respective time coordinates of two arbitrary points in the external weak field, arises due to symmetry considerations. If the null geodesic between clocks passes through the source mass, the TGR effect is periodic over a circular path at fixed coordinate radius: initially, it rises with increasing arc length between clocks, attaining a local maximum prior to pi radians, and then falls to zero at the antipode to the reference clock (i.e., at pi radians when the null geodesic between the clocks passes through the centroid of the idealized point-source mass. Equations (9) and (11), which are radically different in form and in respective methods of derivation, represent the quantitative prediction of the identical observable, so they must agree, bound by the restriction [δ ≪ r] for the former equation. As has been shown, the former equation rests exclusively on the Einstein equivalence principle, while the latter rests on the new idea of relativistic temporal geometry applied to the gravitational field. According to the tenets of mathematical physics, nearly perfect agreement of these two distinct predictive equations implies that they can be relied upon to indicate that the underlying theory is correct. The possibility of two independently-derived formulas of this nature agreeing by chance to an extremely high degree of accuracy is virtually nil. For example, Eq. (5), which rests exclusively on the Einstein equivalence principle, is in very close agreement with the definitively-accurate general predictive equation for the phenomenon of gravitational time dilation (Eq. 22). This equation, which relates the relative frequency of two ideal clocks at different radial coordinates (r1 , r2) in the field, arises from Eq. (20), here rewritten in the form of Eq. (21). In Eq. (22), variable dt represents the rate of an inertial ideal clock in free space (i.e., r → ∞ , g → 0), and dτn represents the rate of an ideal clock at finite coordinate radius (rn) from an isolated ideal gravitational point source and at relative rest to the inertial reference clock. In canonical GR (the Eq. 21 equality on the left is typically derived from the Schwarzschild solution to the Einstein field equations), this relationship is independent of angular coordinates.

dt 1 dt = z = −1 (21) dτ R E dτ 1− s r

R R 1− S 1− S dt dτ dτ r r ⋅ 2 = 2 = 2 → z = 2 −1 (22) dτ dt dτ R E R 1 1 1− S 1− S r1 r1

The following Mathematica notebook (Math 1) compares the predictions of Eq. (5) with Eq. (22) over a 1-km vertical rise from Earth sea level. The last line indicates an error on the order of one part in ten thousand (~ 0.0175 %) for the result yielded by the familiar approximation (Eq. 5) as compared to the deﬁnitive value calculated according to the accurate canonical general formula (Eq. 22) for this empirically-veriﬁed phenomenon.

The geometric parameters employed in the Math 2 calculations are illustrated in Fig. 10. The quantitative predictions of Eq. (9) and Eq. (11) are compared similarly to the comparison of Eq. (5) and Eq. (22) in Math 1. The last line of the Math 2 notebook indicates that the approximation formula (Eq. 9) predicting the TGR effect over a 1-km horizontal distance on the model geoid yields an error on the order of one part per billion (~1×10−7 %) as compared to the definitive value calculated according to the accurate general formula (Eq. 11). If the clock separation distance (δ ) is increased to 100 km, the error in the approximation rises to about 0.001%, which is still an order of magnitude less than the error in Math 1. To reiterate, there is no possibility of such a quantitative correlation having arisen by chance; it constitutes a virtual ‘proof’ that the mathematical physics is correct and that it accurately predicts an empirical phenomenon.

The Mathematica® EllipticE function (line 12) is described in the following URLs: http://reference.wolfram.com/language/ref/EllipticE.html http://mathworld.wolfram.com/CompleteEllipticIntegraloftheSecondKind.html

Fig. 10

“See now the power of truth; the same experiment which at first glance seemed to show one thing, when more carefully examined, assures us of the contrary.” – Galileo Galilei, Discourses and Mathematical Demonstrations… (1638)

White Dwarf Stars One should expect to find compelling empirical evidence for the predicted relativistic transverse-gravitational-redshift effect in astrophysical phenomena. In particular, the unusually strong surface gravity of white dwarf (WD) stars must cause significant amplification of TGR as compared to normal stars: When it is possible to remove relative motion Doppler shift from the redshift measurement of a white dwarf, the remaining intrinsic redshift, which incorporates both Einstein gravitational redshift and TGR will have been interpreted as an Einstein redshift. Accordingly, the calculated mass of the WD according to the empirical measurement interpreted by Eq. (4) will have been considerably higher than that determined by independent means or as may be expected for the remnant of a progenitor star following a cataclysmic explosion (i.e., nova). It is remarkable that the “relativistic” masses of the white dwarf stars, which one obtains by reduction of the observed redshifts, are (on the average, with large scatter) significantly larger than the “astrophysical” ones… Various attempts to explain this discrepancy have been made in the past, e.g., by asymmetry-induced shifts due to slope of the continuum (Schulz 1977) but this problem still is not solved (see also the review by Weidemann 1979). In velocity units the systematic excess of the observed redshift amounts to 10–15 km s−1 (Shipman and Sass 1980; Shipman 1986) above “residual” redshift (i.e., redshift free of all kinematic effects).7 [ApJ (1987)] The primary method of determining white dwarf mass that has been established over recent decades is comparison of empirical spectra with spectral energy distributions of theoretical stellar atmosphere models. Newer models corroborating gravitational redshift measurements would yield masses considerably in excess of empirical reality as they do not incorporate the empirical TGR effect and these models would not accommodate all classes of white dwarf spectra; unphysical systematic effects in the M(Teff) curve for specific regimes (e.g., cooler stars of DA spectral class) would be inevitable. The TGR effect varies in magnitude, from zero for photons emitted from the center of the observed stellar disk collinear with the local gravitational gradient, to the maximum effect observed at the limb where observed photons are emitted orthogonal to the local gravitational gradient. Due to simple geometric considerations of the increasing annular area of constant-transverse-width rings of increasing mean radius (Fig. 11), the majority of a star’s observed photons, which dominate the spectroscopic redshift measurement,

7 B. Grabowski, J. Madej & J. Halenka, “The Impact of the Pressure Shift of Hydrogen Lines on ‘relativistic’ Masses of White Dwarfs,” ApJ 313, 750 (1987) < online >.

© 2015 A. F. Mayer 22 ver. 15.11.05.Z17 incorporate ‘excess redshift’ (i.e., excess to the Einstein redshift) due to the TGR effect. Additionally, the observed light of any distant star, resolved as a point source, has a corresponding continuous spectrum of redshifted wavelengths (TGR) superimposed on the Einstein redshift. Increasing TGR is incurred by photons originating from the star’s surface at increasing apparent angular offsets from the center of the (unresolved) projected stellar disk. Thus, TGR implies pronounced line broadening, which is a ubiquitous feature of white dwarf stars. This important distinguishing feature of white dwarfs is currently understood to be caused primarily by atmospheric pressure broadening, but there is reason to doubt this interpretation of the observable. Given the immense surface gravity of a white dwarf, it is unlikely that such an adequately-dense “atmosphere” would surround the star; its physical surface can be expected to be a sharp boundary between matter anywhere within the vicinity of the star and the vacuum of space. Although now generally accepted, such an unlikely theoretical ‘atmosphere’ surrounding a white dwarf can be perceived as an ad hoc invention that was required to explain the empirical observables without having the causal TGR effect quantitatively modeled or even suspected. Line broadening of white dwarf spectra due specifically to the modeled continuous spectrum produced by TGR must correlate to the calculated magnitude of the effect, which is a testable prediction. The following brief discussion clarifies the foregoing claims with explanatory detail: Although, with rare exception (e.g., HST image of Betelgeuse), all stars other than the Sun are resolved exclusively as point sources, it is a trivial fact that observed stellar radiation includes photons emitted from the entire projected stellar disk, including those specifically emitted from its center region and those distinctly emitted from the region adjacent to its limb. The majority of observed light emitted by a star originates from the distal area of the disk adjacent to the limb (Fig. 11). If the illustrated area ratios shown in this figure seem unintuitive, one may consider that the area of a disk (i.e., the orthographic projection of a ball representing a radiating star) is calculated by an integration of differential rings having width dr at radius r (Eq. 23). This intuitive and trivial integration makes it obvious that for constant dr the differential surface area of successive rings grows linearly with increasing radius.

R A = ∫ 2πr dr = π R2 (23) 0

Fig. 11

© 2015 A. F. Mayer 23 ver. 15.11.05.Z17 The measured non-Doppler redshift of a distant star resolved as a point source can be expected to reﬂect the majority of emitted photons, which are sourced from the more- distal region of the observed stellar disk, rather than its more-central region. The TGR effect implies a signiﬁcant redshift of these photons superimposed on the Einstein redshift, yet the aggregate is currently assumed to accurately indicate a star’s mass- radius relationship by ostensibly measuring zE and solving for µ/r in Eq. (4). A naïve relationship between the mass and radius of a white dwarf (R ~ M −1/3) may be derived from minimizing the sum of its gravitational potential energy and kinetic energy, which is primarily in the form of degenerate electron motion (top curve in Fig. 12). Although this formula is simplistic, it clearly reveals the unintuitive fact that increasing the mass of a white dwarf reduces its radius. Correcting for the relativistic motion of the degenerate electrons in a white dwarf yields the lower curve in Fig. 12. The upper limit to the mass of an electron-degenerate object (~1.44 solar masses) is the Chandrasekhar limit (MCh), beyond which electron-degeneracy pressure cannot support the object against collapse.8

Fig. 12

The first two columns of Table 1 (solar mass and radius fraction) reflect the “Relativistic” curve in Fig. 12. The Einstein redshift (zE in column 3) reflects Eq. (4) and zt in column 4 reflects Eq. (11) for photons originating at the limb of the star. Consider this excerpt from Grabowski et al. (1987) as quoted above, which is no longer accepted; WD models have been changed to match observations, thus eliminating this “systematic excess”: “In velocity units the systematic excess of the observed redshift amounts to 10 – 15 km s−1”

8 Wikipedia, White Dwarf, “Mass–radius relationship and mass limit”; < online >.

© 2015 A. F. Mayer 24 ver. 15.11.05.Z17 −1 Typical white dwarf stars have an observed gravitational redshift (zobs) of about 33 km s , per recent measurements.9 Given that this empirical measurement of average white dwarf redshift is now commonly interpreted exclusively as an Einstein redshift, the last column of Table 1 is then incorrectly interpreted as the zE column. Consequently, the modeled mass of typical white dwarf stars based on their observed redshift (zobs) is about double the empirical value (~ 0.35 → 0.65 M☉). The misinterpretation of measured WD redshifts due to the unmodeled TGR effect yields unreasonable mass values for these remnants of novae, which must atomize and eject a considerable majority of the progenitor star’s mass at radial velocities far exceeding escape velocity.

Mass (M☉) Radius (R☉) zE (km/s) zt (km/s) ∑zobs (km/s)

0.2 0.021 6.1 8.8 14.9

0.3 0.017 11.3 16.3 27.6

0.4 0.016 16 23 39

0.5 0.014 22.9 32.9 55.8

0.6 0.013 29.6 42.5 72.1

0.7 0.011 40.8 58.6 99.4

0.8 0.01 51.3 73.7 125

0.9 0.009 64.2 92.1 156.3

1 0.008 80.2 115.2 195.4 Table 1

emission Eq. (11) parameters for observed stellar radiation radius (b)

of photons φ1 = 0 path of photon from surface to observer at r → ∞ mb li b = R R φ2 → π/2

path of photon from surface to observer at r → ∞ φ1 φ2 → π/2 b = R cos φ1

Fig. 13

Fig. 13 illustrates the parameters (φ1, φ2, b) employed in Eq. (11). The top illustration shows the location of maximum TGR effect, which occurs at the limb, and applies to the parameters used to generate the zt column of Table 1. Given that the dotted arrow and

9 Ross E. Falcon, D. E. Winget, M. H. Montgomery & Kurtis A. Williams, “A Gravitational Redshift Determination of the Mean Mass of White Dwarfs. DA Stars,” ApJ 757, 116 (2012); < online >.

© 2015 A. F. Mayer 25 ver. 15.11.05.Z17 the solid arrow effectively meet at a point representing the star observed at ‘inﬁnity,’ it should be clear that φ2 → π/2 for observations of radiating astrophysical bodies. As the radius of photon emission on the projected disk of a star (0 ≤ b ≤ R) is reduced, the component of the photon’s trajectory that is transverse to the local gravitational gradient is also reduced; accordingly, the value of φ1 approaches φ2 as b → 0 and the calculated value of zt tends to zero as emission moves from the limb towards the center of the projected stellar disk. Fig. 14 summarizes the geometric nature of the TGR effect whereby observed photons originating from different regions of the radiating body have different non-Doppler redshifts superimposed on the Einstein redshift.

nal g atio rad it ie av n r t g Limb photons have maximum transverse component; distant observer star here the transverse component is zero.

Therefore, TGR implies a center-to-limb variation of λ. The wide and continuous range of wavelengths produced implies observation of line broadening, which will be most pronounced for compact stars having high surface gravity. Fig. 14

K Effect & Trumpler Effect Bright Class B stars (i.e., larger and more massive stars) exhibit an excess redshift as was ﬁrst observed at the Lick Observatory in 1911.10 The first spectroscopic measurements of large numbers of B stars showed that, unlike cooler stars, they appeared to be expanding away from the solar neighborhood. This positive redshift was expressed as a ‘K term’ and is referred to in the literature as the K effect. No satisfactory explanation was ever advanced as to why the entire system of luminous young stars should be receding from the position of the Earth. … The same effect was reported by Trumpler who showed that the brightest, hottest stars in young clusters had redshifts which were systematically positive with respect to their clusters. He obtained a mean excess of +10 km s−1. 11 Interpreted as a Doppler shift, these observations make the inference that recession velocity from the Sun is dependent on the effective temperature of a star, which is at best extremely unlikely. The only alternative is an intrinsic property, but this requires a compelling explanation correlating the measured effective temperature of a star with its measured intrinsic redshift. The effective temperature of a star is strongly correlated to

10 W. W. Campbell, “On the motions of the Brighter Class B Stars,” PASP 23 (136), 85 (1911); < online >. 11 Halton Arp, “Redshifts of high-luminosity stars – The K effect, the Trumpler effect and mass-loss corrections,” MNRAS 258 (4), 800 (1992); < online>.

© 2015 A. F. Mayer 26 ver. 15.11.05.Z17 its mass; hotter stars are more massive. As concerns stellar radiation, the magnitude of the TGR effect is dependent on field strength; while a more massive star has the potential to generate stronger surface gravity, the critical factor is its density. Assuming that TGR is indeed responsible for the observed K effect and Trumpler effect, then the Fig. 15 curves imply a need to reevaluate current theoretical models for stars that are more massive than the Sun, in particular the canonical mass-radius relationship, which models their internal pressure and density. The six graphed curves represent stars of

increasing mass measured in solar masses (M☉). Current theoretical models assume a considerably lower M/R ratio for these massive stars than the Sun; they fall well below the −1 horizontal line representing czt ~ 10 km s . As for white dwarfs, the spectra of massive stars is subject to quantitatively well-deﬁned line broadening induced by TGR, which is a testable prediction. Independent confirmation of TGR (e.g., in a laboratory experiment) would solve this important conundrum and would also revolutionize stellar astrophysics.

For reference only, small circles indicate solar ratio:

M M ! 3 = 3 R R!

Canonical M/R (approx.)

Fig. 15

Center-to-limb variation (CLV) of the solar wavelength The Sun has a previously unexplained center-to-limb increase in emission wavelength superimposed in the Einstein redshift with an excess redshift (zc) of “about 1 km s−1” at the solar limb: A wavelength shift … has been measured in the wavelength region 1950 – 2000 Å. … After correction for the gravitational redshift and for all the known relative motions between sun and observer, the average residual redshift [measured at the solar limb] is 7 mÅ and could be from 5 to 12 mÅ for some individual reference lines. This

© 2015 A. F. Mayer 27 ver. 15.11.05.Z17 corresponds in terms of velocity to an equivalent Doppler-Fizeau shift on the whole [solar] spectrum of about 1 km/s away from the observer [i.e., v ~ 0.007/1975 × c]. 12

An unexplained center-to-limb variation [CLV] of solar wavelength has been known for 75 years. Many theories have been developed in order to explain its origin. Although recent studies reveal a large amount of new information on the solar chromosphere, such as asymmetries of lines and various mass motions in granules, which lead to wavelength shifts, no theory can consistently explain the observed center-to-limb variation. … … the fact that there has been no contradicting observation of the red shift of the FeI lines, have firmly established that the wavelengths of the Fraunhofer lines in the solar spectrum are dependent upon distance from the solar limb. This CLV cannot be a consequence of [conventional] relativity, which predicts that all solar lines must be red shifted by a factor of 2.12 ´10−6 and hence should be independent of the position on the solar disk. … During those past years, observers hoped in vain to discover new facts, but the basic observations of the CLV have not changed in 70 years, as is stated by Howard et al. and Dravins…13 [IEEE Tran. Plasma Sci. (1989)] -1 +1 km s

Einstein redshift + TGR (predicted solar CLV observation less minor solar atmospheric contribution)

Fig. 16

12 D. Samain, “Is the ultraviolet spectrum of the quiet sun redshifted?” A&A 244, 217 (1991); < online >. 13 Marmet, op. cit.; < online > < online2 >.

© 2015 A. F. Mayer 29 ver. 15.11.05.Z17 The Math 3 notebook calculates the magnitude of the TGR effect evaluated at the solar limb (~ 0.92 km s−1 expressed as an equivalent Doppler shift, zc), which corresponds to the empirical observable noted in the preceding quotations from the literature. The same notebook is used to generate the Fig. 16 TGR curve by varying (0 < f ≤ 1). This curve, perhaps slightly modiﬁed by superimposed solar atmospheric effects, represents a precise, testable prediction for modern solar observatories.

Galileo spacecraft radio science experiments The NASA/JPL Galileo spacecraft was launched in October 1989. Precision radio Doppler telemetry was acquired during occultation by Jupiter and during ﬂybys of the Galilean moons: Io, Europa, Ganymede and Calisto. The results of some of these experiments provide strong indication of the TGR effect whereby the Doppler signal incurs an increasing unmodeled redshift (i.e., “drift”) during the approach phase to occultation and an unmodeled blueshift after emerging from occultation. The following quotations from the literature show that this previously-inexplicable anomaly was treated as a nuisance and simply removed from the data. The emphasis has been added. On December 8, 1995, the Galileo spacecraft disappeared behind Jupiter for 3.7 hours. During the 6.2 hours centered on the occultation, the spacecraft [low-gain antenna] radiated a coherent signal at a frequency of 2.3 GHz derived from an ultrastable quartz oscillator (USO) on board. This signal was tracked… We extracted the time history of signal frequency through Fourier analysis of these data. We then obtained residual frequencies by subtracting the frequency variation that would have been observed in the absence of an atmosphere/ionosphere on Jupiter. These residual frequencies exhibit a small long-term drift, of order 10−4 Hz sec−1, presumably from instability of the USO and refraction in the interplanetary plasma and Earth’s ionosphere. We removed this drift through use of a simple function fitted to the frequency residuals over a baseline interval well above Jupiter’s ionosphere. Separate corrections were applied at ingress and egress.14

A search for an atmosphere on Europa was carried out when Galileo was occulted by Europa three times. … For a few minutes before and after the occultations, the S band (2.295 GHz, or about 13 cm wavelength) radio signal from Galileo traversed regions above Europa’s surface in which one could observe the effects of refraction by an atmosphere, or more precisely, an ionosphere (a layer of ions and electrons produced in tenuous regions of the atmosphere by photoionization and magnetospheric particle impact), should one exist on Europa. … Ideally, these residuals should have a zero baseline, which is the portion of the data that is away from the influence of possible ionospheric refraction effects. In reality, because of

14 D. P. Hinson et al., “Jupiter’s ionosphere…,” Geophys. Res. Lett. 24, 2107 (1997); see § ‘Procedure and Results’ in < online > < online2 >.

© 2015 A. F. Mayer 30 ver. 15.11.05.Z17 drift in the USO, effects of the long propagation path through the interplanetary medium, and imperfect knowledge of the frequency transmitted by Galileo and the spacecraft trajectory, this baseline has not only a non-zero mean but also a slope, which over periods of ten minutes can be approximated by linear frequency drift. The bias and linear drift in the residuals were removed by fitting of a straight line to the baseline data…15 In a 2002 paper, the NASA/JPL Pioneer Navigation Team, concerned with spacecraft celestial mechanics and radio science, described a common misstep in science: Procedures have been developed which attempt to excise corrupted data on the basis of objective criteria. There is always a temptation to eliminate data that is not well explained by existing models, to thereby “improve” the agreement between theory and experiment. Such an approach may, of course, eliminate the very data that would indicate deficiencies in the a priori model. This would preclude the discovery of improved models.16

GPS The Global Positioning System (GPS) is a complex system designed to provide precise time and position information to a variety of users. A critical GPS feature is the frequent upload of corrections to the atomic clocks onboard the satellites with a radio uplink from the globally-distributed GPS ground segment.17 Without this feature, positional accuracy of GPS would rapidly decline to the point of making the system functionally useless because the clock error in each satellite increases over time. The following quotation from a DoD publication (c. 2000) is revealing: The current stand-alone GPS does not meet the accuracies needed in many of the above-mentioned areas. At present, it is well known that small anomalies exist in position and time computed from GPS data [1–3]. The origin of these anomalies is not understood. In particular, GPS time transfer data from the U.S. Naval Observatory (USNO) indicate that GPS time (for all satellites) is periodic with respect to the Master Clock, which is the most accurate source of official time for DoD. Furthermore, the time obtained from all GPS satellites appears to speed up and slow down in-phase [3]. In 1997, the periodicity of this effect was approximately equal to the sidereal day and had a peak-to-peak amplitude of ≈20 ns, which translates to 20 ft (≈6 m) of error in light travel time [3]. The existence of these anomalies in GPS data motivates the theoretical investigation reported here.18

15 A. J. Kliore et al., “The Ionosphere of Europa from Galileo Radio Occultations,” Science 277, 355 (1997); < online >. 16 John D. Anderson et al., “Study of the anomalous acceleration of Pioneer-10 and 11,” Phys. Rev. D 65, 082004-9 (2002); < online >. 17 Neil Ashby, “Relativity and the Global Positioning System,” Phys. Today (May 2002), p. 41; < online >. 18 Thomas B. Bahder, “Fermi Coordinates of an Observer Moving in a Circle in Minkowski Space: Apparent Behavior of Clocks,” ARL-TR-2211, (Army Research Laboratory, Adelphi, MD, 2000); http://www.arl.army.mil/arlreports/2000/ARL-TR-2211.pdf

© 2015 A. F. Mayer 31 ver. 15.11.05.Z17 Qualitatively, consider the following in light of the foregoing quotation from the literature: The orbital period of a GPS satellite is about five seconds less than one half of a sidereal day so that each satellite very nearly repeats the same ground track every sidereal day, with the five-second difference designed to account for westward drift of the longitude of the ascending node, which is due primarily to the Earth’s oblateness.19 Consequently, the geometry of the null geodesic between a satellite and ground station has a periodicity of approximately one sidereal day. It follows that the modeled relativistic TGR effect during that period will be sinusoidal, exhibiting four different local minima corresponding to the two visible transits and two invisible ‘anti-transits,’ when the distance (as measured through the intervening Earth) between a ground station and the satellite is a local maximum. At these four points in the orbit, the component of the null geodesic between satellite and ground station that is transverse to the local gravitational gradient attains a local minimum. Relative to a particular ground station, for example the USNO Master Clock, an intrinsically-stable atomic clock onboard any particular GPS satellite will exhibit four distinct cycles of speeding up and slowing down with a total period for the four regularly-repeating sequential cycles corresponding to the time between every second transit relative to a ground station (i.e., about one sidereal day). “GPS time” is a time scale reflecting a statistically-averaged “paper clock,” which is a composite of the time recorded by the active, periodically-corrected atomic clock (1 of 4) onboard each of the twenty-four active GPS constellation satellites orbiting in six different orbital planes spaced 60° apart. Therefore, observed periodicity in GPS time must reflect the average effect of TGR on the GPS constellation, which is an engineering system that has been refined over time by empirical methods to reduce dilution of precision. When a satellite rises above the horizon and its signal is first acquired by a ground station and, again, just before occultation of the signal as it descends below the opposite horizon, the component of the signal path transverse to the local gravitational gradient at each point along this path is maximized. Accordingly, the magnitude of the relativistic TGR effect attains a local maximum at these points in the visible portion of the orbit. Contrariwise, the magnitude of the relativistic TGR effect attains a minimum at transit (i.e., the highest elevation angle of the satellite above the horizon relative to the observing ground station). In the context of canonical general relativity, which does not model the relativistic TGR effect, the apparent relative rate of the satellite clock will be anomalously slow upon signal acquisition, speed up as the satellite rises to transit, and then slow down as it descends toward the horizon after transit. In order for GPS satellite clocks to behave correctly, functionally and according to canonical theory, this empirical sinusoidal “frequency drift” of the clocks must be removed. Because the causal physical principle was unsuspected and unmodeled, this could not have been done precisely, but mathematical techniques exist to produce a function very nearly matching an empirically measured curve. The following are excerpts from an article entitled “Characterization of periodic variations in the GPS satellite clocks.” The emphasis has been added. The purpose of inter-comparing clocks is to characterize their performance so that they may be evaluated without systematic or environmental influences. A simple

19 Penina Axelrad & Kristine M. Larson, “GNSS Solutions: Is it true that the GPS satellite geometry repeats every day shifted by 4 minutes?”, InsideGNSS, July/August 2006, 16 – 17; < online >.

© 2015 A. F. Mayer 32 ver. 15.11.05.Z17 quadratic model is often employed to represent any offset in frequency between the two clocks being compared as well as any slowly varying drift in frequency. … The clock data for each satellite were detrended by fitting and removing a second- order polynomial prior to calculating its periodogram. … In order to mitigate the effect of frequency drift, all data were first detrended by removing a second-order polynomial prior to calculating the MDEV.20 When a GPS satellite is near the horizon, this yields the longest intersection of satellite signal path with Earth’s atmosphere, which is associated with signal delays. Also, this is the signal path most likely to suffer from multipath interference. Accordingly, GPS receivers typically incorporate an “elevation mask,” which disregards the signals from any GPS satellite at less than 15° elevation and up to 20° elevation. This practice also eliminates those signals most affected by TGR. The following notebook (Math 4) calculates the predicted magnitude of the TGR effect between a GPS satellite and a ground station as a function of observed elevation angle from the ground station (Fig. 17). In order to graph a smooth curve, a higher density of sample points are required above 70°. The magnitude of the effect at ninety degrees elevation (i.e., zenith) is zero. The observable manifests as a dynamic, symmetric relativistic time dilation that mimics the relativistic effect of substantial changes in the orbital velocity of the satellite as it rises and falls in the sky relative to the observer. Such changes are obviously impossible given its stable, almost-circular orbit.21 It is useful to conceptualize the effect calculated by Eq. (11) as a radial pseudo-velocity (czt). Assuming any attempt to mitigate the TGR effect by empirical methods, the standard approach is “ﬁtting and removing a second-order polynomial.” The last line of Math 4 shows the result of ﬁtting a second-order polynomial curve to the ‘data points’ used to graph the TGR curve. These ‘data’ are the equivalent of perfect measurements, assuming an accurately-modeled empirical effect. The polynomial is graphed as the thin red line in Fig. 18 in comparison to the TGR curve. Note the use of a linear vertical scale in contrast to the log scale used in Fig. 17. This polynomial exhibits a conspicuous sinusoidal deviation from the curve representing the TGR effect, which is assumed to model empirical behavior. Even if this polynomial were effectively removed from the anomalous empirical behavior of GPS satellite instruments in accordance with the operational requirements of the system, a sinusoidal residual remains. Reiterating the most relevant points from the quote at the beginning of this section. In particular, GPS time transfer data from the U.S. Naval Observatory (USNO) indicate that GPS time (for all satellites) is periodic with respect to the Master Clock, which is the most accurate source of official time for DoD. Furthermore, the time obtained from all GPS satellites appears to speed up and slow down in-phase [3].

20 Kenneth L. Senior, Jim R. Ray & Ronald L. Beard, “Characterization of periodic variations in the GPS satellite clocks,” GPS Solut. 12, 211 (2008); < online >. — MDEV: “modiﬁed Allan deviation” 21 GPS constellation SV orbital eccentricity is in the range (0.00022… < e < 0.02263…) with e̅ ≈ 0.00760 per the YUMA almanac for GPS Week 843 representing a total of 32 GPS satellites, c. October 2015.

Eq. (11) is based on physical principles reflecting empirical reality. By inspection, it is not feasible to flawlessly model the TGR effect by empirical methods in order to eliminate its deleterious effect on GPS. Achieving optimum performance of the GPS system requires the correct mathematical model of every empirical phenomenon having an effect on GPS clocks and signals. GPS is an embodiment of modern mathematical physics, so to the extent that the mathematics does not adequately reflect empirical reality, there will be negative consequences for the system’s operational performance.

© 2015 A. F. Mayer 34 ver. 15.11.05.Z17 A critical design feature of GPS is that all SVs (i.e., GPS satellites) have an almost- circular orbit at very nearly the same altitude. In accord with canonical general relativity, a key assumption arising from this feature is that all SV clocks run at the same rate relative to the ground, their relative rate being inﬂuenced exclusively by their gravitational potential and their relative velocity; minor variations in their orbits are precisely measured and have been accounted for. The predicted TGR effect challenges this assumption. Prior to initial implementation and empirical testing, when GPS was originally conceived, it is unlikely that its designers anticipated the need for the frequently- uploaded clock corrections that proved to be necessary in order for the system to function. Canonically, these frequent clock corrections should be unnecessary given the intrinsic stability of the onboard atomic clocks and the similarity of the SV orbits; in practice, the unmodeled TGR effect necessitates them. The graphs’ horizontal scales assume a GPS SV orbit with transit at zenith relative to the observer’s ground station. The top scale, correlated to ψ and expressed in hours and minutes, reflects the time the SV is visible from horizon to zenith. It should be clear that the graphed czt(ψ) curve is valid for any observed SV orbit, where transit typically occurs below zenith (ψmax. < 90°). Note that czt is expressed in cm s−1 where a value of 30 cm s−1 −9 (just above the maximum value of the plotted curve at ψ = 0°) corresponds to zt ≈ 10 .

cz t(ψ) disallowed) < 15° ψ (SVs @ Typical max.elevation mask Typical Typical GPS receiver elevation mask Typical

Fig. 17

Because TGR causes a GPS satellite clock to tick slower relative to a ground clock as an inverse function of observed SV elevation angle, the phenomenon produces a dynamic clock bias. This ‘bad clock’ error is particularly pronounced at low SV elevation angle prior to the ﬁrst SV clock correction applied within the visible portion of the orbit.

© 2015 A. F. Mayer 35 ver. 15.11.05.Z17 The phenomenon mimics atmospheric signal propagation delay that equally affects all signal frequencies (e.g., L1, L2 and L5). Such delay is also expected to increase as an inverse function of SV elevation angle. Currently, TGR-induced clock bias must be erroneously modeled, at least in part, as a component of atmospheric refractive delay, which is inherently uncertain and difﬁcult to model. After accounting for TGR, the actual uncertain magnitude of atmospheric effects will be considerably smaller than previously thought, so one can anticipate that the precision of GPS may be signiﬁcantly improved.

Typical receiver elevation mask (ψ < 15°)

-10 Prior to correction, this zt residual (~10 ) over a period of ~10 min. would yield an accumulated anomalous pseudo-range error of >10 m (clock error >30 ns likely interpreted as a signal delay). Note: low-elevation GPS signal delays that are inexplicably “large” (confdential data) have been privately reported by a team at NASA Goddard Space Flight Center. cz ( t ψ) A sinusoidal residual remains after removing the polynomial.

Fig. 18

Although the TGR effect is induced by the gravitational field and involves no relative motion, it is convenient to conceive of the effect as a radial pseudo-velocity interpreted as a pseudo-expansion between two reference points; all observables are consistent with such a velocity (ztc), as if it were real. Thus, in addition to the measurable first-order Doppler shift (zt) there is an associated time delay (Eq. 24) for the propagation of electromagnetic signals where (d/c) is light travel time for points separated by a distance d in an inertial reference frame. This time delay is graphed in Fig. 18, which considers both change in range to the satellite and change in zt as a function of elevation angle. The definition of d in Eq. (24) references Fig. 7. This graph makes it clear that the TGR effect needs to be taken into consideration as concerns use of GPS for accurate timing and clock synchronization, which has become an important function of the system.

zt d Δt = • d = asin(ϕ2 ) − Rsin(ϕ1 ) (24) c

The plotted curve is the unmodeled on-way delay of a radar round trip; at the horizon, the light travel distance is c∙Δt = 2 cm farther away than the slightly imperfect geometry modeled

Typical max. elevation mask max. elevation Typical by conventional GR.

Note: This curve does not include the efect of TRG-induced clock error. A measured apparent GPS signal-propagation delay due to any such clock error, which is highlighted in Fig. 18, will

disallowed) be much greater. < 15° ψ (SVs @ (SVs Typical GPS receiver elevation mask elevation GPS receiver Typical

Target GPS timing accuracy (0.01 nanoseconds)

Fig. 19

The need for new experiments What makes general relativity a good physical theory it that it is falsiﬁable. In its present form, the theory maintains that two ideal clocks at relative rest and at equal gravitational potential are synchronous. Similarly, textbooks maintain that the round-trip transmission of electromagnetic radiation between any two points at relative rest in a gravitational ﬁeld incurs a net-zero change in wavelength (i.e., photon energy). These predictions are challenged by the claimed phenomenon of relativistic transverse gravitational redshift. A conceivable objection to conducting new experiments is the claim that ‘null results’ of prior experiments demonstrate that the TGR effect does not exist. An experiment with a null result (e.g., Michelson-Morley) has true value only when there was an expectation of a non-null result, which was diligently sought. The commonly-assumed objectivity of modern science is unrealistic: widely accepted scientific dogma has the power to put

© 2015 A. F. Mayer 37 ver. 15.11.05.Z17 rigid boundaries on cognition, to literally control human perception so as to confuse the difference between empirical data and biased interpretation of that data, and to severely limit socially-acceptable scientific discourse. The extent to which this occurs today should not be underestimated. Clinical tests have revealed that perception is controlled both by expectation and by social constraints that contradict empirical reality. Given perfect awareness of an anomaly that is universally considered to be ‘impossible,’ it is common for people not to report such an anomaly or even to overtly deny the anomaly for fear of reputational repercussions. In addition, we do not know about any reported anomalous results that were censored by the peer review process, ostensibly to preserve the ‘quality’ of published information. New information that poses a threat to the status quo often does not meet those subjective ‘quality’ requirements. The reported results of all physical experiments and empirical observations should be considered in the context of expectations. In the past, observation of the TGR effect was universally assumed to be physically impossible, not only because it is inconsistent with canonical general relativity, but because it naïvely appears to be inconsistent with the first law of thermodynamics. Published results (i.e., interpretations) of all relevant prior experiments and empirical observations were influenced by this strong expectation, and thus are inherently scientifically unreliable with respect to providing evidence for or against the TGR phenomenon. Resistance to conducting new experiments could possibly arise in some segments of the experimental physics community. The majority of professional scientists take pride in what they perceive to be their scientific objectivity, but many (and perhaps most) are oblivious to the overriding psychological effect of expectation. Reporting observation of the TGR effect will demonstrate a lack of scientific objectivity in the prior work of many individuals and the institutions they represent, including many of those who are most capable of conducting the required new experiments. Accordingly, experimentalists should include those who do not have a strong personal and professional incentive to report a null result in order to demonstrate their prior scientific objectivity. The only valid authority in physics is the accurately reported results of well-defined, repeatable empirical tests with realistic (i.e., accurate) error bars, which are conducted by numerous teams of impartial experimentalists. The Math 2 notebook, can be expected to motivate many experimental physicists with the wherewithal to rigorously test the empirical prediction with a new experiment because this simple notebook changes expectations concerning the outcome of the experiment based on the tenets of mathematical physics. Such experimentation offers the real and thrilling possibility of falsifying a fundamental pillar of modern physics, thereby instigating a major scientific revolution, which is arguably the highest goal of experimental physics. A valid null result, although not nearly as exciting, would serve the important function of falsifying the theoretical ideas and empirical predictions presented in this report, however, this is unlikely due to the rigor of the underlying theoretical ideas, which rest on first principles. Under the circumstances, and given the history of expunging or editing data conflicting with long-accepted and widely-respected canonical theory, it may be better personal and scientific practice to challenge and verify any claimed null result appearing in the literature, rather than to simply trust the source.

© 2015 A. F. Mayer 38 ver. 15.11.05.Z17 Cosmology Disregarding the local gravitational fields of respective host clusters, galaxies are clearly immersed in a homogeneous and isotropic universal (i.e., cosmological) gravitational field produced by the sum-total of gravitating cosmological mass-energy. According to the cosmological principle, which reflects symmetry considerations, no galaxy has a preferred cosmological location, so all galaxies are at the same ‘gravitational potential’ with respect to this ubiquitous field. This is similar to the fact that no point on a spherical shell of constant coordinate radius in a conventional static symmetric gravitational field having an idealized point-like source has a preferred location, and that all such points are at the same gravitational potential relative to this source. Referring back to Fig. 1, which reflects the basic underlying principles of general relativity, review the simple fact that the gravitational field produces a distance-induced bilateral redshift between distinct ideal clocks at the same coordinate radius in the field. Thus, according to these same principles, one may anticipate a redshift-distance relationship between galaxies that is exclusively induced by the cosmological distance between them. Accordingly, the widely-accepted idea that a general expansion of space between galaxies is required to explain the observed cosmological redshift is immediately put in jeopardy. In addition to the foregoing, the revolutionary concept of relativistic temporal geometry presented herein, as a fundamental interpretation of relativity theory, has immediate application to cosmology. Only one basic underlying logical assumption is required in addition to the cosmological principle: the Universe has a finite boundaryless volume that contains a finite amount of universally-conserved mass-energy. The cosmological principle asserts that, at cosmological scale, this mass-energy is uniformly distributed throughout this finite volume and the resulting uniform large-scale mass density implies cosmologically-uniform spatial curvature. The geometric symmetry associated with such curvature implies that cosmic space may be accurately modeled by the volumetric ‘surface area’ (S3) of a Riemannian 3-sphere. A key insight provided by the concept of relativistic temporal geometry is that no universal “conformal time” coordinate can be associated with this manifold: An arbitrary 2-dimensional ‘slice’ of finite, boundaryless cosmic space is modeled by a 2-sphere and the neighborhood of each point on this surface represents a unique localized free-falling inertial reference frame associated with a geometrically-unique local proper time coordinate. Because the redshift-distance relationship implied by the geometric relationship between such time coordinates clearly defies both conventional understanding and prior perception of the phenomenon, a new analysis of astrophysical data, motivated by this new expectation, was required. Starting with the first Hubble diagram (1929) published in a prestigious scientific journal (Proceedings of the National Academy of Sciences of the United States of America), a significant amount of seemingly reliable empirical evidence using many different observations, instruments and techniques has been published by numerous authorities in support of the ‘Hubble law’ (i.e., a linear cosmological redshift-distance relationship). However, based on existing and easily-reproducible analysis of various data sets of unprecedented size and quality yielded by the Sloan Digital Sky Survey (SDSS),22 all

22 Sloan Digital Sky Survey Data Release 12 (6 January 2015); < online >.

© 2015 A. F. Mayer 39 ver. 15.11.05.Z17 prior empirical evidence published in support of a linear redshift-distance relationship is revealed to be patently false: When confronted with SDSS data of unparalleled objective reliability, all of the predictive curves generated by the Lambda Cold Dark Matter (LCDM) “concordance cosmological model” fail spectacularly.23 Moreover, any variation of the numerous free parameters associated with that widely-accepted model cannot ameliorate this predictive failure as the dominant feature of the model is a redshift-distance relationship that is quite obviously radically incorrect. In contrast, the predictive cosmological equations of a new cosmological model (NCM) associated with the new concepts introduced herein all correlate to the data with astonishing accuracy. The aforementioned data analysis, which is presented in an online slide deck, conflicts with expectations of reality and requires detailed personal review and technical criticism: http://www.SensibleUniverse.net/Cosmos Upon review of the above, one realizes that the pursuit of science in recent decades was not nearly as objective as generally thought to be the case. A lesson to be learned is that modern science is strongly affected by the Asch effect (psychology), defined as “a form of conformity in which a group majority influences individual judgments.” Together with Carl W. Wirtz (1922) and Ludvik Silberstein (1924), Georges Lemaître was among the first cosmologists to hypothesize that galaxy redshifts were caused by general cosmic expansion,24 later explicitly proposing a primordial cosmic creation event, now commonly known as the “Big-Bang.”25 Based on analysis of redshift-distance estimates published by Edwin Hubble in 1926,26 Lemaître’s explicit seminal paper on the subject gave an estimate for the current-epoch expansion rate of 625 km s-1 Mpc-1.27 Lemaître and Hubble initially met in 1925 at Caltech/Mount Wilson28,29 and again in 1928 when they both attended the International Astronomical Union (IAU) conference.30 In 1929, Hubble published his famous “Hubble diagram,” a graph of empirical galaxy redshift-distance measurements including the suggestion of a linear redshift-distance -1 -1 relationship having a slope (“Hubble constant,” H0) of 500 km s Mpc .31 This now famous and widely-accepted purported relationship became known as the “Hubble law.”

23 Alexander F. Mayer, The SDSS Renaissance: The End of the ‘Dark Age’ in Cosmology, Part I, (21 October 2015); < online >. 24 Giora Shaviv, “Did Edwin Hubble plagiarize?”; arXiv:1107.0442 [physics.hist-ph]; < online >. 25 G. Lemaître, “Evolution of the Expanding Universe,” PNAS 20(1) 12 (1934); < online >. 26 Edwin Hubble, “Extragalactic nebulae,” ApJ 66, 321 (1926); < online >. 27 M. l’Abbé G. Lemaître, “Un Univers Homogéne de Masse Constante et de Rayon Croissant…,” Annales de la Société Scientiﬁque de Bruxelles A47, 49 (1927); < online >. 28 Jeremiah P. Ostriker & Simon Mitton, Heart of Darkness: Unraveling the Mysteries of the Invisible Universe, Princeton: Princeton Univ. Press, 2013, p. 68. 29 John Farrell, The Day Without Yesterday: Lemaître, Einstein, and the Birth of Modern Cosmology, New York: Thunder’s Mouth Press, 2005, p. 78. 30 Sidney van den Bergh, “Discovery of the Expansion of the Universe,” J. Roy. Astron. Soc. Can. 105(5), 197 (2011); < online >. 31 Edwin Hubble, “A Relation Between Distance and Radial Velocity Among Extra-Galactic Nebulae,” PNAS 15, 168 (1929), p. 172; < online >.

© 2015 A. F. Mayer 40 ver. 15.11.05.Z17 It was later realized that the inverse of H0 yields a close approximation to the modeled age of the Universe (“Hubble time,” T ). Hubble’s 1929 ‘measurement’ of H0 = 500 implies T < 2 Gyr, which is clearly not sensible. A lower value for expansion rate yields a greater cosmic age with the current-value “1% concordance Hubble constant”32 (H0 = 69.6 ± 0.7) yielding a modeled cosmic age of T ≈ 14 Gyr. Although it is virtually certain that Hubble was familiar with Lemaître’s hypothesis of an expanding universe in 1929, he failed to acknowledge it, thus giving the impression that his graph reflected a purely empirical discovery that was not influenced by a preexisting suggestive theoretical model. According to the known sequence of historical events and Hubble’s much-exaggerated estimate of a conceivable cosmic expansion rate, it seems likely that Hubble arbitrarily fit a linear relationship to the data in order to claim empirical discovery of an extremely significant physical phenomenon. This tactic had its intended effect as Hubble became the most famous astronomer in history, both in his own time and after his death, even to the present day. This achievement was likely aided by the fact that Hubble’s wife was a millionaire socialite, whose wealth and influence during the economic environment of the Great Depression (1929 – 1939) probably contributed to the numerous glowing articles in the popular press about America’s “great astronomer.” Additionally, as he was tall, athletic, handsome and a talented conversationalist, Hubble was a natural celebrity, apparently with many obsequious admirers. Lemaître’s work was at least in part motivated by biblical creationist ideation according to his unpublished 1921 essay, God’s First Three Declarations: “an attempt to describe scientifically the first verses of Genesis.”33 This motivation, combined with mathematical talent and an avid interest in astronomy, led Lemaître to interpret the general theory of relativity in the context of cosmology, a new field of study initiated in 1917 with seminal papers by Albert Einstein and Willem de Sitter. In February, Einstein pioneered the field with a paper proposing the addition of a cosmological constant to the field equations.34 In November, de Sitter published the third and final paper of a series in which he reified Einstein’s exact solution to the field equations, expressing it as an explicit cosmological metric (Eq. 25), also deriving a similar exact solution (Eq. 26), which incorporated a distance-dependent cosine term as a coefficient to the time variable.35 r ds2 = −dr2 − R2 sin2 ⎡dψ 2 + sin2 ψ dθ 2 ⎤ + c2dt 2 (25) R ⎣ ⎦

r r ds2 = −dr2 − R2 sin2 ⎡dψ 2 + sin2 ψ dθ 2 ⎤ + cos2 c2dt 2 (26) R ⎣ ⎦ R

32 C. L. Bennet, D. Larson, and J. L. Weiland, “The 1% Concordance Hubble Constant,” ApJ 794, 135 (2014); < online >. 33 Helge Kragh, Matter and Spirit in the Universe: Scientiﬁc and Religious Preludes to Modern Cosmology. London: Imperial College Press, 2004, p. 141; < online >. 34 Albert Einstein, “Cosmological Considerations on the General Theory of Relativity” (1917) in The Principle of Relativity, New York: Dover, 1952, p. 175. 35 Willem de Sitter, “Einstein's theory of gravitation and its astronomical consequences. Third paper,” MNRAS 78, 3 (1917); < online >.

© 2015 A. F. Mayer 41 ver. 15.11.05.Z17 The cosmological-distance-induced relativistic time dilation modeled by this cosine term in the metric became known as the “de Sitter effect.” To his credit, Hubble mentioned this theoretical phenomenon as a possible alternate explanation of galaxy redshifts in the last paragraph of his famous 1929 ‘Hubble diagram’ paper: The outstanding feature, however, is the possibility that the velocity-distance relation may represent the de Sitter effect, and hence that numerical data may be introduced into discussions of the general curvature of space. … it may be emphasized that the linear relation found in the present discussion is a first approximation representing a restricted range in distance.36 In the context of relativistic temporal geometry, whereby the time variable in any such metric has inherent geometric properties, the explicit cosine term in Eq. (26) expresses the fundamental geometric relationship (i.e., ubiquitous local orthogonality) between the local time variable and local spatial variables. Selecting an arbitrary 2-dimensional slice of uniformly-curved three-space by setting θ = 0, an illuminating model of the resulting submanifold is that of a ﬁnite, boundaryless spatial spherical surface with radius R. Clarity of the mathematics is improved by the following variable substitution, whereby the normalized (R = 1) distance r of an arbitrary point from an observer at the origin of coordinates is expressed in radians by the angular parameter [0 ≤ χ ≤ π].

2 2 2 2 2 2 ⎡ r ⎤ dτ = −dχ − sin χ (dψ ) + cos χ (dt ) ⎢ dτ ≡ ds χ ≡ R = 1 c = 1 ⎥ (27) ⎣⎢ R ⎦⎥

The neighborhood of every unique point p on this sphere, which may be visualized as a minute square patch (dx, dy)p at coordinate ( χ, ψ)p [0 ≤ ψ ≤ 2π] represents a free falling inertial reference frame. The applicable Minkowski metric, reﬂecting Einstein’s strong equivalence principle, requires orthogonality of local differential space (dx, dy)p and time (dτp) coordinates. Accordingly, the strictly-local geometric deﬁnition of the local proper time dimension is represented by the local vertical (i.e., a temporal vector coordinate collinear with the geometric radial coordinate, R). As such, the neighborhood of each distinct point on the surface is associated with a distinct local proper time coordinate. The fundamental non-parallelism of such spatially-separated time coordinates naturally implies a symmetric relativistic time dilation similar to the schematic representation seen in Fig. 8, but extending over the surface of a sphere. Relative to an observer referencing local time (dt), the rate of a remote ideal clock (dτ) is a function of the cosmological distance represented by the angular parameter χ between the clocks. dt 1 = (28) dτ cos χ

Symmetry considerations imply that no observer has a preferred frame of reference; this distance-induced time dilation effect is bilateral; each observer measures the remote ideal clock, which may be instantiated by a photon having a known arbitrary rest-frame

36 Hubble, op. cit., p. 173.

dt 1 ⎡ π ⎤ z ≡ −1 = −1 0 ≤ χ ≤ (29) dτ cos χ ⎣⎢ 2 ⎦⎥

A marked peculiarity of de Sitter’s solution to the ﬁeld equations leading to the modeled “de Sitter effect” is that this solution assumes a zero cosmic energy density, which may seem to be nonsensical. However, Eq. (28) implies time reversal between antipodes; equal energy-density magnitudes for any two arbitrary cosmological hyper-hemispheres, each representing half of the Universe, are opposite in sign. Then locally, ρ > 0 everywhere, but the net cosmic energy density from a relativistic perspective is zero.

Conclusion Simple and intuitive ideas accompanied by elegant predictive mathematics are the hallmark of good physics, meaning physics that describes with reasonable accuracy how the physical Universe actually works, also suggesting how that description of reality can be reliably veriﬁed by repeatable experiments. A “repeatable experiment” means one that is conceived with the expectation that it will be repeated, and that it is actually repeated numerous times by different impartial experimentalists. Such an expectation necessarily puts economic and time-to-results constraints on such experiments.

Acknowledgements I thank the following individuals for their valuable friendship and encouragement: Mr. Michael Yang http://www.columbususa.com (@ NASA GSFC) Dr. Michael Fiske http://aemea.org Mr. Tom Phinney* https://www.isa.org/ustag-iec-sc65c/ Mr. Tadeusz Glinkiewicz http://www.put.edu.pl/institution Mr. Mark Anderson http://markawriter.com Dr. Arjen Dijksman http://blog.espci.fr/qdots/people/dijksman/ Dr. John Manoyan http://gahmusa.org/index.php/foundation-leaders/ Mr. Neree Dastous http://www.sf-ﬁre.org Dr. Selim Shahriar http://tinyurl.com/sshahriar (salient M.I.T. classmate)

* I thank Mr. Phinney for contributing his technical editorial expertise to this report. He has been the primary author or editor of over 15,000 pages of international computer communications standards. In 2003 he was recognized by ISA as one of the 50 most inﬂuential people in modern world history credited with advancing automation, instrumentation, and control technologies. (The ﬁrst of these 50 was James Watt, recognized for the invention of the steam-engine governor.) In 2005 Tom was the recipient of the IEC’s 1906 award, which recognizes major contributions to furthering the interests of worldwide electro-technology standardization. He is a Honeywell Senior Fellow, an ISA Fellow, and the US Technical Advisor to ANSI for digital communications within the industrial process measurement and control industries (IEC/SC 65C).

“eiπ +1 = 0 : Analysis Incarnate” – John W. Morgan

Denying the validity of [log(-1) ∈ ℂ]… shatters the foundation of all analysis, which consists principally in the generality of the rules and operations which are deemed true, whatever the nature which one supposes for the quantities to which they are applied. – Leonhard Euler (n.b., this notion was denied!)37

It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. – Georg Friedrich Bernhard Riemann

The views on space and time which I wish to lay before you have sprung from the soil of experimental physics. Therein lies their strength. Their tendency is radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality. – Hermann Minkowski

The basic idea is to present the essentials of relativity from the Minkowskian point of view, that is, in terms of the geometry of space-time … because it is to me (and I think to many others) the key that unlocks many mysteries. My ambition has been to make space-time a real workshop for physicists, and not a museum visited occasionally with a feeling of awe. – John Lighton Synge

However successful the theory of a four-dimensional world may be, it is difficult to ignore a voice inside us which whispers: “At the back of your mind, you know that a fourth dimension is all nonsense.” I fancy that that voice must ofter have had a busy time in the past history of physics. What nonsense to say that this solid table on which I am writing is a collection of electrons moving with prodigious speeds in empty spaces, which relatively to electronic dimensions are as wide as the spaces between the planets in the solar system! What nonsense to say that the thin air is trying to crush my body with a load of 14 lbs. to the square inch! What nonsense that the star cluster which I see through the telescope, obviously there now, is a glimpse into a past age 50,000 years ago! Let us not be beguiled by this voice. It is discredited. – Arthur Stanley Eddington

37 Misguided mathematicians insisted that either log(-1) = 0 (e.g., D’Alembert & John Bernoulli), or that it simply did not exist (e.g., Leibniz); Morris Kline, Mathematical Thought From Ancient to Modern Times, Volume 2, (Oxford University Press, 1990), pp. 406-411.

© 2015 A. F. Mayer 44 ver. 15.11.05.Z17 Addendum: Derivation of Eq. 11 (see page 12) Einstein’s empirically-verified “sufficient approximation” formula for the bending of light in a weak gravitational is derived from the field equations:38 4µ 2R α = = S (30) bc2 b This angle evaluates to about 0.6 milliarcseconds on the geoid and 1.75 arcseconds at the mean Solar radius. Accordingly, evaluating the radial distance between the center of gravity and a point on the trajectory of light through a typical weak astrophysical field very closely approximates a linear photon trajectory, yielding a simple relationship: b r = (31) cosϕ Eq. 31 defines the distance r between a point on a line and a point not on this line, where reference angle φ = 0 corresponds to the perpendicular distance b. Referencing Fig. 9, consider two arbitrary points separated by the angle dϕ on a spherical surface with coordinate radius r. The respective geometric representations of the distinct local proper time coordinates at these two points constitute adjacent generatrices (g) of a right circular cone with directrix 2πr and aperture 2η. From this geometric picture we have the relationship r sinη = (32) g Combining this with the definition of η (Eq. 18) yields

r R r 3 = S → g = (33) g r RS

The two generatrices constitute two equal legs of an isosceles triangle with base r·dϕ. Deﬁne the vertex angle of this triangle as the inﬁnitesimal dχ. The geometry implies

r dϕ R R dχ = = r dϕ S = S dϕ (34) g r 3 r Plugging in the deﬁnition of r = f (φ) from Eq. (31) into the above yields

dχ R cosϕ = S (35) dϕ b

38 Derived simply and resting exclusively on first principles, a neoteric general solution for gravitational lensing in a spherically-symmetric static field, which closely approximates fields of typical astrophysical bodies, yields a trajectory defined to have eccentricity e = b/RS. Accordingly, Einstein’s 1916 formula is −1 an accurate approximation in the weak field of the exact general predictive equation: α = 2sin (RS/b).

RS RS ⎛ ϕ ⎞ χ = cosϕ dϕ = 2 E 2 (36) b ∫ b ⎝⎜ 2 ⎠⎟

Being analogous to angle ζ appearing in the Fig. 8 schematic, the angle χ in Eq. (36) represents a symmetric divergence in spacetime between distinct local proper time coordinates that is independent of an asymmetric divergence (Eq. 18) associated with any change in coordinate radius between such coordinates. This symmetric divergence implies a symmetric time dilation between respective locations that is quantiﬁed by the geometric projection of the remote time coordinate unit vector on the local time coordinate unit vector (i.e., the inverse dot product): dt = sec χ (37) dτ By deﬁnition, there is a corresponding redshift:

zt = sec χ −1 (38) Eq. (11), repeated below as Eq. (39), is the resulting empirical prediction for the transverse gravitational redshift in terms of known values where angles φ1 and φ2 can take on any values within the limits speciﬁed, according to any two arbitrary endpoints.

⎡ ϕ2 ⎤ ⎧ ⎫ RS ⎛ ϕ ⎞ ⎤ ⎪ ⎡ π π ⎤ ⎪ zt = sec ⎢2 E ⎜ 2⎟ ⎥ ⎥ −1 ⎨ [b > 0] & − ≤ ϕ ≤ ⎬ (39) ⎢ b ⎝ 2 ⎠ ⎥ ⎣⎢ 2 2 ⎦⎥ ⎣ ⎦ϕ1 ⎦ ⎩⎪ ⎭⎪

−π/2 +π/2 @ −∞ φ1 φ2 @ +∞ b 1 2 φ = 0 Fig. 20