Abstract
In this section we discuss the principle of equivalence. We shall see that, besides allowing the extension of the principle of relativity to a generic, not necessarily inertial, frame of reference, it allows to define gravity, in a relativistic framework, as a property of the four-dimensional space-time geometry.
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Notes
- 1.
It goes without saying that we are referring to two spherical bodies or to bodies whose dimensions are negligible with respect to their distance r.
- 2.
We recall that the ratio between the electric and gravitational forces between two protons is of the order of \(10^{38}\).
- 3.
The inertial motion inside a free falling system is a well known fact nowadays, think about the absence of weight of astronauts inside orbiting spacecrafts.
- 4.
We recall that the relative acceleration \(\mathbf{{a}}_R\) is given in general by the sum of four terms, the translation acceleration \(\mathbf {a'}\), the centripetal acceleration \(\mathbf{{a}}_{(\textit{centr})}\), the Coriolis acceleration \(\mathbf{{a}}_{(\textit{Cor})}\) and a further term proportional to the angular acceleration of \(S'\) with respect to S.
- 5.
Linearity was then a consequence of the requirement that the principle of inertia holds in both the old and the transformed frames: A motion which is uniform with respect to one of them cannot be seen as accelerated with respect to the other.
- 6.
In the proximity of the event horizon of a black hole tidal forces are so strong as to completely disintegrate any body falling inside.
- 7.
More correctly one should also consider the contribution of the sun which is not much less than that of the moon. For the sake of simplicity we just illustrate the main contribution due to the moon.
- 8.
This crude estimate must be considered, together with the correction due to the sun, just a mean value. It does not take into account resonance phenomena due to the earth rotation and the shape of the oceanic depths which can locally alter in a sensible way the rough computation leading to (3.24).
- 9.
In this section only we denote by \(\xi ^i\) the Cartesian coordinates, while the notation \(x^i\) is used for generic curvilinear coordinates.
- 10.
Recall that the space-time (four-dimensional) distance was conventionally defined as the negative of the proper distance, see Eq. (2.45).
- 11.
From now on, we use Greek indices to label four dimensional space-time coordinates and Latin ones for the coordinates in Euclidean space.
- 12.
We shall call Minkowskian the Cartesian rectangular coordinates in the four-dimensional Minkowski space-time, with metric \(\eta _{\alpha \beta }\).
- 13.
Here and in the following of this chapter we use the word tensor, whose precise meaning will be given in Chap. 4, in a loose sense, that is as a quantity carrying indices and whose transformation properties are fixed in terms of the change of coordinates (or of reference frame). In this chapter the transformation of coordinates considered are either cartesian orthogonal, or Lorentzian or even arbitrary, as explained below.
- 14.
When dealing with problems which exhibit some degree of symmetry it may, however, be more useful to use curvilinear coordinates, like spherical, cylindrical, etc.
- 15.
By large domain we mean a domain whose extension is finite or infinite.
- 16.
Being the plane flat, we can describe it by Cartesian coordinates.
- 17.
Here and in the following by locally we mean that our statement is valid in an infinitesimal neighborhood of a point where higher order terms can be neglected.
- 18.
Alternatively one can use Cartesian coordinates at each point using the local tangent plane; in this case, however, one needs a quantity, called connection, which relates the local geometries associated with different tangent planes.
- 19.
A more precise definition of curvature will be given in the next section.
- 20.
Note that this implies that the free falling frames are the inertial frames described by special relativity. However, in the presence of a gravitational field, they can be only defined locally, since only locally the effects of gravity can be canceled.
- 21.
As we shall see in the sequel of this chapter the space-time metric \(g_{\mu \nu }(x)\) is related to the gravitational potential rather than to the gravitational field.
- 22.
A maximal circle is the shortest path joining two points on the sphere. It can be obtained by intersecting the spherical surface with a plane determined by the two points and the center of the sphere.
- 23.
It can be shown that the signature of the metric, that is the number of positive and negative eigenvalues of the matrix \(g_{\mu \nu }(x)\), is the same at each point of a manifold. Since at a given point the metric can be taken to coincide with \(\eta _{\alpha \beta }\) this explains the meaning of the statement in the text.
- 24.
Note that since the closed path is infinitesimal we are allowed to use the in the tangent hyper-plane the usual flat geometry of special relativity and therefore the reference to \(v^\mu \) as a Lorentz vector is appropriate.
- 25.
With respect to the notations used in Chap. 2, we have changed notation for the locally inertial coordinates from \(x^\mu \) to \(\xi ^\alpha \) (\(\alpha =0,1,2,3\)) since in the present setting the latter describe the locally inertial coordinates of special relativity, while the former describe a general frame, for example the coordinates used in the “laboratory” frame, where the gravitational field is present.
- 26.
As usual the value at infinity of \(\phi \) is set equal to zero.
- 27.
We also note that the given error increases day by day, since it is a cumulative effect.
- 28.
Here covariant means with respect to general coordinate transformations.
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D’Auria, R., Trigiante, M. (2016). The Equivalence Principle. In: From Special Relativity to Feynman Diagrams. UNITEXT for Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-22014-7_3
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DOI: https://doi.org/10.1007/978-3-319-22014-7_3
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