Giovanni Gherardi Il paradiso degli Alberti
Kraszewski JI 2 Lubonie t.2
Gordon Dickson Space Winners
Dare Tessa Spindle Cove 03.2 Dama o póśÂ‚nocy t. 2
041.Brown_Sandra_Ksiaze_i_dziewczyna
Morgan Raye Kr悜‚lewski potomek
Stan aktywnoÂści fiz w stylu życia studentów awf w gdańsku
Isaac Asimov's Caliban 1 Caliban
Alan Dean Foster Sagramanda
Dom otwarty BaćąĂ˘Â€Âšuckiego
[ Pobierz całość w formacie PDF ]
more general, in every gravitational field, a clock will go more quickly or less quickly, according to
the position in which the clock is situated (at rest). For this reason it is not possible to obtain a
reasonable definition of time with the aid of clocks which are arranged at rest with respect to the
body of reference. A similar difficulty presents itself when we attempt to apply our earlier definition
of simultaneity in such a case, but I do not wish to go any farther into this question.
49
Relativity: The Special and General Theory
Moreover, at this stage the definition of the space co-ordinates also presents insurmountable
difficulties. If the observer applies his standard measuring-rod (a rod which is short as compared
with the radius of the disc) tangentially to the edge of the disc, then, as judged from the Galileian
system, the length of this rod will be less than I, since, according to Section 12, moving bodies
suffer a shortening in the direction of the motion. On the other hand, the measaring-rod will not
experience a shortening in length, as judged from K, if it is applied to the disc in the direction of the
radius. If, then, the observer first measures the circumference of the disc with his measuring-rod
and then the diameter of the disc, on dividing the one by the other, he will not obtain as quotient the
familiar number À = 3.14 . . ., but a larger number,2) whereas of course, for a disc which is at rest
with respect to K, this operation would yield À exactly. This proves that the propositions of
Euclidean geometry cannot hold exactly on the rotating disc, nor in general in a gravitational field,
at least if we attribute the length I to the rod in all positions and in every orientation. Hence the idea
of a straight line also loses its meaning. We are therefore not in a position to define exactly the
co-ordinates x, y, z relative to the disc by means of the method used in discussing the special
theory, and as long as the co- ordinates and times of events have not been defined, we cannot
assign an exact meaning to the natural laws in which these occur.
Thus all our previous conclusions based on general relativity would appear to be called in question.
In reality we must make a subtle detour in order to be able to apply the postulate of general
relativity exactly. I shall prepare the reader for this in the following paragraphs.
Next: Euclidean and Non-Euclidean Continuum
Footnotes
1)
The field disappears at the centre of the disc and increases proportionally to the distance from
the centre as we proceed outwards.
2)
Throughout this consideration we have to use the Galileian (non-rotating) system K as
reference-body, since we may only assume the validity of the results of the special theory of
relativity relative to K (relative to K1 a gravitational field prevails).
Relativity: The Special and General Theory
50
Relativity: The Special and General Theory
Albert Einstein: Relativity
Part II: The General Theory of Relativity
Euclidean and Non-Euclidean Continuum
The surface of a marble table is spread out in front of me. I can get from any one point on this table
to any other point by passing continuously from one point to a " neighbouring " one, and repeating
this process a (large) number of times, or, in other words, by going from point to point without
executing "jumps." I am sure the reader will appreciate with sufficient clearness what I mean here
by " neighbouring " and by " jumps " (if he is not too pedantic). We express this property of the
surface by describing the latter as a continuum.
Let us now imagine that a large number of little rods of equal length have been made, their lengths
being small compared with the dimensions of the marble slab. When I say they are of equal length,
I mean that one can be laid on any other without the ends overlapping. We next lay four of these
little rods on the marble slab so that they constitute a quadrilateral figure (a square), the diagonals
of which are equally long. To ensure the equality of the diagonals, we make use of a little
testing-rod. To this square we add similar ones, each of which has one rod in common with the
first. We proceed in like manner with each of these squares until finally the whole marble slab is
laid out with squares. The arrangement is such, that each side of a square belongs to two squares
and each corner to four squares.
It is a veritable wander that we can carry out this business without getting into the greatest
difficulties. We only need to think of the following. If at any moment three squares meet at a corner,
then two sides of the fourth square are already laid, and, as a consequence, the arrangement of
the remaining two sides of the square is already completely determined. But I am now no longer
able to adjust the quadrilateral so that its diagonals may be equal. If they are equal of their own
accord, then this is an especial favour of the marble slab and of the little rods, about which I can [ Pobierz całość w formacie PDF ]
