Synchronous frame

Proper time in general relativity

In the special theory of relativity, choice of coordinates is limited by the requirement for a special kind of spacetime metric: the Minkowski metric. In the general theory of relativity, there is no such requirement so that the choice of reference frame is not limited: the three space coordinates x¹, x², x³ can take any values that define the positions of bodies in space while the time coordinate x⁰ can be measured by clocks with any possible adjustment. The problem thus arises how one can determine the real distances and time intervals by the values of x⁰, x¹, x², x³.

First one must determine the true time (proper time), written by the symbol τ, with a coordinate x⁰. Consider two very close events that occur practically in the same point of space. The interval ds between these two events is cdτ where dτ is the proper time interval that separates them. Substituting x¹ = x² = x³ = 0 (making all space coordinates equal to zero) in the general expression for the metric ds² = g_ik dxⁱ dx^k, one obtains

$ds^2 = c^2 d\tau^2 = g_{00} \left ( dx^0 \right )^2,\,$

so that

$d\tau = \frac {1}{c}\sqrt{g_{00}}\,dx^0,$

(eq. 1)

or, for the time between any two events in the same point of space

$\tau = \frac {1}{c} \int \sqrt{g_{00}}\,dx^0.$

(eq. 2)

The relationship eq. 2 defines the proper time between events in the same place through changes in the time coordinate x⁰. Note that, according to the above formulae, g₀₀ is positive:

$g_{00} > 0.\,$

(eq. 3)

One must make a difference between the condition eq. 3 and the condition made when choosing the signature (the signs of the principal values of the g_ik tensor). A g_ik tensor that does not satisfy the signature condition does not correspond to any real gravitational field, that is, to any real spacetime metric. If g_ik does not satisfy the condition eq. 3, it means only that the respective reference frame cannot be defined by real bodies; if the signature condition is fulfilled then by a proper coordinate transformation one can make g₀₀ positive (an example of such frame is the rotating reference frame).

Synchronization over the whole space

In the special relativity theory, the space distance element dl is defined as the intervals between two very close events that occur at the same moment of time. In the general relativity theory this cannot be done, that is, one cannot define dl by just substituting dx⁰ = 0 in ds. The reason for this is the different dependence between proper time and time coordinate x⁰ in different points of space.

To find dl in this case, one can first synchronize time over the whole space in the following way (Fig. 1): Send a light signal from some space point B with coordinates x^α + dx^α into a very close point A with coordinates x^α and then immediately reflect back the signal from A to B. The time necessary for this operation (measured in point B), multiplied by c is, obviously, the doubled distance between the two points.

The squared interval, with separated space and time coordinates, is:

$ds^2 = g_{\alpha\beta}\, dx^\alpha\, dx^\beta + 2g_{0\alpha}\, dx^0\, dx^\alpha + g_{00} \left ( dx^0 \right )^2,$

(eq. 4)

where, as usual, a repeated Greek index within a term means summation by values 1, 2, 3. The interval between the events of signal arrival in point A and its immediate reflection back is zero (two events in the same time at the same point). Equation ds² = 0 solved for dx⁰ gives two roots:

$dx^{0(1)} = \frac{1}{g_{00}} \left ( -g_{0\alpha}\, dx^\alpha - \sqrt{\left ( g_{0\alpha}g_{0\beta} - g_{\alpha \beta}g_{00} \right )\, dx^\alpha \,dx^\beta} \right ),$

$dx^{0(2)} = \frac{1}{g_{00}} \left ( -g_{0\alpha}\, dx^\alpha + \sqrt{\left ( g_{0\alpha}g_{0\beta} - g_{\alpha \beta}g_{00} \right ) \,dx^\alpha \,dx^\beta} \right ),$

(eq. 5)

which correspond to the propagation of the signal in both directions between A and B. If x⁰ is the moment of arrival/reflection of the signal in A, the moments of signal departure from B and its arrival back in B correspond, respectively, to x⁰ + dx^{0 (1)} and x⁰ + dx^{0 (2)}. The solid lines on Fig. 1 are the world lines of points A and B with coordinates x^α and x^α + dx^α, respectively, while the dashed lines are the world lines of the signals. Fig. 1 supposes that dx^{0 (2)} is positive and dx^{0 (1)} is negative, which, however, is not necessarily the case: dx^{0 (1)} and dx^{0 (2)} may have the same sign. The fact that in the latter case the value x⁰ (A) in the moment of signal arrival at A may be less than the value x⁰ (B) in the moment of signal departure from B does not contain a contradiction because clocks in different points of space are not supposed to be synchronized. It is clear that the full "time" interval between departure and arrival of the signal in point B is

$dx^{0(2)} - dx^{0(1)} = \frac{2}{g_{00}} \sqrt{\left ( g_{0\alpha}g_{0\beta} - g_{\alpha \beta}g_{00} \right ) \,dx^\alpha \,dx^\beta}.$

The respective proper time interval is obtained from the above relationship according to eq. 1 by multiplication by $\scriptstyle{\sqrt{g_{00}}/c}$, and the distance dl between the two points – by additional multiplication by c/2. As a result:

$dl^2 = \left ( -g_{\alpha \beta} + \frac{g_{0\alpha}g_{0\beta}}{g_{00}} \right )\, dx^\alpha \,dx^\beta.$

(eq. 6)

This is the required relationship that defines distance through the space coordinate elements.

Space metric tensor

Let one rewrite eq. 6 in the form

$dl^2 = \gamma_{\alpha \beta} \,dx^\alpha\, dx^\beta,\,$

(eq. 7)

where

$\gamma_{\alpha \beta} = -g_{\alpha \beta} + \frac{g_{0\alpha}g_{0\beta}}{g_{00}}$

(eq. 8)

is the three-dimensional metric tensor that determines the metric, that is, the geometrical properties of space. Equations eq. 8 give the relationships between the metric of the three-dimensional space and the metric of the four-dimensional spacetime.

In general, however, the metric g_ik depends on x⁰ so that the space metric eq. 7 changes with time. Therefore, it doesn't make sense to integrate dl: this integral depends on the choice of world line between the two points on which it is taken. It follows that in general relativity the distance between two bodies cannot be determined in general; this distance is determined only for infinitesimally close points. Distance can be determined also for finite space regions only in such reference frames in which g_ik does not depend on time and therefore the integral ∫dl along the space curve acquires some definite sense.

The tensor –γ_αβ is inverse to the contravariant 3-dimensional tensor g^αβ. Indeed, writing equation g^ikg_kl = $\scriptstyle{\delta_l^i}$ in components, one has:

$g^{\alpha \beta}g_{\beta \gamma} + g^{\alpha 0}g_{0 \gamma} = \delta_\gamma^\alpha,$

$g^{\alpha \beta}g_{\beta 0} + g^{\alpha 0}g_{00} = 0,\,$

(eqs. 9)

$g^{0 \beta}g_{\beta 0} + g^{00}g_{00} = 1.\,$

Determine g^α0 from the second equation and substitute in the first to obtain

$-g^{\alpha \beta} \gamma_{\beta \gamma} = \delta_\gamma^\alpha,$

(eq. 10)

which was to be demonstrated. This result can be presented otherwise by saying that g^αβ are components of a contravariant 3-dimensional tensor corresponding to metric eq. 7:

$\gamma^{\alpha \beta} = -g^{\alpha \beta}.\,$

(eq. 11)

The determinants g and γ composed of elements g_ik and γ_αβ, respectively, are related to each other by the simple relationship:

$-g = g_{00}\gamma.\,$

(eq. 12)

In many applications, it is convenient to define a 3-dimensional vector g with covariant components

$g_\alpha = -\frac{g_{0 \alpha}}{g_{00}}.$

(eq. 13)

Considering g as a vector in space with metric eq. 7, its contravariant components can be written as g^α = γ^αβg_β. Using eq. 11 and the second of eqs. 9, it is easy to see that

$g^\alpha = \gamma^{\alpha \beta} g_\beta = -g^{0 \alpha}.\,$

(eq. 14)

From the third of eqs. 9, it follows

$g^{00} = \frac{1}{g_{00}} - g_\alpha g^\alpha.$

(eq. 15)

Simultaneity in general relativity

Synchronization of clocks located at different space points means that events happening at different places can be measured as simultaneous if those clocks show the same times. Let us see if this is possible in general relativity (in curved space).

It is obvious that such synchronization should be done by exchange of light signals between points. Consider again propagation of signals between infinitesimally close points A and B in Fig. 1. The clock reading in B which is simultaneous with the moment x⁰ in A lies in the middle between the moments of sending and receiving the signal in B; this is the moment

$x^0 + \Delta x^0 = x^0 + \tfrac{1}{2} \left ( dx^{0(2)} + dx^{0(1)} \right ).$

Substitute here eq. 5 to find the difference in "time" x⁰ between two simultaneous events occurring in infinitesimally close points as

$\Delta x^0 = -\frac{g_{0 \alpha}\, dx^\alpha}{g_{00}} \equiv g_\alpha \,dx^\alpha.$

(eq. 16)

This relationship allows clock synchronization in any infinitesimally small space volume. By continuing such synchronization further from point A, one can synchronize clocks, that is, determine simultaneity of events along any open line. The synchronization condition can be written in another form by multiplying eq. 16 by g₀₀ and bringing terms to the left hand side

$\Delta x_0 = g_{0i} dx^i = 0\,$

(eq. 17)

or, the "covariant differential" dx₀ between two infinitesimally close points should be zero.

However, it is impossible, in general, to synchronize clocks along a closed contour: starting out along the contour and returning to the starting point one would obtain a Δx⁰ value different from zero. Thus, unambiguous synchronization of clocks over the whole space is impossible. An exception are reference frames in which all components g_0α are zeros.

Note the inability to synchronize all clocks is a property of the reference frame and not of the spacetime itself. It is always possible in infinitely many ways in any gravitational field to choose the reference frame so that the three g_0α become zeros and thus enable a complete synchronization of clocks. To this class are assigned cases where g_0α can be made zeros by a simple change in the time coordinate which does not involve a choice of a system of objects that define the space coordinates.

In the special relativity theory, too, proper time elapses differently for clocks moving relatively to each other. In general relativity, proper time is different even in the same reference frame at different points of space. This means that the interval of proper time between two events occurring at some space point and the time interval between the events simultaneous with those at another space point are, in general, different from one another.

Synchronous frame

As concluded from eq. 17, the condition that allows clock synchronization in different space points is that metric tensor components g_0α are zeros. If, in addition, g₀₀ = 1, then the time coordinate x⁰ = t is the proper time in each space point (with c = 1). A reference frame that satisfies the conditions

$g_{00} = 1, \quad g_{0 \alpha} = 0,\,$

(eq. 18)

is called synchronous frame. The interval element in this system is given by the expression

$ds^2 = dt^2 - g_{\alpha \beta}\, dx^\alpha\, dx^\beta,\,$

(eq. 19)

with the space metric tensor components identical (with opposite sign) to the components g_αβ:

$\gamma_{\alpha \beta} = -g_{\alpha \beta}.\,$

(eq. 20)

In synchronous frame time, lines are geodesics in the four-dimensional spacetime. These lines are normal to the hypersurfaces t = const. Indeed, the 4-vector normal to this hypersurface n_i = ∂t/∂xⁱ has covariant components n_α = 0, n₀ = 1. The respective contravariant components at conditions eq. 18 are again n^α = 0, n⁰ = 1, that is, coincide with the components of the 4-vector u ⁱ tangential to the time lines.

Conversely, these properties can be used to construct synchronous frame in any spacetime. To this end, choose some time-like hypersurface as an origin, such that has in every point a normal along the time line (lies inside the light cone with an apex in that point); all interval elements on this hypersurface are space-like. Then draw a family of geodesics normal to this hypersurface. Choose these lines as time coordinate lines and define the time coordinate t as the length s of the geodesic measured with a beginning at the hypersurface; the result is a synchronous frame.

Such a construct, and hence, choice of synchronous frame, is always possible. Moreover, such choice is not unique. Metric of type eq. 19 allows any transformation of space coordinates that does not depend on time and, additionally, a transformation brought about by the arbitrary choice of hypersurface used for this geometric construct.

An analytic transformation to synchronous frame can be done with the use of the Hamilton–Jacobi equation. The principle of this method is based on the fact that particle trajectories in gravitational fields are geodesics.

The Hamilton–Jacobi equation for a particle (with a unit mass) in a gravitational field is .

References

• Landau, Lev D.; Lifshitz, Evgeny M. (1988), Classical Theory of Fields (7th Russian ed.), Moscow: Nauka, ISBN 5-02-014420-7 Vol. 2 of the Course of Theoretical Physics.