Slicing a Light Sphere by Another Frame's Simultaneity Gives an Ellipse of Eccentricity β with the Emission Point at a Focus ── The radius in each direction is exactly the Doppler factor for that direction, the two vertices are k and 1/k, the semi-major axis is γ, the semi-minor axis is cT, and the centre is where the moving observer stands ── the separator is whose simultaneity does the cutting ── [Paper 1036]

Abstract

Light emitted at the origin is said to spread as a sphere in every frame. This paper counts what comes out when that sphere is cut by the simultaneity of another frame. No new theorem and no new law is claimed. Scope of this paper (scope note): No new theorem and no new law is claimed──the Lorentz transformation, the invariance of the light cone, the relativity of simultaneity, the relativistic Doppler effect and the fact that a plane section of a cone is a conic are all standard (Einstein 1905, Rindler 2006). No dynamics is imported──nothing sits at the focus. The polar equation has the same form as a Kepler orbit because both are ellipses written with a focus at the origin, not because a force is involved. Visual appearance is not discussed──what is treated here is a figure in coordinates cut by a simultaneity surface; the image formed by light that actually arrives belongs to Paper 901. The convention on the one-way speed of light is untouched──only a pair of inertial frames with standard synchrony is used, and the synchronisation convention of Paper 24 is not varied. Acceleration is not treated──two inertial frames only; no observer that turns around. The values are examples──the six values of β from 0.1 to 0.99 and cT=1 are for comparison and no real apparatus is claimed. Relation to earlier papers:Paper 61 read c as the light sphere r=ct and drew volume and the inverse-square law from it──here it is counted that this sphere ceases to be a sphere under another frame’s simultaneity while the light cone does not change. Paper 118 showed that coordinate time dt has no domain of its own and becomes a clock only once attached to a congruence of world lines──here that choice of label is shown, in a single figure, to make the same light either a sphere or an ellipse. Paper 491 showed that the classical Doppler effect is two formulas and that relativity is their geometric mean──here that geometric mean is the near vertex of the ellipse, and the product of the vertices is the square of the semi-minor axis. Paper 1026 showed that a receding clock tower looks stopped because of light delay and that relativity rather speeds the image up by γ──here that (1−β)γ appears as the near vertex. Paper 901 showed that a moving sphere does not look contracted──that is the image made by arriving light, this is the slice by a simultaneity surface: two different quantities about the same moving sphere. What is added is obtaining by three independent routes that the eccentricity of the slice is exactly β and that a focus is the emission point, showing that the radius in each direction is exactly the Doppler factor for that direction, obtaining that the two vertices are k and 1/k with product equal to the square of the semi-minor axis, that the centre coincides with the position of the moving observer, and counting that the volume ratio is γ in every dimension. First, and this is the core. The light sphere is not an invariant; the light cone is──when the points cut by the simultaneity surface of a frame S’ moving at speed β are read in the original frame S, the radius is not constant: at β=0.6 it runs from 0.500000000000 to 2.000000000000. Yet for all six values of β the points lie on the same light cone, within 10 to the minus 13 (Section 2). Second, the slice is an ellipse whose eccentricity is exactly β──three independent routes agree: a least-squares conic fit, the sum of distances from the two foci, and the polar equation. At β=0.6 the eccentricity is 0.600000000000, the semi-major axis 1.250000000000 (=γ), the semi-minor axis 1.000000000000 (=cT) and the semi-latus rectum 0.800000000000 (=cT/γ), and one of the two foci is the emission point itself (Section 2). Third, the two vertices are the relativistic Doppler factors──at β=0.6 the far vertex is 2.000000000000=k and the near vertex 0.500000000000=1/k, and their product is the square of the semi-minor axis, 1.000000000000. The near vertex is the same number that Paper 1026 obtained as the advance of the receding image, (1−β)γ, and the geometric mean 0.577350269190 of the two classical formulas 0.500000000000 and 0.666666666667 of Paper 491 (at β=0.5) sits here as well (Section 3). Fourth, the radius in each direction is exactly the Doppler factor for that direction──the radius r(θ) agrees with 1/(γ(1−β cosθ)) within 10 to the minus 12 for all six values of β. The ellipse is therefore a polar plot of the Doppler factor. Sideways the radius equals the semi-latus rectum cT/γ exactly, 0.800000000000 at β=0.6, the direction in which only time dilation is left (Section 4). Fifth, the centre is where the moving observer stands, and the volume is γ times larger in every dimension──the centre 0.750000000000 (β=0.6) coincides with the position βγcT of the observer in S at that time. Numerical integration gives a three-dimensional volume of 5.235987756, which is 1.250000000 times (=γ) the sphere’s 4.188790205. Since only the long axis is stretched, the ratio stays γ from d=2 to d=5 (Section 5). Sixth, the separator is whose simultaneity does the cutting──cut by the same frame’s simultaneity the radius is constant and the figure is a sphere; cut by another frame’s it is an ellipse. The simultaneity surface tilts even when one merely walks: the now at 2.5 million light years moves by 4.230357 days at 5 km per hour and by 248.338468875 years at the orbital speed of the Earth. What moved is the surface, not the light cone (Section 6 and Section 7). Slicing a light sphere by another frame’s simultaneity gives an ellipse of eccentricity β with the emission point at a focus. At β=0.6 the semi-major axis is 1.250000000000, the semi-minor axis 1.000000000000, the semi-latus rectum 0.800000000000 and the centre 0.750000000000; the two vertices 2.000000000000 and 0.500000000000 are the relativistic Doppler factors k and 1/k, and their product is the square of the semi-minor axis, 1.000000000000. The radius in each direction is exactly the Doppler factor of that direction, the three-dimensional volume is 1.250000000 times the sphere, and this ratio is γ in every dimension──the centre is where the moving observer stands, and that observer, measuring with their own simultaneity, finds cT in every direction. The separator is whose simultaneity does the cutting. Placed among the earlier papers──Paper 61 read c as the light sphere r=ct, Paper 491 gave the two classical formulas and their geometric mean, Paper 1026 the advance (1−β)γ of the receding image, and Paper 118 that coordinate time is a label. Here those three numbers are set on one ellipse as its vertices and semi-latus rectum, and it is counted that the choice of label is what decides between sphere and ellipse. To be honest──not one formula is new; what is new is only putting three numbers on a single slice of one light cone. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: special relativity, relativity of simultaneity, light cone, ellipse, Doppler factor, Lorentz transformation. ----- 原点で光った光は、どの系でも球に広がると言われる。本稿は、その球を「別の系の同時」で切ると何が出るかを数える。新しい定理も法則も主張しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──ローレンツ変換、光円錐の不変性、同時の相対性、相対論的ドップラー効果、平面で切った円錐が円錐曲線になることは、いずれも既知である(アインシュタイン 1905、リンドラー 2006)。力学を持ち込まない──焦点に何かが在るわけではない。ケプラー軌道と同じ形の極方程式が出るのは、どちらも焦点を原点に取った楕円だからであって、力の話ではない。見かけの形を論じない──本稿が扱うのは同時面で切った座標の上の図形であり、目に届く光がつくる像は論文901 の側である。片道の光速の規約に入らない──標準同期を置いた二つの慣性系の組だけを使い、同期の取り決め(論文24)は動かさない。加速を扱わない──二つの慣性系だけを置き、向きを変える観測者は扱わない。値は例である──β=0.1 から 0.99 までの六つと cT=1 は比べるための値で、実在の装置を主張しない。既刊との関係:論文61 は、c を光球 r=ct として読み、そこから体積と逆二乗を出した──ここでは、その球が別の系の同時では球でなくなることと、それでも光円錐は変わらないことを数える。論文118 は、座標時間 dt が定義域を持たないラベルであり、世界線の族に貼り付けて初めて時計になることを示した──ここでは、そのラベルの選び方が同じ光を球にも楕円にもすることを、一枚の図形で見せる。論文491 は、古典のドップラーが二つの式であり、相対論がその幾何平均であることを示した──ここでは、その幾何平均が楕円の近い頂点であり、頂点どうしの積が短半径の二乗であることを数える。論文1026 は、離れる時計台の針が止まって見えるのは光の遅れで、相対論はむしろ像を γ 倍だけ速めることを示した──ここでは、その (1−β)γ が楕円の近い頂点として現れる。論文901 は、動く球が縮んで見えないことを示した──あちらは目に届く光の像、こちらは同時面の切り口であり、同じ「動く球」の別の量である。加えたのは、切り口の離心率がちょうど β であり焦点が発光点であることを三つの独立な道で出したこと、各方向の動径がその方向のドップラー因子そのものであることを示したこと、二つの頂点が k と 1/k で積が短半径の二乗になることを出したこと、中心が動く観測者の位置に一致すること、体積の比がどの次元でも γ になることを数えたことである。 第一に、これが本稿の芯である。光の球は不変量ではなく、不変なのは光円錐である──速さ β で動く系 S’ の同時面で切った点を、もとの系 S で見ると、動径は一定ではなく、β=0.6 では 0.500000000000 から 2.000000000000 まで開く。それでも六つの β すべてで、点は同じ光円錐の上に在る(ずれは 10 の −13 乗以下)(第2節)。 第二に、その切り口は、離心率がちょうど β の楕円である──円錐曲線の当てはめ・二焦点からの距離の和・極方程式という三つの独立な道が同じ数を返した。β=0.6 で離心率 0.600000000000・長半径 1.250000000000(=γ)・短半径 1.000000000000(=cT)・半通径 0.800000000000(=cT/γ)であり、二つの焦点のうち一つは発光点そのものである(第2節)。 第三に、二つの頂点は、相対論のドップラー因子である──β=0.6 で遠い頂点 2.000000000000=k、近い頂点 0.500000000000=1/k であり、二つの積は短半径の二乗 1.000000000000 である。近い頂点は論文1026 が離れる像の進みとして出した (1−β)γ と同じ数で、論文491 の古典の二つの式 0.500000000000 と 0.666666666667 の幾何平均 0.577350269190(β=0.5)もここに乗る(第3節)。 第四に、各方向の動径は、その方向のドップラー因子そのものである──動径 r(θ) は 1/(γ(1−β cosθ)) に六つの β すべてで 10 の −12 乗以内で一致した。すなわちこの楕円は、ドップラー因子を向きごとに極座標へ描いた図である。横向きでは動径は半通径 cT/γ に厳密に等しく、β=0.6 で 0.800000000000──時間の遅れだけが残る向きである(第4節)。 第五に、中心は動く観測者の位置であり、体積はどの次元でも γ 倍になる──楕円の中心 0.750000000000(β=0.6)は、S でその時刻に観測者が居る位置 βγcT に一致した。三次元の体積は数値積分で 5.235987756 となり、球の 4.188790205 の 1.250000000 倍(=γ)である。長軸だけが γ 倍になるので、この比は d=2 から 5 まで γ のままである(第5節)。 第六に、分離子は、切る面が誰の同時かである──同じ系の同時で切れば動径は一定で球、別の系の同時で切れば楕円になる。同時面はこちらが歩くだけでも傾き、250 万光年先の「いま」は時速 5 km で 4.230357 日、地球の公転の速さで 248.338468875 年動く。動いたのは面であって、光円錐ではない(第6節・第7節)。 光の球をよその系の同時で切ると、発光点を焦点とする離心率 β の楕円になる。β=0.6 で長半径 1.250000000000・短半径 1.000000000000・半通径 0.800000000000・中心 0.750000000000 であり、二つの頂点 2.000000000000 と 0.500000000000 は相対論のドップラー因子 k と 1/k、その積は短半径の二乗 1.000000000000 である。各方向の動径はその方向のドップラー因子そのもので、三次元の体積は球の 1.250000000 倍、この比はどの次元でも γ である──中心は動く観測者の居る位置で、その人が自分の同時で測れば四方どこでも cT である。分離子は、切る面が誰の同時かである。既刊の中に置けば──論文61 は光球 r=ct を、論文491 は古典の二つの式とその幾何平均を、論文10