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Principles of Time and Space Hiroshige Goto Principles of Time and Space New Octonions and Relativity Third Edition Hiroshige Goto KIYOTAKI Fukuoka Japan This is the third edition of the Japanese book (ISBN 978-4-9906664-0-8) published in September 2012. The text can be read at http://gotoclinic.org/space-time.pdf. Download is free for personal use. The author's e-mail address is [email protected]. Copyright c 2014 by Hiroshige Goto Published by KIYOTAKI 1-5-2-1301 Kiyotaki, Moji, Kitakyushu, Fukuoka 801-0833, Japan e-mail: [email protected] This book can be purchased only at Amazon.co.jp (www.amazon.co.jp). ISBN 978-4-9906664-2-2 Printed in Japan on acid-free paper This book is dedicated to my beloved wife Rieko Preface to the Revised Edition The quaternion is a complex number that extends into four dimensions. In this book, the quaternion is transformed into a number in the curved four-dimensional space{time and the transformed quaternion is called the new octonion. Because the new octonion consists of four real numbers and four imaginary numbers, four kinds are thought to exist (similar to the real and imaginary numbers), each in our space{time. In addition, if the structure of the four-dimensional space{time is examined using the new octonion, space is considered to have a double structure. In other words, two four-dimensional space{time structures overlap. Furthermore, when using the new octonion, the conclusions of special relativity can be explained without contradiction if the axiom that mass is the time component of the unit world line is imposed. This leads to the conclusion that in our world, mass and energy are expressed as imaginary numbers. An algebraic theorem states that a numerical system with the algebraic opera- tions of addition, subtraction, multiplication, and division is composed only of real numbers, complex numbers, quaternions, and octonions. Therefore, it is natural to assume that the physical laws of four-dimensional space{time must be described with quaternions. William Rowan Hamilton, who discovered the quaternion in 1843, dedicated the latter half of his life to applying quaternions to physics but without success. However, the concept of vectors came to light through his efforts; the concept of vector led to the development of the concept of tensors. Tensors are in- dispensable to mathematics in modern physics and engineering; in general relativity, black holes were predicted and described using tensors. Because the tensor is a mathematical tool by which curved four-dimensional space{time is described, it is widely thought that using the quaternion to study space{time would represent a backward motion|an unnecessary and fruitless en- deavor. However, tensor has a complex matrix form. The purpose of natural science is to obtain simple laws from complex phenomena. I feel that the usage of a tensor is significantly complex for describing physical laws. This study starts by proposing that a simple algebra such as the quaternion may indeed be suitable to describe the laws of space{time. Therefore, I have tried to create an algebra using only ad- dition, subtraction, multiplication, and division to study curved four-dimensional space{time. vii Hamilton did not succeed in applying quaternions to physics because the theory of relativity did not exist at that time. Hamilton's quaternion is the mathematics of flat space{time; however, space{time curves in the theory of relativity. Physical laws must be described with mathematics that matches the curved space{time. In other words, if the quaternion is not altered to suit the demands of the curved space{time, physical laws will not be described by quaternions. First, the complex number is transformed to fit the mathematics of Lorentz trans- formations and is called the new complex number. Similar to the complex num- ber, the quaternion is transformed into the curved space{time quaternion. This is dubbed an octonion because it includes eight numbers. The Graves' octonion consists of one real number and seven imaginary numbers, but the octonion of the curved space{time consists of four real numbers and four imaginary numbers. Therefore, it is termed a new octonion. If the space{time structure is examined with the new octonion and the mechanics of special relativity is recalculated, new interpretations of the physical principles and double structure of space{time are suggested. However, the space{time diagram suggested by the new octonion may disagree with actual space{time even if the new octonion is mathematically consistent. It is well known that we cannot use Euclidean geometry, which is consistent in flat space, to study curved space{time. The actual space{time may not have a double structure even if it is suggested by the new octonion. Future problems include whether the new octonion is mathematics without contradiction and whether the actual space-time is identical to that suggested by the new octonion. The desire to learn the truth is a motivational force of science. Steven Paul Jobs, one of the founders of Apple Computer, Inc., was quoted as saying \Stay hungry, stay foolish," which means that we should always seek better things and not be tethered to preconceived notions. The physics described by tensors is accomplished. However, it is a scientific action to seek a different mathematics to describe curved space{time. Because the new octonion is simple mathematics, it can be understood even by a high school student. It is my wish that people who are interested in physics and mathematics and who want to learn the truth must acknowledge this book and use the new octonion in their studies. Hiroshige Goto August 2012 viii Abstract The contents of this book are summarized here. Because the technical terms are explained in the text, they are used here without explanation. Complex numbers have been expanded to Hamilton's quaternion in four dimen- sions. This study aims to transform the quaternion into mathematics suitable for describing curved four-dimensional space{time. An algebraic theorem states that a numerical system with the algebraic operations of addition, subtraction, multi- plication, and division has only real numbers, complex numbers, quaternions, and octonions. Therefore, the physical principles of four-dimensional space{time should be describable using quaternions. First, it is found that the division A= jAj of a complex number is the coordinate transformation if A is a complex number and A is its complex conjugate. (Chap. 2) Therefore, it is considered that the Lorentz transformations for special relativity t − (v=c2)x t0 = p ; 1 − v2=c2 x − vt x0 = p 1 − v2=c2 can be obtained if A= jAj is applied in two-dimensional space{time. The result is that the sign of the velocity of light squared of the Lorentz transformations changes from minus to plus, presumably because complex numbers are the mathematics of flat space{time. However, the space{time is curved. Therefore, new complex numbers must be created to describe the curved space{time. It is assumed that a fourth imaginary number h exists in addition to the three imaginary numbers i; j; k of Hamilton's quaternions. The algorithms are as follows: i2 = j2 = k2 = h2 = −1; ij = −ji = k; jk = −kj = i; ki = −ik = j; hi = ih; hj = jh; hk = kh: ix A new complex number ah + bi is created from the complex number a + bi and equations identical to the Lorentz transformations are obtained when the coordinate transformation A= jAj is done with this new complex number. This shows that the new complex number gives mathematics of the curved space{time. (Chap. 3) In addition, it is considered that time advances only in the positive direction because a contradiction occurs if time t is assumed as negative when we use the coordinate transformation A= jAj to obtain the Lorentz transformations. (Chap. 3) Herein, we examine the new complex number to determine whether it leads to ac- curate mathematics. Time dilation, proper time, length contraction, world distance, and the twin paradox of special relativity in two-dimensional space{time are proven with the new complex numbers and new complex planes. Without light emission and examples of a clock and ruler, results identical to those of special relativity are obtained using only the new complex number and the equation of the world line. (Chap. 6, Chap. 7, Chap. 8, Chap. 9) Next, by using the fourth imaginary number h, a new quaternion ah+bi+cj +dk in the curved space{time is created from Hamilton's quaternion a + bi + cj + dk. Furthermore, the new quaternion is used in the coordinate transformation A= jAj and the Lorentz transformations in the curved four-dimensional space{time are ob- tained. These equations are called the new Lorentz transformations. If observer B moves with constant velocity v in the x-direction with respect to observer A at rest, y0 = y; z0 = z in special relativity. However, y0 and z0 are functions of the three variables y; z, and v, and the x motion changes the y and z distances if these are calculated using the new quaternion mathematics without assuming the isotropy of space{time. These equations are as follows: y + (v=c)zh y0 = p ; 1 − v2=c2 z − (v=c)yh z0 = p : 1 − v2=c2 Because of the homogeneity of space and the fact that y0 = f(v)y and z0 = g(v)z were assumed in special relativity, these results were not obtained. Because a con- stant velocity of light is obtained by the velocity-transformation equations and because the world distance calculated by these transformations is invariant, the accuracy of the new Lorentz transformations is proven. (Chap. 10) x In addition, unlike the definition of world distance of special relativity, i.e., s2 = (ct)2 − x2 − y2 − z2; the definition of world distance calculated by the new quaternion is s2 = −(ct)2 + x2 + y2 + z2: (Chap:10) Here, we prove that the new quaternion has eight numbers that consist of a real number a, imaginary numbers h; i; j; k, and products hi; hj; hk of two imaginary numbers.
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