Notes on GTR
STR begins with the assumption that we can find a reference-body K such that Galileo’s law of inertia holds for it. Then, we claimed that any reference-body moving at a constant velocity with respect to this one will be equivalent to K for the purpose of formulating the laws of physics (and more generally, for describing the world accurately).
GTR begins by modifying and extending this claim to the following:
All bodies of reference, K, K’, etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature) whatever may be their state of motion (p. 69).
However, this way of putting it needs a bit of fixing up, which comes later (with
Gaussian coordinate systems).
Moving in this direction in physics seems to be blocked by the fact that, in an accelerating body of reference (i.e. a body of reference not moving at a constant velocity with respect to a Galilean reference-body), the law of inertia is violated: Consider a train that has applied its brakes, so that it’s now slowing down with respect to the ground frame. If someone on this train drops an object and observes its path as it falls, she will not see it fall directly to the floor at her feet under the influence of gravity. Instead, as the train slows, the falling object will also continue its forward motion, accelerating
(treating the train as our reference body) towards the front of the train. So the law of falling bodies, it seems, would have to look different in an accelerating frame: we account for the object’s fall towards the floor by appeal to Newton’s law of gravity, but its accelerating motion towards the front of the train seems inexplicable on the ordinary laws unless we re-describe it from the point of view of a Galilean reference-body, relative to which it does fall straight down towards the centre of the earth. Einstein concludes:
At all events it is clear that the Galilean law does not hold with respect to the nonuniformly moving carriage. Because of this, we feel compelled at the present juncture to grand a kind of absolute physical reality to non-uniform motion, in opposition to the general principle of relativity. (p. 70)
But gravity itself is already a rather mysterious thing. Newton once said of gravity,
“hypotheses non fingo,” (I feign no hypotheses), meaning that while it was clear to him that there was such a force, he had no account of why objects are attracted to each other.
The power of this supposed force to explain the motions of falling bodies, the planets, the tides etc., is convincing evidence (given measurements carefully ruling out alternative accounts of the force & its behaviour) of its existence. But one important property of this force is particularly worth noting: It is absolutely independent of the nature of the bodies it acts upon—in a given gravitational field, all bodies undergo the same acceleration.
Another way to express this same fact is to say that inertial mass (i.e. the mass that appears in the equation F = ma) is equal (up to a constant we can set to 1 by choosing our

units) to gravitational mass (the mass that appears in the equation F g
= M (gravitational field strength).
In classical physics this equality is simply posited. But in GTR Einstein aims to make inertia and gravity both expressions of “ the same quality of a body” (p. 74). In chapter twenty, Einstein proposes a thought experiment that makes his commitment to this idea plain: We’re asked to consider a man (sic) in an enclosed room that is being pulled (and so accelerated) by a rope attached to its roof, at a constant rate. As a result, any unsupported object in the room falls to the floor with a constant acceleration; further, no matter what the object is made of, it accelerates towards the floor at the same rate. But the man in the box is free to think of his circumstances in another way: He supposes that the box is suspended in a constant (unvarying) gravitational field. Once again, it will follow that unsupported objects will fall to the floor at a constant acceleration, and the same acceleration will apply to all objects regardless of their makeup. So the predictions our experimenter makes about the behaviour of things in the box, based on his gravitational hypothesis, are the same as the ones we make based on our accelerational hypothesis.
Here we have an argument for the principle of general relativity: With the help of his gravitational hypothesis, the man in the box is able to use the box as a “stationary” system of reference for doing physics (describing motions, determining the basic laws, etc.) even though the box constitutes an accelerated framework. A fundamental feature of general relativity does the key job here—the equivalence hypothesis, which asserts that gravitational fields are indistinguishable from the effects of accelerations. And the equivalence hypothesis implies that inertial and gravitational mass must be the same, that they somehow express the same fact about the bodies that have them, since we can transform descriptions that account for the behaviour of objects by appeal to inertia into equivalent descriptions that account for it by appeal to a gravitational field.
In Chapter twenty-one Einstein builds a case for saying that there’s something unsatisfactory about how classical mechanics and special relativity work—specifically, this is about the special place that Galilean reference bodies have in these theories. Even ignoring gravity, it’s worth asking what’s wrong with a reference body that is continually turning about some axis (such as the earth itself). After all, such a turning reference body doesn’t even need to be acted on to keep it turning. The only difference between such a reference body and a Galilean one is that distant, widely separated objects move at a fixed velocity with respect to the Galilean reference body, but they undergo an accelerated motion (as though they were in a gravitational field of a certain description) in the rotating one. But there is no clear or observable difference between the referencesystems themselves that could account for this difference in the motions of the bodies.
General relativity promises to solve this problem, since it allows us to arrive at the same physical laws using either reference-system.
In Chapters twenty-two and twenty-three Einstein applies special relativity to a Galilean reference-system, and then uses the relation between this Galilean reference-system and a

rotating reference system to infer some of the effects of gravity on space-time measurements. First, light must travel in a curved path within a gravitational field.
Second, from the point of view of the Galilean reference system, a measuring rod stays the same length when it is oriented radially out from the axis of rotation, but when it is oriented perpendicularly, its motion causes it to contract. Since the rotating reference body regards this contraction as due to the gravitational field, we can conclude that a gravitational field causes a compression of objects placed in it, perpendicular to the direction of the field, and that increases with the strength of the field. (The field gets stronger as we move out from the axis of rotation, as the velocity of the rotating system relative to the Galilean system gets higher.) Similarly, from the point of view of the
Galilean system, a clock (fixed with respect to the rotating system) will run slower the further out from the axis of the rotating system it is. So the stronger a gravitational field a clock is in, the slower it will run.
This has important implications—the special relativity standard of simultaneity, and the simple approach to measuring distances in space and time using the constancy of the speed of light, can’t be applied in GTR—we’re going to have to reconsider how we specify coordinates in space-time (and how we use them to describe motions…).
This brings us to the relation between Euclidean and Non-Euclidean geometries.
Euclid’s geometry began with five axioms:
A1: Through any two points there is a straight line.
A2: A straight line may be extended indefinitely in a straight line.
A3: Given any point and any radius, a circle may be drawn with that radius and that point as a center.
A4: All right angles are equal to each other.
A5: Given a line and a point not on that line, exactly one (i.e. one and only one) line can be drawn through the point and parallel to the given line.
The Tradition : From the time of Euclid, geometry was regarded as the best example of just how human knowledge of the world should work. Beginning with the axioms (and definitions), Euclidean geometers set forth to prove all the theorems of geometry. Since the truth of the axioms was generally regarded as self-evident, and since the theorems followed from those axioms (as we could see simply by following through each proof), geometry seemed to show that we could know substantial facts about the world (about the results of certain measurements, for example) by means of “pure reason”. Thus geometry seemed to support the idea of a priori knowledge, that is, the idea that we can know things about the world without depending on experience to tell us about them. The notion that we have such knowledge (and its employment in the construction of systems of metaphysics) is one of the central themes of rationalism , one of the two influential forms of philosophy to emerge in early modern times. (Descartes, in particular, held that we had innate ideas of the basic natures or properties that things could possess, and that we could clearly and distinctly grasp how these natures had to be related to each other without any need for experience to show us.)

The sticking point
: But from very early on, geometers were concerned about Euclid’s fifth axiom. This axiom says that given any line and a point not on that line, one and only one line can be drawn through the point, parallel to the line. Unlike the other four
(which say simple and straightforward things like “Given any two points, one and only one (straight) line can be drawn connecting them.” And “A circle of any radius can be drawn with any given point as its centre.”, this “parallels” axiom is complex enough and hard enough to be certain of that many thought it really ought to be proven as a theorem, rather than left as an axiom.
Efforts to prove the parallel postulate: Efforts to prove the axiom go way back, including many of the great Arab geometers, and continue up to Saccheri, a Jesuit priest who actually thought he had succeeded in “cleansing Euclid of all blemish.” Saccheri took the approach of trying to prove the axiom by reductio. That is, he assumed that it was false and tried to derive a contradiction. There are (of course) two ways that it could be false. There could be no parallel lines at all, through the given point, or there could be more than one. From both of these, Saccheri tried to show that some contradiction would follow. And in fact, from the first (the “no parallels” assumption) he was able to do it.
The other axioms do require that there be at least one parallel (though they can be adjusted to remove this impediment). But from the second, “many parallels” assumption, though he did derive some odd theorems (one of which he mistook for a contradiction), in fact he derived no contradiction at all. In fact, as has been shown since, there are systems of geometric axioms that include the “no parallels” axiom, and others that include the
“many parallels” axiom, which are provably consistent IF Euclid’s axioms are consistent.
Non-Euclidean Geometry : The upshot of all this was the recognition, early in the nineteenth century, that there were non-Euclidean systems of geometry. The geometrical truth, it seemed, could no longer be decided a priori, by pure reason. There were now three different “classes” of geometry: Bolyai-Lobachevskii, Riemannian, and Euclidean.
Riemann, in particular, systematized the whole range of geometries for “curved” space.
These geometries are Euclidean in the limit, that is, by confining ourselves to a small enough area, we can make the geometry as close to Euclidean as we want. But as the area gets larger, the geometry differs more and more from Euclid’s.
Geometry Bolyai-
Lobachevskii
Parallels Many
Euclid Riemann
One None
C/D for a circle
Sum of interior angles for a triangle
>

< 180 o
=

= 180 0
<

> 180 o
Gauss’ Test:
Gauss proposed to test which geometry applies (at least on the scale of the earth) by measuring the angles contained in a large triangle formed by shining lights on the peaks of three mountains. The results (unsurprisingly) came out to 180 o
, within

experimental error. But this result’s interpretation needs some real care. After all, the method used seems to assume that light rays travel in straight lines. If they don’t, then the figure enclosed by the light rays would not really be a triangle, and the sum of the angles contained in it could differ from the sum of the angles in a real triangle. Of course we could test whether the light was travelling in straight lines by finding some other way to measure whether the lines the light followed were straight. For example, we could try to measure the length of the paths the light followed. If these paths were the shortest paths from peak to peak, then (since a line is the shortest distance between any two points) we would have to say the lines were straight after all. But when we make this measurement, we are assuming that nothing is distorting our measuring rods, making one path seem shorter when in fact it’s really longer. And if the distortion is caused by a universal force , i.e., a force which distorts the results of all measuring devices to exactly the same extent, then there is no way we can use our observations to rule out the possibility that such a force has affected our measurements.
Poincarré’s Conventionalism : Poincarré’s response to this problem was to accept a form of conventionalism. According to conventionalism, there is no objectively right geometry for physical space. Instead, we can choose (conventionally) any geometry we want. Of course, to make that geometry fit with actual measurement results, we may have to invoke universal forces. These forces can be chosen in such a way as to “adjust” whatever measurements (corrected for any differential forces affecting their outcomes) we make to fit our choice of geometry. Given that we can make this choice, and fix any problems using universal forces, Poincarré suggested that we ought to use Euclidean geometry because it is so simple and convenient. But that is not to say that physical space is really Euclidean. According to Poincarré, space is not really one way or another; it’s our choice of geometry that decides how we will go about measuring the world, not our measurements that determine the right geometry.
Einstein vs. Lorentz
: a brief aside on special relativity. Einstein’s interpretation of the
Lorentz transformations is an interesting switch. Lorentz interpreted his transformations as the effect of a universal force, caused by motion through the ether, which would prevent us from detecting that motion in any way. But Einstein derives them from the principle of relativity (for Galilean frames) and the constancy of the velocity of light for all observers. So for Lorentz a universal force has distorted the results of our measurements, but for Einstein, our measurements are just as they should be, and the transformations simply express how our measurements will compare with the measurements of someone moving with respect to us (at some constant velocity).
Einstein has dispensed with the ether, which Lorentz claimed was there but couldn’t be measured. So in a clear sense, Einstein has a more economical theory.
General Relativity implies that we will get non-Euclidean results for measurements made in non-Galilean frames, and that (since we can’t remove all gravitational fields by appeal to accelerations) in general, the geometry of space-time will be non-Euclidean. To see where this leads us we need to reconsider how we go about arriving at a coordinatesystem for space-time. In special relativity, we assumed that we could use measuring rods and clocks, synchronized according to Einstein’s criterion of simultaneity, to set up

a working coordinate system. And the system that results is a Euclidean one: the spacetime interval between two events obeys the standard form of a four-dimensional
Euclidean distance: ds
2
= dx
1
2
+ dx
2
2
+ dx
3
2
+ dx
4
2
That is, the square of the “distance” between two points in a four-dimensional Euclidean geometry is the sum of the squares of the distances in each of the four “directions” the geometry allows. We can get the space-time interval into this form by rewriting the usual equation a bit (making c=1 by choosing units for space and time that give this result, and treating time distances as “imaginary,” i.e. as coming in units of the square root of -1, i ): ds 2 = dx
1
2 + dx
2
2 + dx
3
2 – c 2 dt 2 so if c=1, ds
2
= dx
1
2
+ dx
2
2
+ dx
3
2
+ ( i dt)
2
But General Relativity forces us to consider a much wider range of coordinate systems, which are called Gaussian coordinates. A two-dimensional Gaussian coordinate system results when we “cover” a two-dimensional surface with two families of (in general curved) lines such that: every line of one family intercepts each line of the other family somewhere, no two lines of one family intersect each other at all, and real numbers are assigned to each line of each family in order (so that for every line between two other lines in a family, a real between the reals assigned the other two is assigned to it).
Such a coordinate system makes distances a bit trickier to calculate. In such a system we have to account for the “curviness” of space-time, which means moving along one axis of the coordinate system is not independent of moving along other axes. This curvature is captured by introducing some further terms into the distance calculation:
For a two-dimensional Gaussian coordinate system we get: ds 2
= g
11 dx
1
2 + 2g
12 dx
1 dx
2
+ g
22 dx
2
2
That is, the square of the “distance” between two points in such a coordinate system is a constant, g
11
, times the square of the distance along the x
1
family of lines (the difference in real numbers assigned to the x
1
lines that cross the two points) plus twice a constant, g
12
, times the product of the distance along the x
1
family with the distance along x
2 family, plus a constant, g
22
, times the square of the distance along the x
1
family of lines.
The constants reflect the “curviness” and how it stretches or compresses distances & links distances along the x
1
family of lines to distances along the x
2
family of lines.
This is not exactly pretty, but as math goes it’s clean and straightforward. And it works so long as (and ONLY so long as) the geometry of the space(-time) gets closer and closer to being Euclidean as we shrink the region of the space under consideration. (This seems like a small price to pay for being able to impose some coordinates on space-time.
Consider a curved surface, like the surface of a car’s fender or a billiard ball—as we consider smaller and smaller regions of the surface, it will look more and more like a small piece of a flat Euclidean plane. So it seems natural to suppose that this assumption will hold.)

Given this abstract account of how to impose a coordinate system on space-time, Einstein makes it a little more physical-sounding by appeal to the “reference-mollusc” (a deformable/stretchable, curvy and irregular body) with irregular clocks attached to all of its points—irregular in the sense that their rates need not be constant, but they must match: two such clocks placed close together can be synchronized, and the closer they are kept, the closer their readings will stay.(p. 110) But the re-statement of the general principle of relativity sticks with the generalized Gaussian coordinate systems:
All Gaussian coordinate systems are essentially equivalent, for formulation of the general laws of nature. (p. 108)
This leads us to the final form of Einstein’s theory of gravity: First we consider the special “gravitational” fields that result when we consider an accelerating frame of reference and calculate (using the Lorentz transformations) its measurement results based on the results obtained in a Galilean frame that can be applied to the same phenomena.
Then we generalize the resulting account of gravity, under 3 further constraints:
(a) The generalization must satisfy the general postulate of relativity. (see just above)
(b) If there is any matter in the domain under consideration, only its inertial mass, i.e. its energy, is relevant to its contribution to the gravitational field. (This is required to ensure equivalence will hold—if something else mattered, then not everything would have to be equally accelerated by a gravitational field.)
(c) The gravitational field together with matter must obey conservation of mass/energy and conservation of impulse (= momentum)
The result agrees with Newton when the fields are weak (and weak, here, covers anything close to the fields we typically experience). But when they get strong, significant differences with Newton arise. The rotation of the ellipse of Mercury’s orbit
(“anomalous precession of the perihelion of Mercury”) over time (= 43 arc-seconds per century more than Newton’s gravity and classical mechanics produce) is such a difference—general relativity predicts this deviation from the classical result quite precisely. Other successful tests include curvature of starlight as it passes by the sun during an eclipse (Eddington, 1919), red-shifting of spectral lines from massive stars (= gravitational clock slowing) (Adams, 1924), and the decay of the orbits of a pair of closely-orbiting pulsars, due to the energy lost to gravitational radiation.
A final puzzle :
Suppose you’re in an airplane about to begin its take-off run. A child next to you has a helium balloon tied to a string and floating above her. What will the balloon do as the take-off run begins?
1.
Shift toward the back of the plane.
2.
Shift toward the front of the plane.
3.
Stay floating where it is.
As a hint, try applying the principle of equivalence together with the normal behaviour of helium balloons floating in the air within a gravitational field.