# Reading Help Relativity: The Special and General Theory

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spectrum was definitely established by Adams in 1924, by observations `

` on the dense companion of Sirius, for which the effect is about thirty `

` times greater than for the Sun. R.W.L. -- translator `

` `

` `

` `

` APPENDIX IV `

` `

` THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY `

` (SUPPLEMENTARY TO SECTION 32) `

` `

` `

` Since the publication of the first edition of this little book, our `

` knowledge about the structure of space in the large (" cosmological `

` problem ") has had an important development, which ought to be `

` mentioned even in a popular presentation of the subject. `

` `

` My original considerations on the subject were based on two `

` hypotheses: `

` `

` (1) There exists an average density of matter in the whole of space `

` which is everywhere the same and different from zero. `

` `

` (2) The magnitude (" radius ") of space is independent of time. `

` `

` Both these hypotheses proved to be consistent, according to the `

` general theory of relativity, but only after a hypothetical term was `

` added to the field equations, a term which was not required by the `

` theory as such nor did it seem natural from a theoretical point of `

` view (" cosmological term of the field equations "). `

` `

` Hypothesis (2) appeared unavoidable to me at the time, since I thought `

` that one would get into bottomless speculations if one departed from `

` it. `

` `

` However, already in the 'twenties, the Russian mathematician Friedman `

` showed that a different hypothesis was natural from a purely `

` theoretical point of view. He realized that it was possible to `

` preserve hypothesis (1) without introducing the less natural `

` cosmological term into the field equations of gravitation, if one was `

` ready to drop hypothesis (2). Namely, the original field equations `

` admit a solution in which the " world radius " depends on time `

` (expanding space). In that sense one can say, according to Friedman, `

` that the theory demands an expansion of space. `

` `

` A few years later Hubble showed, by a special investigation of the `

` extra-galactic nebulae (" milky ways "), that the spectral lines `

` emitted showed a red shift which increased regularly with the distance `

` of the nebulae. This can be interpreted in regard to our present `

` knowledge only in the sense of Doppler's principle, as an expansive `

` motion of the system of stars in the large -- as required, according `

` to Friedman, by the field equations of gravitation. Hubble's discovery `

` can, therefore, be considered to some extent as a confirmation of the `

` theory. `

` `

` There does arise, however, a strange difficulty. The interpretation of `

` the galactic line-shift discovered by Hubble as an expansion (which `

` can hardly be doubted from a theoretical point of view), leads to an `

` origin of this expansion which lies " only " about 10^9 years ago, `

` while physical astronomy makes it appear likely that the development `

` of individual stars and systems of stars takes considerably longer. It `

` is in no way known how this incongruity is to be overcome. `

` `

` I further want to rernark that the theory of expanding space, together `

` with the empirical data of astronomy, permit no decision to be reached `

` about the finite or infinite character of (three-dimensional) space, `

` while the original " static " hypothesis of space yielded the closure `

` (finiteness) of space. `

` `

` `

` K = co-ordinate system `

` x, y = two-dimensional co-ordinates `

` x, y, z = three-dimensional co-ordinates `

` x, y, z, t = four-dimensional co-ordinates `

` `

` t = time `

` I = distance `

` v = velocity `

` `

` F = force `

` G = gravitational field `

` `

` `

` `

` `

`

` on the dense companion of Sirius, for which the effect is about thirty `

` times greater than for the Sun. R.W.L. -- translator `

` `

` `

` `

` APPENDIX IV `

` `

` THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY `

` (SUPPLEMENTARY TO SECTION 32) `

` `

` `

` Since the publication of the first edition of this little book, our `

` knowledge about the structure of space in the large (" cosmological `

` problem ") has had an important development, which ought to be `

` mentioned even in a popular presentation of the subject. `

` `

` My original considerations on the subject were based on two `

` hypotheses: `

` `

` (1) There exists an average density of matter in the whole of space `

` which is everywhere the same and different from zero. `

` `

` (2) The magnitude (" radius ") of space is independent of time. `

` `

` Both these hypotheses proved to be consistent, according to the `

` general theory of relativity, but only after a hypothetical term was `

` added to the field equations, a term which was not required by the `

` theory as such nor did it seem natural from a theoretical point of `

` view (" cosmological term of the field equations "). `

` `

` Hypothesis (2) appeared unavoidable to me at the time, since I thought `

` that one would get into bottomless speculations if one departed from `

` it. `

` `

` However, already in the 'twenties, the Russian mathematician Friedman `

` showed that a different hypothesis was natural from a purely `

` theoretical point of view. He realized that it was possible to `

` preserve hypothesis (1) without introducing the less natural `

` cosmological term into the field equations of gravitation, if one was `

` ready to drop hypothesis (2). Namely, the original field equations `

` admit a solution in which the " world radius " depends on time `

` (expanding space). In that sense one can say, according to Friedman, `

` that the theory demands an expansion of space. `

` `

` A few years later Hubble showed, by a special investigation of the `

` extra-galactic nebulae (" milky ways "), that the spectral lines `

` emitted showed a red shift which increased regularly with the distance `

` of the nebulae. This can be interpreted in regard to our present `

` knowledge only in the sense of Doppler's principle, as an expansive `

` motion of the system of stars in the large -- as required, according `

` to Friedman, by the field equations of gravitation. Hubble's discovery `

` can, therefore, be considered to some extent as a confirmation of the `

` theory. `

` `

` There does arise, however, a strange difficulty. The interpretation of `

` the galactic line-shift discovered by Hubble as an expansion (which `

` can hardly be doubted from a theoretical point of view), leads to an `

` origin of this expansion which lies " only " about 10^9 years ago, `

` while physical astronomy makes it appear likely that the development `

` of individual stars and systems of stars takes considerably longer. It `

` is in no way known how this incongruity is to be overcome. `

` `

` I further want to rernark that the theory of expanding space, together `

` with the empirical data of astronomy, permit no decision to be reached `

` about the finite or infinite character of (three-dimensional) space, `

` while the original " static " hypothesis of space yielded the closure `

` (finiteness) of space. `

` `

` `

` K = co-ordinate system `

` x, y = two-dimensional co-ordinates `

` x, y, z = three-dimensional co-ordinates `

` x, y, z, t = four-dimensional co-ordinates `

` `

` t = time `

` I = distance `

` v = velocity `

` `

` F = force `

` G = gravitational field `

` `

` `

` `

` `

`