What do you think?
Rate this book
Space, Time, Matter
"The standard treatise on the general theory of relativity." — Nature
"Whatever the future may bring, Professor Weyl's book will remain a classic of physics." — British Journal for Philosophy and Science
Reflecting the revolution in scientific and philosophic thought which accompanied the Einstein relativity theories, Dr. Weyl has probed deeply into the notions of space, time, and matter. A rigorous examination of the state of our knowledge of the world following these developments is undertaken with this guiding principle: that although further scientific thought may take us far beyond our present conception of the world, we may never again return to the previous narrow and restricted scheme.
Although a degree of mathematical sophistication is presupposed, Dr. Weyl develops all the tensor calculus necessary to his exposition. He then proceeds to an analysis of the concept of Euclidean space and the spatial conceptions of Riemann. From this the nature of the amalgamation of space and time is derived. This leads to an exposition and examination of Einstein's general theory of relativity and the concomitant theory of gravitation. A detailed investigation follows devoted to gravitational waves, a rigorous solution of the problem of one body, laws of conservation, and the energy of gravitation. Dr. Weyl's introduction of the concept of tensor-density as a magnitude of quantity (contrasted with tensors which are considered to be magnitudes of intensity) is a major step toward a clearer understanding of the relationships among space, time, and matter.
"Whatever the future may bring, Professor Weyl's book will remain a classic of physics." — British Journal for Philosophy and Science
Reflecting the revolution in scientific and philosophic thought which accompanied the Einstein relativity theories, Dr. Weyl has probed deeply into the notions of space, time, and matter. A rigorous examination of the state of our knowledge of the world following these developments is undertaken with this guiding principle: that although further scientific thought may take us far beyond our present conception of the world, we may never again return to the previous narrow and restricted scheme.
Although a degree of mathematical sophistication is presupposed, Dr. Weyl develops all the tensor calculus necessary to his exposition. He then proceeds to an analysis of the concept of Euclidean space and the spatial conceptions of Riemann. From this the nature of the amalgamation of space and time is derived. This leads to an exposition and examination of Einstein's general theory of relativity and the concomitant theory of gravitation. A detailed investigation follows devoted to gravitational waves, a rigorous solution of the problem of one body, laws of conservation, and the energy of gravitation. Dr. Weyl's introduction of the concept of tensor-density as a magnitude of quantity (contrasted with tensors which are considered to be magnitudes of intensity) is a major step toward a clearer understanding of the relationships among space, time, and matter.
368 pages, Paperback
First published January 1, 1919
23 people are currently reading
268 people want to read
About the author
Hermann Weyl
110 books57 followersHermann Klaus Hugo Weyl (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years.
Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the 'universalism' of Henri Poincaré or Hilbert, Weyl came as close as anyone. Michael Atiyah, in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him (The Mathematical Intelligencer (1984), vol.6 no.1).
Source: http://en.wikipedia.org/wiki/Hermann_...
Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the 'universalism' of Henri Poincaré or Hilbert, Weyl came as close as anyone. Michael Atiyah, in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him (The Mathematical Intelligencer (1984), vol.6 no.1).
Source: http://en.wikipedia.org/wiki/Hermann_...
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 8 of 8 reviews
July 21, 2017
Weyl's book is most famous for introducing gauge theory, which was later reborn in the form of phase transformations in quantum theory. Weyl did not live guite long enough to hear of the latter being applied by Yang and Mills, though he socially interacted with Yang in his last year at Princeton.
Einstein and Pauli both criticized Weyl's original unified theory based on general relativity using a length gauge, both as implying false empirical consequences (Einstein), since it implied tiny changes of length dependent on path and as untestable (Pauli). (Obviously it could not both be empirically false and non-empirical.) Yet Eddington and later Einstein himself revived similar theories. Eddington claimed that the length differences were to tiny as to be undetectable, but also that his own gauge theory could be thought of not as literal space/time structure but as a geometrization of an abstract background theory for specific space/time structures.
Thomas Ryckman's The Reign of Relativity: Philosophy in Physics 1915-1925 (Oxford Studies in the Philosophy of Science)has an excellent eighty page discussion of these ideas of Weyl in relativity, as well as chapters on those of Eddington in the 1920s.
Another novelty is Weyl's suggestion that General Relativity could be tied to observation via the conformal structure as representing light cones and the projective structure as particles in free fall. This alternative to the rods and clocks approach, on the basis of which Weyl was criticized, has been developed by Ehlers (who edited the new German edition of this work Raum, Zeit, Materie: Vorlesungen über allgemeine Relativitätstheorie (German Edition)) Pirani and Schild.
Weyl also introduces what he calls "tensor densities" which Shouten called "Weyl tensors" and Synge and Schild call oriented tensors, often called twisted tensors. These are analogous to and include "axial vectors."
Weyl's introduction of the "affine connection" after criticism of Levi-Civita's notion of parallelism led the way to further notions of connections and generalization of the notion of connection as such by Elie Cartan and others.
These are but a few of the intellectual gems in this work.
The philosophical parts are, unfortunately, almost uniformly mistranslated. The phenomenological introduction is re-translated in Kockelmans and Kisiel, eds. Phenomenology and the Natural Sciences (SPEP). (Courant suggested Weyl as successor to Husserl in the philosophy chair at Goettingen!) This together with the misprints in formulas, makes it desirable that the whole book be retranslated.
Einstein and Pauli both criticized Weyl's original unified theory based on general relativity using a length gauge, both as implying false empirical consequences (Einstein), since it implied tiny changes of length dependent on path and as untestable (Pauli). (Obviously it could not both be empirically false and non-empirical.) Yet Eddington and later Einstein himself revived similar theories. Eddington claimed that the length differences were to tiny as to be undetectable, but also that his own gauge theory could be thought of not as literal space/time structure but as a geometrization of an abstract background theory for specific space/time structures.
Thomas Ryckman's The Reign of Relativity: Philosophy in Physics 1915-1925 (Oxford Studies in the Philosophy of Science)has an excellent eighty page discussion of these ideas of Weyl in relativity, as well as chapters on those of Eddington in the 1920s.
Another novelty is Weyl's suggestion that General Relativity could be tied to observation via the conformal structure as representing light cones and the projective structure as particles in free fall. This alternative to the rods and clocks approach, on the basis of which Weyl was criticized, has been developed by Ehlers (who edited the new German edition of this work Raum, Zeit, Materie: Vorlesungen über allgemeine Relativitätstheorie (German Edition)) Pirani and Schild.
Weyl also introduces what he calls "tensor densities" which Shouten called "Weyl tensors" and Synge and Schild call oriented tensors, often called twisted tensors. These are analogous to and include "axial vectors."
Weyl's introduction of the "affine connection" after criticism of Levi-Civita's notion of parallelism led the way to further notions of connections and generalization of the notion of connection as such by Elie Cartan and others.
These are but a few of the intellectual gems in this work.
The philosophical parts are, unfortunately, almost uniformly mistranslated. The phenomenological introduction is re-translated in Kockelmans and Kisiel, eds. Phenomenology and the Natural Sciences (SPEP). (Courant suggested Weyl as successor to Husserl in the philosophy chair at Goettingen!) This together with the misprints in formulas, makes it desirable that the whole book be retranslated.
December 28, 2016
I give this book 5/5 stars for the intrinsic intellectual tour-de-force masterpiece that it represents and for historical interest of this work. However, I do not recommend it for those who are learning relativity theory. This is not an appropriate exposition. It's a heavily difficult philosophical-mathematical-physical investigation, obviously written in style that is too hard for this generation to follow, unfortunately. It's too deep and too rich in content, but obscure in many aspects. If you want to dive into Weyl's genius, it is a gem. Otherwise, it's frustrating. I feel a mixture of both. I look forward to return to this book, nevertheless.
April 3, 2014
Having enjoyed two of Weyl's other books in English translation, I had high hopes for this one; but I am afraid to say that it is unreadable. I gave up at the following sentence, around page 60:
The fact that a tensor which has been derived from them by mathematical operations and is an invariant (i.e. dependent on them alone and not upon the co-ordinate system) is equal to zero is what, in general, the expression of a physical law amounts to.I did wonder why Weyl, in the foreword to Philosophy of Mathematics and Natural Science, made such a point of stressing that it had been translated correctly.
June 30, 2025
Most often, one peruses older scientific literature mainly out of an antiquarian interest. But in the case of Albert Einstein’s contemporary and intellectual equal, the gifted German mathematician Hermann Weyl (1885-1955), one encounters all the excitement of an intellectual exploration into an unknown territory. Such will be true, indeed, of Weyl’s landmark work Raum-Zeit-Materie, the first edition of which appeared as early as 1918 and which is now available in an eighth edition, released by Springer-Verlag in 1993.
What, in our view, constitutes Weyl’s greatness? Set aside his early major contributions to the theory of Riemann surfaces and his later ground-breaking research into the representation theory of Lie groups, which happens to be so relevant to the problem of symmetries in quantum mechanics and which was recognized as such at the time, and overlook his thoughtful mature contributions to the philosophy of mathematics and of the natural sciences in order to focus on the youthful piece at hand, viz., Raum-Zeit-Materie itself.
Here, our hero exposes non-Euclidean geometry and its relevance to physics, which was first recognized a few years before by Albert Einstein and his collaborator Marcel Grossmann. Yet the theoretical physicist Einstein was content to absorb the absolute tensor calculus in the form in which he found it, as previously developed from the late nineteenth century onwards by the mathematicians Ricci, Christoffel, Levi-Civita and so forth, while the more enterprising Weyl demonstrates his precocity by posing the problem afresh and seeking a logically clear formulation from first principles. Central to Weyl’s argument in Part II is the idea of parallel transport (originally due to Levi-Civita) and its connection with curvature. What for Einstein is but a symbol with which to perform calculations in a tensorial algebra acquires here a deep and intuitive grounding. Part III turns to the application of these purely geometrical concepts to the relativity of space and time as disclosed, first, in the special theory of relativity and, again, at a deeper level, in the general theory of relativity. Another subject of Part III is Gustav Mie and his matter theory, what was a current topic of active research but which is little known today as it seems to have lost out to quantum mechanics as the focal concern among physicists at the research frontier.
The last Part IV of Raum-Zeit-Materie unfolds Weyl’s genial idea of a gauge theory. Although one needs to approach it bearing the precaution in mind that Weyl’s version of a pure infinitesimal geometry involving a conformal factor in the metric – despite its undeniable brilliance – doesn’t quite work as a theory of electromagnetism, as his sympathetic colleague Einstein was reluctantly to determine. Nevertheless, it would inspire later generations of physicists to develop the non-abelian Yang-Mills gauge theory now foundational to the electroweak and chromodynamic forces as embodied in the standard model.
Let us go deeper, and dwell on the role of epistemological foundations in the early chapters (for the speculations and more detailed formulae of the later chapters, however thought-provoking, have largely been overtaken by time). Weyl chooses to preface his axiomatic development of Riemannian geometry with a nine-page introduction of a strictly philosophical nature. What one may take away from it is a concern to ground the constructions to follow in a visualizable mental picture that appeals to spatial intuition, evident everywhere in what he later does. This can explain the synthetic power of Weyl’s general approach to geometry and offers thereby an object lesson in mathematical method. For one can well learn from it how productively to tackle other problems or, indeed, by pondering what Weyl does more carefully, perhaps hit upon something that he may have missed. The reader keen to follow up on these topics may consult our review here of the conference proceedings edited by Julian Bernard and Carlos Lobo and appearing in print as Weyl and the Problem of Space: From Science to Philosophy (Springer-Verlag, 2019).
Thus, to conclude, today’s young aspiring theorist should look upon Raum-Zeit-Materie as an exemplary training ground in how to engage in original thought. Perhaps someone will take Weyl’s example to heart and, by returning to first principles, come up with a productive starting point for a successful quantized theory of gravity, still lacking in our day despite decades of feverish work on the problem by a generation of physicists schooled, unlike Weyl himself, in the American slapdash style of heuristic reasoning and therefore chary of entering into any profound probing of our received ideas of space and time.
What, in our view, constitutes Weyl’s greatness? Set aside his early major contributions to the theory of Riemann surfaces and his later ground-breaking research into the representation theory of Lie groups, which happens to be so relevant to the problem of symmetries in quantum mechanics and which was recognized as such at the time, and overlook his thoughtful mature contributions to the philosophy of mathematics and of the natural sciences in order to focus on the youthful piece at hand, viz., Raum-Zeit-Materie itself.
Here, our hero exposes non-Euclidean geometry and its relevance to physics, which was first recognized a few years before by Albert Einstein and his collaborator Marcel Grossmann. Yet the theoretical physicist Einstein was content to absorb the absolute tensor calculus in the form in which he found it, as previously developed from the late nineteenth century onwards by the mathematicians Ricci, Christoffel, Levi-Civita and so forth, while the more enterprising Weyl demonstrates his precocity by posing the problem afresh and seeking a logically clear formulation from first principles. Central to Weyl’s argument in Part II is the idea of parallel transport (originally due to Levi-Civita) and its connection with curvature. What for Einstein is but a symbol with which to perform calculations in a tensorial algebra acquires here a deep and intuitive grounding. Part III turns to the application of these purely geometrical concepts to the relativity of space and time as disclosed, first, in the special theory of relativity and, again, at a deeper level, in the general theory of relativity. Another subject of Part III is Gustav Mie and his matter theory, what was a current topic of active research but which is little known today as it seems to have lost out to quantum mechanics as the focal concern among physicists at the research frontier.
The last Part IV of Raum-Zeit-Materie unfolds Weyl’s genial idea of a gauge theory. Although one needs to approach it bearing the precaution in mind that Weyl’s version of a pure infinitesimal geometry involving a conformal factor in the metric – despite its undeniable brilliance – doesn’t quite work as a theory of electromagnetism, as his sympathetic colleague Einstein was reluctantly to determine. Nevertheless, it would inspire later generations of physicists to develop the non-abelian Yang-Mills gauge theory now foundational to the electroweak and chromodynamic forces as embodied in the standard model.
Let us go deeper, and dwell on the role of epistemological foundations in the early chapters (for the speculations and more detailed formulae of the later chapters, however thought-provoking, have largely been overtaken by time). Weyl chooses to preface his axiomatic development of Riemannian geometry with a nine-page introduction of a strictly philosophical nature. What one may take away from it is a concern to ground the constructions to follow in a visualizable mental picture that appeals to spatial intuition, evident everywhere in what he later does. This can explain the synthetic power of Weyl’s general approach to geometry and offers thereby an object lesson in mathematical method. For one can well learn from it how productively to tackle other problems or, indeed, by pondering what Weyl does more carefully, perhaps hit upon something that he may have missed. The reader keen to follow up on these topics may consult our review here of the conference proceedings edited by Julian Bernard and Carlos Lobo and appearing in print as Weyl and the Problem of Space: From Science to Philosophy (Springer-Verlag, 2019).
Thus, to conclude, today’s young aspiring theorist should look upon Raum-Zeit-Materie as an exemplary training ground in how to engage in original thought. Perhaps someone will take Weyl’s example to heart and, by returning to first principles, come up with a productive starting point for a successful quantized theory of gravity, still lacking in our day despite decades of feverish work on the problem by a generation of physicists schooled, unlike Weyl himself, in the American slapdash style of heuristic reasoning and therefore chary of entering into any profound probing of our received ideas of space and time.
Want to read
December 20, 2023"Reality", wrote Hermann Weyl (1922,
217), "is not a three-dimensional Euclidean space but rather a four-dimensional world, in which space and time are linked together indissolubly' (emphasis in original):
However deep the chasm may be that separates the intuitive nature of space from that of time in our experience, nothing of this qualitative difference enters into the objective world which physics endeavours to crystallise out of direct experience. It is a four-dimensional continuum, which is neither "time" nor "space".
Minkowski christened the four-dimensional manifold the 'absolute world'. Weyl gladly adopted the same terminology. It probably was the Harvard psychologist William James who first coined the term
'block universe' in 1884. James thus spoke of a "solid" and "iron block" when describing a deterministic universe.”
217), "is not a three-dimensional Euclidean space but rather a four-dimensional world, in which space and time are linked together indissolubly' (emphasis in original):
However deep the chasm may be that separates the intuitive nature of space from that of time in our experience, nothing of this qualitative difference enters into the objective world which physics endeavours to crystallise out of direct experience. It is a four-dimensional continuum, which is neither "time" nor "space".
Minkowski christened the four-dimensional manifold the 'absolute world'. Weyl gladly adopted the same terminology. It probably was the Harvard psychologist William James who first coined the term
'block universe' in 1884. James thus spoke of a "solid" and "iron block" when describing a deterministic universe.”
November 24, 2021
Two stars because the math was correct.
H. Weyl was a terrible writer and the translators were terrible also. Weyl mostly does not explain what he is trying to show. He just drops in notation as if the notation is as common as an addition symbol.
I read the first chapter in 1990. It was terrible then. Twenty years later and a lot of mathematics under the bridge, it still remains a terrible book. I would not recommend it to anyone. And why does Dover keep publishing it?
In a single word review: YUCK.
H. Weyl was a terrible writer and the translators were terrible also. Weyl mostly does not explain what he is trying to show. He just drops in notation as if the notation is as common as an addition symbol.
I read the first chapter in 1990. It was terrible then. Twenty years later and a lot of mathematics under the bridge, it still remains a terrible book. I would not recommend it to anyone. And why does Dover keep publishing it?
In a single word review: YUCK.
October 19, 2020
Old language usage makes it hard to follow. Covers theory of general relativity.
February 5, 2015
Very good on space, ok on time.
Displaying 1 - 8 of 8 reviews