Suchergebnisse
Suchergebnisse:
In physics, spacetime is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects such as how different observers perceive where and when events occur.
- LOUIS WERNER
- INVARIANCE; VECTORS AND TENSORS
- to write '(x'k).
- INVARIANCE; VECTORS AND TENSORS
- of SI
- Adx*
- jAdx =jA>dx>
- ^ = A' A,
- DERIVATIVES
- 8A'k_8xtcxmdAl
- And since this must hold for any vector, we must have
- And since this must hold for any vector, we must have
- AkB By cancelling the terms Ak
- dx[ c2xm dxm dx'kdx'{
- THE NOTION OF PARALLEL TRANSFER
- 42 SPACE-TIME STRUCTURE
- THE QUESTION OF INTEGRABILITY
- CONNEXION
- r*ft = IV (8.6)
- Bk;i = tki.
- GEODESIC COORDINATES
- Bk;i = tki.
- GEODESIC COORDINATES
- THE MEANING OF THE METRIC ACCORDING TO THE SPECIAL THEORY OF RELATIVITY
- THE ELEMENTARY NOTION OF CONSERVATION LAWS
- SH_ 8
- = f.f</*4
- We compute SI and perform, after the pattern indicated before, the partial integrations needed to put it into the form
- (Rik-igikR);i = j
- DISCUSSION OF THE PRECEDING THEORIES
IN GRATITUDE FOR HIS INESTIMABLE AID CONTENTS INTRODUCTION page i PART I: THE UNCONNECTED MANIFOLD
We envisage a (four-dimensional) continuum whose points are distinguished from each other by allotting a quadruplet of con-tinuous labels xv x2> xz, #4 to each of them. However, this first labelling shall have no prerogative over any other one x'z = x'z(xx... #4), *J = aifo ... *4), j where the x'k are four continuous, differentiate functions of th...
Given (in one frame) two points, P, with coordinates xki and P, with coordinates xkJ the difference
We envisage a (four-dimensional) continuum whose points are distinguished from each other by allotting a quadruplet of con-tinuous labels xv x2> xz, #4 to each of them. However, this first labelling shall have no prerogative over any other one x'z = x'z(xx... #4), *J = aifo ... *4), j where the x'k are four continuous, differentiate functions of th...
CHAPTER II INTEGRALS. DENSITIES. DERIVATIVES INTEGRALS. DENSITIES The subject-matter of the previous chapter is called tensor algebra. It is characterized by the fact that only relations between invariants, vectors or tensors referring to the same point of the continuum are contemplated. From the point of view taken here,f algebraic relations betwe...
/ • (always over an invariantly fixed domain), would that be an in-variant? Obviously not. Though there is no objection to adding invariants that refer to different points, yet we know that on trans-forming thus
to hold, in other words for the integral to be an invariant, the ' transformation law' for A would have to be not
that is, it would by definition have to take on as a factor the functional determinant appearing in the transformed integral in consequence of the transformation of the 'product of the differentials'. We give a quantity behaving in that way the name of scalar density. It has become customary to denote a density by a Gothic t To make the integral in...
For shortness we shall henceforth occasionally indicate the derivative with respect to xk by a lower index k, preceded by a comma. Except in the case of an invariant, the derivative of a tensor-component, as for example, has no proper meaning, because it results from subtracting tensors referring to different points, viz. the Ak in the point xt fro...
dxi dxk dxt dxm dx\\dxk (Z.IO) which is exactly the same as (2.14), only for an arbitrary Ax (not just, as there, a gradient). Again we see that Alm behaves like covariant second-rank tensor, except for the additional term, containing the non-differentiated Ax and the second derivatives of the transformation. This again has the effect that our array...
**;. = 0. So the mixed unity tensor, regarded as a field, has the invariant derivative zero with respect to any affinity. Now envisage the invariant product AkBk of two arbitrary vector-fields. According to the two guiding prin-ciples laid down we want thus AkB By cancelling the terms Ak iBk, we get AkB*tt = AkB*ii + A This we write exchanging the ...
**;. = 0. So the mixed unity tensor, regarded as a field, has the invariant derivative zero with respect to any affinity. Now envisage the invariant product AkBk of two arbitrary vector-fields. According to the two guiding prin-ciples laid down we want thus
iBk, we get AkB*tt = AkB*ii + A This we write exchanging the dummies ky n in the last term
W5} We wish, by a suitable choice of the transformation, to make all the F' vanish at one point—say for simplicity at the point xk = o. We choose the transformation so that at this point the inverse transformation has the analytical development where we assume af6^ = aklm, because obviously nothing would be gained if we did not. We get from (4.5), ...
There is an alternative way of introducing the notion of affine connexion and invariant derivative. In view of the fundamental character of these notions in all our considerations we shall indicate this alternative. The array of derivatives dAk/dxi does not con- °Q(xt+dxi) stitute an invariant entity, because they are formed by the ' inadmissible* ...
But please take care! The equation holds at the other point for the parallel-transferred tensors. But the tensors may be field-tensors. And their actual values at the other point need not and, as a rule, will not be those obtained by parallel-transfer. So the equation need not hold for the field-tensors at the neighbouring point! That is the more n...
Easily the most interesting and vital point that occurs in the study of affine connexion and parallel-transfer is this. If you envisage a tensorial entity, e.g. a contravariant vector Ak at a point P (not necessarily a member of a' field'), and carry it by continual parallel-transfer around a closed circuit C back to P, the entity does not in gener...
Given an affinity F*/m, let us envisage at a point P(xk) a line element dxky leading from P to the point P'(xk + dxk). Transfer dxk (being a vector in P) according to the connexion F from P to P', and let the result be d'xk (being a vector in Pf). Transfer this vector from P' to P (xk + dxk + d'xk). The result, d"xk transfer to P'"(xk + dxk + d'xk ...
Then we are left with i?w = o, (8.7) as the general equation to impose on the Ps where there is no matter. This is in one respect very satisfactory, in another respect not quite so. Let us first speak of the satisfactory point. It is, that the vanishing of a tensor of the second rank in empty space is just what we would expect as the mathematical d...
Similarly and for the same reasons 'latinizing' and 'gothicizing', i.e. dividing or multiplying by *Jg> may be effected under the semicolon. It deserves to be mentioned that gik, though it was originally defined in a different way, actually is the fundamental tensor gik> with both its subscripts raised, exactly after our convention. If only one ind...
We have in Chapter vi proved that a frame always exists in which all the components of a given symmetric affinity vanish at a given point. We call this geodesic coordinates at that point. From (9.4) this means, in the present case, that all the derivatives gikj vanish at that point or that the gik are stationary there. It is sometimes very convenie...
Similarly and for the same reasons 'latinizing' and 'gothicizing', i.e. dividing or multiplying by *Jg> may be effected under the semicolon. It deserves to be mentioned that gik, though it was originally defined in a different way, actually is the fundamental tensor gik> with both its subscripts raised, exactly after our convention. If only one ind...
We have in Chapter vi proved that a frame always exists in which all the components of a given symmetric affinity vanish at a given point. We call this geodesic coordinates at that point. From (9.4) this means, in the present case, that all the derivatives gikj vanish at that point or that the gik are stationary there. It is sometimes very convenie...
Our geometrical construction—a four-dimensional continuum with affine and metrical connexion—is to serve as a model of the real physical world. What physical interpretation are we to give to the * line-element' ds—the infinitesimal invariant determined by every pair of infinitely neighbouring points xk and xk + dxk ? In the beginning of Chapter ix ...
We have proposed as the field-laws of gravitation RM = o in empty space-time, w = TH where there is * matter',) TH being the stress-energy-momentum tensor of matter. (What it means exactly will be discussed forthwith.) One must be very careful in adopting field equations at pleasure, just as much as, or even more than, when writing down a set of al...
Now if you multiply the first equation by/i, the second by git etc. and add them up, you get (remember the meaning of 2): Now since £ = -J-, -— = -—-, the S over the first two terms will J% dxt dxk dxi give simply dH\\dxi provided H does not contain xi explicitly. (We have assumed this explicit dependence only in order to drop it now: to show that t...
For both theories—or versions—conservation identities (and a few others intimately connected with them) can be derived very much on the same lines as was done in Chapter xi for Einstein's theory. I will not deal with them here, but refer the reader to my paper in Proc. R. Irish Acad. 52, A, p. 1, 1948. For all that I know, no special solution has y...
For both theories—or versions—conservation identities (and a few others intimately connected with them) can be derived very much on the same lines as was done in Chapter xi for Einstein's theory. I will not deal with them here, but refer the reader to my paper in Proc. R. Irish Acad. 52, A, p. 1, 1948. For all that I know, no special solution has y...
For both theories—or versions—conservation identities (and a few others intimately connected with them) can be derived very much on the same lines as was done in Chapter xi for Einstein's theory. I will not deal with them here, but refer the reader to my paper in Proc. R. Irish Acad. 52, A, p. 1, 1948. For all that I know, no special solution has y...
For both theories—or versions—conservation identities (and a few others intimately connected with them) can be derived very much on the same lines as was done in Chapter xi for Einstein's theory. I will not deal with them here, but refer the reader to my paper in Proc. R. Irish Acad. 52, A, p. 1, 1948. For all that I know, no special solution has y...
1. Feb. 2022 · February 1, 2022. 15 min read. What Is Spacetime Really Made Of? Spacetime may emerge from a more fundamental reality. Figuring out how could unlock the most urgent goal in physics—a quantum...
31. März 2024 · Space-time, in physical science, single concept that recognizes the union of space and time, first proposed by the mathematician Hermann Minkowski in 1908 as a way to reformulate Albert Einstein’s special theory of relativity (1905). Learn more about space-time in this article.
- The Editors of Encyclopaedia Britannica
Learning Objectives. Discuss the three models of spacetime. Aristotelian spacetime. Galilean spacetime. Einstein’s spacetime. “ The test of a first-rate intelligence is the ability to hold two opposing ideas in mind at the same time and still retain the ability to function. ” —F. Scott Fitzgerald.
13. Nov. 2020 · Summary. This exploration of the global structure of spacetime within the context of general relativity examines the causal and singular structures of spacetime, revealing some of the curious possibilities that are compatible with the theory, such as 'time travel' and 'holes' of various types. Investigations into the epistemic and modal ...
The Large Scale Structure of Space–Time is a 1973 treatise on the theoretical physics of spacetime by the physicist Stephen Hawking and the mathematician George Ellis. It is intended for specialists in general relativity rather than newcomers.
