# Gravitational force of several massive bodies, from the viewpoint of general relativity

According to Wald's GR, "absolute gravitation force has no meaning". The text goes on to describe two cases: one where a gravitational force can be defined, and one in which it cannot. I'd like to understand why there is a difference.

Case 1:

...in the curved spacetime geometry in the vicinity of the Earth, the orbits of time translation symmetry differ from the geodesics of the metric. We could use the time translation symmetry of this example to define a preferred set of background observers. We then could define the gravitational force field of the Earth to be minus the acceleration a body must undergo in order to remain stationary. Thus, in this case a well defined meaning can be assigned to gravity as a force field.

Case 2:

However, in the absence of time translation symmetry--e.g. in a case where there are several massive bodies in relative motion--there exists no natural set of curves whose comparison with geodesics could be used to define gravitational force.

Why not? Even though the system is not time invariant, at any instant in time there is a fixed configuration of masses in space. There is no essential difference between a single mass (e.g. Earth) and a fixed configuration of masses. Therefore at each instant in time a gravitational field can be defined, so we have a time-dependent gravitational field overall. Is there anything wrong or insufficient with this viewpoint?

Perhaps I'm not fully understanding the necessary condition for existence of a "preferred set of background observers". What does "preferred" really mean? Also, what does Wald mean by a "natural" set of curves?

Any clarification will be much appreciated.

• What do you mean by "instant of time"? Which time? If the spacetime is stationary there is natural choice(at least locally).
– MBN
Nov 27 '14 at 9:21

This is discussed in section 4.3 in my 1984 edition. The quote supplied can't really be understood in isolation - you need to consider the whole section.

Wald's point is that in general relativity there are no inertial observers because in general spacetime is nowhere flat. In Newtonian physics or special relativity acceleration can be measured relative to an inertial observer, and all inertial observers will agree on the measured acceleration. But in GR since there are no inertial observers there is no reference against which to measure acceleration.

For a time independant geometry like the Schwarzschild metric there is a next best thing, because we can take an observer at a fixed spatial position and measure the acceleration compared to that observer. To see how this is done look at the question What is the weight equation through general relativity?.

But take a simple time dependant system like the Earth and the Moon. We can fix our observer relative to the Earth, but then the observer won't be fixed relative to the Moon. So Earthbound and Moonbound observers will reach different conclusions about values of acceleration and hence force.

• For the Earth-Moon system, I'm not completely sure what the problem is. To be concrete, suppose a spaceship is falling towards the Earth. Is it essentially because of the relativity of simultaneity that we cannot speak of THE acceleration of the spaceship, according to both the Earth observer and the Moon observer? In other words, the Earth observer may say "At 12am the spaceship is at distance A, Moon is distance B, and acceleration is X". But the Moon observer will not think that the spaceship is located (B-A) from him (vectorially), so his acceleration measurement will not be X. Correct?
– yjc
Nov 27 '14 at 22:29

According to Wald's GR [...] e.g. in a case where there are several massive bodies in relative motion--there exists no natural set of curves whose comparison with geodesics could be used to define gravitational force.

Why not?

Why not, indeed.
What could be more "natural" than to make the required comparison ("with geodesics") for each participant (and its corresponding "curve" or trajectory) separately; thereby defining "gravitational force" (or more inclusively: "gravito-inertial force") in general, and evaluating it case by case, separately for each participant and each trial (and without concern for any possibility of "defining a preferred set of background observers").

Specificly, given (a section of) the trajectory of participant $A$ as an ordered set of coincidence events $\{ \varepsilon_{A O} ..., \varepsilon_{A Q} ..., \varepsilon_{A X} \}$ in which $A$ had taken part (having met and passed participants $O$, $Q$ and $W$, among others, in this order),
and given the ("geodesic"-based) values of interval ratios between pairs of the corresponding events in which $A$ had taken part in this trial,
i.e. the real number values of ratios

$$\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]}{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]}$$

for all pairs of participants (such as for instance $P$ and $Q$, and also including $O$ and $X$) whom $A$ had met in the course of the trial,

then the magnitude $|~\mathbf a_A[~Q~]~|$ of $A$'s acceleration at the (event of) the meeting and passing of participant $Q$ can thereby be expressed as

\begin{array}{ll} |~\mathbf a_A[~Q~]~| := \frac{c}{\sqrt{\stackrel{~}{|~s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]~|}}} \times {\text{Limit}}_{ \large{\left\{\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]} \rightarrow 0 \right\} }} &~ \cr \scriptsize{ \left[ ~~ \sqrt{ \stackrel{~}{\frac{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]} }} \times \sqrt{ \eqalign{ \stackrel{~}{ \left( \frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]} \right)}~ \left( \frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]} \right)~ + \left( \frac{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]}{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]} \right)~ + \left( \frac{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]} \right)~ \\ - 2 \left( \frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]} \right)~ - 2 \left( \frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]} \right)~ - 2 } } ~~~ \right] }; \end{array}

and the average direction of $A$'s acceleration throughout a trial from event $\varepsilon_{A O}$ until event $\varepsilon_{A X}$ would be expressed in terms of families of suitable "geodesic" participants; namely as "towards" any one participant $B$ (if there exists one), for each pair of events in which $B$ took part as well, such as $\varepsilon_{A B P} \equiv \varepsilon_{A P}$ and $\varepsilon_{A B W} \equiv \varepsilon_{A W}$, for which

$$\sqrt{ \frac{s^2[~\varepsilon_{A B P}, \varepsilon_{B Y}~]}{s^2[~\varepsilon_{A B P}, \varepsilon_{A B W}~]} } + \sqrt{ \frac{s^2[~\varepsilon_{B Y}, \varepsilon_{A B W}~]}{s^2[~\varepsilon_{A B P}, \varepsilon_{A B W}~]} } = 1,$$

and the instantaneous direction of $A$'s acceleration at the (event of) the meeting and passing of participant $Q$ would be expressed by the partial ordering of such families, wrt. a partial ordering of trials which all include event $\varepsilon_{A Q}$.

The corresponding "gravito-inertial force applicable to" participant $A$, in the trial under consideration, at the event $\varepsilon_{A Q}$ of having met and passed participant $Q$, would be "in the opposite direction", of magnitude

$$m_A~|~\mathbf a_A[~Q~]~|,$$

where $m_A$ is "the mass" of participant $A$.

Quite separately from these considerations we can of course distinguish

Case 1: [...] in the vicinity of the Earth [...] a preferred set of background observers [...] remain stationary

... i.e. one set (or even several different sets) of observers/participants who remained chronometrically rigid to each other, and

Case 2: [...] several massive bodies in relative motion

where no such "rigid sets" of participants may exist.