# A Geometry and integral laws of physics

#### victorvmotti

Reading the English translation of Einstein's seminal paper on GR.

http://einsteinpapers.press.princeton.edu/vol6-trans/90?ajax

This paragraph below on p78 doesn't make much sense to me.

Could you provide a second English translation or even adding math notation.

"Before Maxwell, the laws of nature with respect to their space dependence were in principle integral laws; this is to say that in elementary laws the distances between finitely distinct points did occur. Euclidean geometry is the basis for this description of nature. This geometry means originally only the essence of conclusions from geometric axioms; in this regard it has no physical content. But geometry becomes a physical science by adding the statement that two points of a "rigid" body shall have a distinct distance from each other that is independent of the position of the body. After this amendment, the theorems of this amended geometry are (in a physical sense) either factually true or not true. It is geometry in this extended sense which forms the basis of physics. Seen from this aspect, the theorems of geometry are to be looked as integral laws of physics insofar as they deal with distance of points *at a finite range*."

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#### pervect

Staff Emeritus
I would interpret this as EInstein saying that geometry is pure mathematics, but when one introduces the concept of rigid bodies, presumably based on the then current definition of the meter, that geometry then becomes subject to experimental tests as one can actually build physical representations of the meter via copying the prototype meter bars. I believe the appropriate meter standard would be the 1889 artifact definition of the meter - see http://physics.nist.gov/cuu/Units/meter.html

I don't quite understand the reference to "integral laws, though.

Of course in curved space-times, the very existence of "rigid bodies" become problematical. Additionally, one is most likely using a more modern definition of the meter that's not based on a physical artifact (but is still experimentally realizable).

#### Ibix

I don't quite understand the reference to "integral laws, though.
Do you think he means that Euclidean geometry is integral to the Newtonian laws? That Newtonian physics implicitly assumes that an ideal rigid rod behaves exactly as a Euclidean line segment? So really just a remark leading in to the idea of the whole Newtonian edifice being untenable on a pseudo-Riemannian background.

#### robphy

Homework Helper
Gold Member
The next paragraph goes on to talk about replacing "action at a distance" theories with theories of "local action".

#### vanhees71

Gold Member
First of all, there's also the German original available,

http://einsteinpapers.press.princeton.edu/vol6-doc/101?ajax

and I find the English translation very accurate (I'm a German native speaker, using the usual non-native speaker physicist's "broken English" to communicate physics ;-))

I think the point Einstein is making here that there's a principle paradigm change from Newton's laws which where "action-at-a-distance laws" to Maxwell's equations of the electromagnetic field, which is a "local-interaction theory", and that for the points at a finite distance and that the geometry (as a mathematical construct) is assumed to be the valid description of physical space (given by the possibility to define distances by rigid bodies which provide a fixed distance between two fixed points independent of their state of motion). This is, however, an assumption subject to physics, i.e., experimental check. It's not a priori said that Euclidean geometry must be the correct description of space in nature, and this gives the physicist the freedom to use other mathematical (geometrical) constructions to describe space (and as is clear already from special relativity then also time or, in modern terms, a more general space-time manifold).

#### victorvmotti

Specifically I do not get the points here

integral laws; this is to say that in elementary laws the distances between finitely distinct points did occur.

For example?

Seen from this aspect, the theorems of geometry are to be looked as integral laws of physics

What is the definition of an integral law?

#### robphy

Homework Helper
Gold Member
\begin.maybe
Maybe an "integral law" refers to treating a distributed object or system of particles as if it could be treated a single rigid object...e.g. referring to some kind of center of an object. Then one can make statements like Coulomb's law. (In geometry one would make reference to the center of the circle rather than just the individual points of the circle.)
\end.maybe

#### pervect

Staff Emeritus
Specifically I do not get the points here

integral laws; this is to say that in elementary laws the distances between finitely distinct points did occur.

For example?

Seen from this aspect, the theorems of geometry are to be looked as integral laws of physics

What is the definition of an integral law?
I don't get this part either, especially the relevance of the distance between finitely distinct points.

It's useful and possibly related to Einstein's point to observe that the group structure of Newtonian space-time is something like $E(3) \times R$ (I'm not sure where to look this up to make sure I have the language and details exactly right). Here E(3) is the 3 dimensional Euclidean group, representing the geometry of the space. The group structure of special relativity is different, however, it is the Poincare group (or the Lorentz group, if one does not include translations). The later is SO(1,3) - which is not derived from the Euclidean group - and wiki gives the former as "the semidirect product of $R^{1,3}$ and SO(1,3)" which is also not derived from the Euclidean group.

I'm not positive if one can quite say that the "integral geometry" in Einstein's sense is the geometry of the underlying group. Perhaps one can interpret the remark about the distance between finitely distinct points as relating to the existence of a group structure?

#### PeterDonis

Mentor
What is the definition of an integral law?
A law that isn't a local differential law. A local differential law is a law that involves only quantities and their rates of change at a single spacetime point (event). For example, the tensor equations used in GR are local differential laws.

Newton's law of gravity, however, is not a local differential law, because it involves a finite distance $R$ between the two gravitating objects. Similarly, Coulomb's Law in its original form was not a local differential law, for the same reason.

Of course we now know that Coulomb's Law can be reinterpreted as a particular approximation to Maxwell's Equations, which are local differential laws. In this approximation (roughly, all charges are stationary and there is no current), we can obtain the distance $R$ that appears in Coulomb's Law by integrating a line element over Euclidean space; as I understand it, this is why Einstein refers to laws like the original form of Coulomb's Law as integral laws.

#### PeterDonis

Mentor
I don't get this part either, especially the relevance of the distance between finitely distinct points.
I think what he means is that, from the local differential point of view (which, remember, is Einstein's point of view--he believed that all of physics could be expressed as local differential laws), the "distance between finitely distinct points" is not a fundamental quantity; it is obtained (per my previous post) by integrating some line element along some curve. But in a curved manifold, the value of such an integral is path-dependent, and Einstein was showing that making gravity consistent with SR requires a curved manifold.