GR: "Proper Distance" Meaning and Usage

[PAllen]

I don't know in what way you think a "frame in which the events are not simultaneous" does not define a foliation. Further, the "t" direction is the one orthogonal to the space foliation you chose, so "dt nonnegative" is a concept that depends on the foliation. There is no absolute definition of positive time direction for spacelike intervals.

Of course, but I was meaning you don't need to assume any special foliation; any at all will do, and the proper distance (as I've defined it) computed in it will match spatial distance in a foliation where the events are simulaneous (so I conjecture).

Also, when not thinking about issues specific to choice of foliation, I tend to think in terms of 4-coordinate patches. Of course, any such coordinate patch defines a foliation, I just often don't think of it that way. Any such patch has a t direction. Which event is later in t, of course varies with the patch chosen, but in any patch there is positive t ordering of the events. The substance of my conjecture is it doesn't matter which of these (coordinate systems) you pick - you will always get the same result.

 

[PAllen]

Actually, I now believe that interval computed without any foliation assumption, in any frame, will be such a 3-distance in *any* foliation in which the end events are simultaneous, not just some such foliation.

Let me clarify this. I believe it will be a 3-distance in any frame in which the end events are simultaneous and the minimal interval geodesic is an amissable path (meaning has dt = zero along the path; if dt positive were allowed, dt negative would be required to achieve the simaneity, and the would make the geodesic an inadmissable path in such a coordinate system).

Similarly, I need to clarify that I claim you can compute a unique proper distance in any coordinate system (with appropriately transformed metric) using my definition, the local minimum property for the spacelike geodesics will only be true for those geodesics that are admissable paths in that coordinate system (admissable paths are those with dt >= 0 everywhere, for the given coordinate system).

 

[Ich]

Of course, but I was meaning you don't need to assume any special foliation; any at all will do, and the proper distance (as I've defined it) computed in it will match spatial distance in a foliation where the events are simulaneous (so I conjecture).

But that's not true. You can connect two events with arbitrary almost-lightlike curves, and it's easy to find for each curve a coordinate system where all the events on the curve happen simultaneously, or in ascending, or descending order. The "dt nonnegative" requirement is really completely arbitrary.

in any patch there is positive t ordering of the events

Not if they are spacelike separated.

 

[PAllen]

But that's not true. You can connect two events with arbitrary almost-lightlike curves, and it's easy to find for each curve a coordinate system where all the events on the curve happen simultaneously, or in ascending, or descending order. The "dt nonnegative" requirement is really completely arbitrary.

Of course. And my claim is if you follow my definition for proper distance, you can carry it out and get the same value for any of them. The resulting proper distance will be a 3-distance in a coordinates system where dt=0 along the minimal geodesic (the geodesics are all computed with the 4-metric using the affine definition; interval along them is measured using the (+++-) signature metric). Proper distance will only 'look' like a distance in a dt=0 foliation; otherwise it looks like a mix of events at different times.

The dt nonnegative requirement is sort of a side conjecture. The affine geodesics are local minima in 4-interval only if the geodesic is an admissable curve in that coordinate system, and it is only among admissable paths that it is local minimum.

Not if they are spacelike separated.

The end events can are either simultaneous or one of them has a higher t coordinate. I don't care about the intervening events for my definition of proper distance. I only care about this for the statement about under what conditions a geodesic is a local minimum.

 

[yuiop]

I am looking for a practical operational method to measure proper distance in GR to try and get an intuitive feel for the concept. One difficulty is that using a rod as a ruler is problematic in GR because even in free fall the ruler is subject to tidal forces and its proper length is being physically changed making it useless as a measuring device. Would the following proposed method work for measuring proper distance in free fall?

1) Attach a master clock to the centre of gravity of the falling object.
2) Attach further clocks above and below the master clock at intervals defined in (3).
3) Arrange the clocks so that the radar distance between any two clocks is the same and the clocks are spaced close enough to each other so that the radar distance measured from either end of the unit gap is approximately the same to an agreed accuracy.
4) Attach a mechanism to each clock that adjusts its spatial separation from its neighbouring clocks so that the unit gap length as defined in (3) is actively maintained to provide continuous active calibration of the ruler.

In the above set up only the master clock will be inertial in the sense that it experiences no proper acceleration. Since all the secondary clocks experience proper acceleration and since the proper acceleration is a function of the proper distance from the master clock it would be interesting to find out is the proper distance can be defined in terms of proper acceleration. If the above method does define a mechanism to measure proper distance in the free falling frame, the change in proper length of a natural free falling object due to tidal forces could in principle be measured by such a device attached to the object.

Secondly, would anyone here agree that the "Fermi normal distance" described by Pervect in this thread: https://www.physicsforums.com/showthread.php?t=435999 is in fact a proper distance in the free falling frame? (Note: I do not think that Pervect has claimed the Fermi normal distance is the proper distance in the free falling frame, but I suspect it might be.)

 

[Passionflower]

Secondly, would anyone here agree that the "Fermi normal distance" described by Pervect in this thread: https://www.physicsforums.com/showthread.php?t=435999 is in fact a proper distance in the free falling frame? (Note: I do not think that Pervect has claimed the Fermi normal distance is the proper distance in the free falling frame, but I suspect it might be.)

As I argued several times in Schwarzschild coordinates the integrand to obtain the proper distance for a free falling observer (from infinity) is simply dr. So then for instance the proper distance to the EH for rs=1 is simply r-1.

 

[JesseM]

Given two events in GR that are space like in their relation (one is outside the light cones of the other), the greatest lower bound of the spacelike intervals of geodesics connecting them is a meaninful definition of their proper distance.

But this definition requires that you already have a definition of what constitutes a spacelike "geodesic" (as opposed to any ol' spacelike path), right? Are you using the "affine definition" you referred to earlier? If so, does that definition actually allow for multiple spacelike geodesics between a given pair of events in any arbitrary curved spacetimes, or are there some classes of spacetime where the affine definition gives a unique spacelike "geodesic" between a given pair of spacelike-separated events, in which case there'd seem to be no point in talking about the "greatest lower bound" of "geodesics connecting them"?

 

[JesseM]

As I argued several times in Schwarzschild coordinates the integrand to obtain the proper distance for a free falling observer (from infinity) is simply dr. So then for instance the proper distance to the EH for rs=1 is simply r-1.

When you refer to "proper distance for a free falling observer" (as opposed to some other observer), are you assuming some particular definition of simultaneity used by a free falling observer, like the one used in Fermi normal coordinates or the one used in Schwarzschild coordinates, so that you're talking about the minimal distance on a spacelike path that lies entirely within a surface of simultaneity? If not, what's the physical meaning of talking about proper distance "for" any given observer?

 

[PAllen]

But this definition requires that you already have a definition of what constitutes a spacelike "geodesic" (as opposed to any ol' spacelike path), right? Are you using the "affine definition" you referred to earlier? If so, does that definition actually allow for multiple spacelike geodesics between a given pair of events in any arbitrary curved spacetimes, or are there some classes of spacetime where the affine definition gives a unique spacelike "geodesic" between a given pair of spacelike-separated events, in which case there'd seem to be no point in talking about the "greatest lower bound" of "geodesics connecting them"?

Good questions, I often have trouble with precision in wording. I plan to post a clearer set of self contained definitions on this later tonight.

1) Yes, for space-like geodesic I am thinking of the affine definition. Note this is a path in space-time. Pervect claims (and I agree) that if you mechanically use variation on the interval of a path, you get the same equations anyway. So it doesn't really matter.

2) Any definition of geodesic in a curved geometry allows multiple geodesics between two points. Other threads here have given examples different orbits (with special alignments) that provide different inertial paths between a pair of spacetime events. In my first long entry on this I gave the 4-space example of pole points a on a 4-shpere. This yields a 3 continuous parameter family of geodesics between these special pairs of points (all with the same length). I have no idea what the worst case is for spacelike geodesics in GR, but I wanted to define things generally. An earlier poster had addressed this issue using greatest lower bound; I liked that approach. My guess is that most common situations in GR have a small finite number of geodesics between a random pair of events, maybe only one for many event pairs in a hyperbolic geometry. However, in a closed geometry, at least two geodesics seems hard to avoid (one may be *extremely* long).

 

[Passionflower]

When you refer to "proper distance for a free falling observer" (as opposed to some other observer), are you assuming some particular definition of simultaneity used by a free falling observer, like the one used in Fermi normal coordinates or the one used in Schwarzschild coordinates, so that you're talking about the minimal distance on a spacelike path that lies entirely within a surface of simultaneity? If not, what's the physical meaning of talking about proper distance "for" any given observer?

Let's take for example the distance to the EH for a stationary observer at Schwarzschild coordinate r where m=1/2.

The integrand to get the distance is:

$$(1-r^{-1})^{-1/2}$$

The velocity v(r) (as measured by a stationary observer) for an observer free falling from infinity is:

$$v = \sqrt {{r}^{-1}}$$

So, for this observer that distance has to be Lorentz contracted.

The Lorentz factor becomes:

$$\gamma = (1-v^2)^{-1/2} = (1-r^{-1})^{-1/2}$$

But then applying this to the integrand will simply cancel everything out.

So d(r) is simply r-1.

For other observers it is obviously different because the Lorentz factor is different. Not all free falling observers have the same velocity at a given v(r). But if we know their velocity wrt a stationary observer we can calculate the Lorentz factor and thus calculate the proper distance.

These things are straightforward in a Schwarzschild solution because this spacetime is static. How I understand it for non stationary spacetimes the concept of proper distance really does not make much sense.

 

[JesseM]

Let's take for example the distance to the EH for a stationary observer at Schwarzschild coordinate r where m=1/2.

The integrand to get the distance is:

$$(1-r^{-1})^{-1/2}$$

The velocity v(r) (as measured by a stationary observer) for an observer free falling from infinity is:

$$v = \sqrt {{r}^{-1}}$$

So, for this observer that distance has to be Lorentz contracted.

Lorentz contraction is a phenomenon observed in inertial frames, so I think some physical justification is needed to apply it to a situation where we're talking about proper distance at a single instant in Schwarzschild coordinates. Perhaps we can do it as follows. I know we can conceptualize the proper distance by imagining a line of closely-spaced (really infinitesimally-spaced) observers at different distances from the horizon, extending from just outside the horizon to the radius r, and at a single instant in Schwarzschild coordinates each observer uses a freefalling ruler instantaneously at rest relative to themselves to measure the distance between them and their slightly farther neighbor, then we sum up all these local distance measures and that's the proper distance. So then if we have a freefalling observer passing by all these fixed-distance observers in sequence, then each time the freefalling observer passes between a new set of fixed-distance observers he can measure the distance between them using his own freefalling ruler, and the distance he measures will be smaller than the distance they themselves measured by the Lorentz factor, with v equal to the velocity of the other pair of observers in his own local inertial frame at the moment he's between them. So if he sums up all his own measurements of distance between successive pairs of fixed-distance observers, then that sum is basically what you mean when you talk about the distance for the freefalling observer, right?

If so, my only question would be whether the velocity you gave for the freefalling observer relative to the stationary observer, $$v = \sqrt {{r}^{-1}}$$, was the velocity of one in the other's local inertial frame at the moment they passed, or whether it's the coordinate velocity of the freefalling observer in Schwarzschild coordinates, since the two might be different and applying the Lorentz factor only makes sense if it's the first one.

 

[PAllen]

For other observers it is obviously different because the Lorentz factor is different. Not all free falling observers have the same velocity at a given v(r). But if we know their velocity wrt a stationary observer we can calculate the Lorentz factor and thus calculate the proper distance.

These things are straightforward in a Schwarzschild solution because this spacetime is static. How I understand it for non stationary spacetimes the concept of proper distance really does not make much sense.

Most of this thread concerns a totally different concept of proper distance: can you, and if so, how, and what meaning you can give, to a unique, observer independent distance between two events. People often speak of the the proper time between two events with timelike separation. Despite the fact that different world lines connecting them will have different proper time, one speaks of the 'the proper time' between the by specifying the proper time along the shortest time geodesic between them. This is completely invariant, and observer independent. The discussion on this thread is to what degree one can define a similar proper distance between two events with space like separation, without specifying any particular foliation.

 

[pervect]

While Passionflower has not (to my knowledge) given a direct answer to the fundamental question of "what curve his distance is measured over" or "what definition of simultaneity is he using", it appears to me from his discussion that what he is doing is defining a time-like congruence of "preferred" observers (those falling in from infinity) - and considering the space-like hypersurface that's orthogonal to those observers in order to perform his space-time split.

It would greatly aid the discussion if Passionflower could confirm or deny this impression. (Even asking for a less technical wording of the question would be a bit of a breakthrough compared to the current lack of response.)

Clearly, since our results are different, what Passionflower is computing is not equivalent to what I'm computing. The distance to the event horizon in Fermi Normal coordinates is NOT simply the value of the Schwarzschild R coordinate.

There is also a question of terminology here. The consensus of the previous discussion in this thread was that the term "proper distance" simply means the distance measured along some space-like curve, nothing more, nothing less. So there are some semantic issues here as well when people ask for "the" proper distance, as though it were unique. It's not unique.

The question of whether or not the Fermi-Walker distance is the same as that meaured by a Born rigid ruler is an interesting one - I don't have a definite answer at this point.

Note that in asking for the distance to the event horizon, we aren't asking for the distance between a pair of point. The event horizon is a null surface - you can think of it as the worldline of an outgoing photon, in fact.

So what in essence we are asking is very similar to the question of "how far away is that photon from me now", more than "what is the distance between two points".

 

[Passionflower]

Most of this thread concerns a totally different concept of proper distance: can you, and if so, how, and what meaning you can give, to a unique, observer independent distance between two events. People often speak of the the proper time between two events with timelike separation. Despite the fact that different world lines connecting them will have different proper time, one speaks of the 'the proper time' between the by specifying the proper time along the shortest time geodesic between them. This is completely invariant, and observer independent. The discussion on this thread is to what degree one can define a similar proper distance between two events with space like separation, without specifying any particular foliation.

Distance is obviously not observer independent. This is even true in SR where we have flat spacetime.

I simply gave an answer to a direct question. I answered this question. Do you think there is anything wrong with the formulas or the answer? If so, please say that. If not, I am not sure what your point, or perhaps objection, is.

I take it that you do realize that a Schwarzschild solution is static and thus distances for stationary observers do not change in time. I take it you also understand that objects having a velocity wrt those stationary observers will observe those distances differently.

If so, my only question would be whether the velocity you gave for the freefalling observer relative to the stationary observer, $$v = \sqrt {{r}^{-1}}$$, was the velocity of one in the other's local inertial frame at the moment they passed, or whether it's the coordinate velocity of the freefalling observer in Schwarzschild coordinates, since the two might be different and applying the Lorentz factor only makes sense if it's the first one.

The velocity (sometimes called proper velocity) is the velocity of a free falling observer from infinity with respect to the stationary observer, e.g. v_ff(r) wrt o_stationary(r).

Coordinate velocity on the other hand is the velocity measured by the 'observer at infinity' and is, as you say as well, distinct from (local) proper velocity wrt a stationary observer.

The distance to the event horizon in Fermi Normal coordinates is NOT simply the value of the Schwarzschild R coordinate.

Well it should be very simple to resolve this issue as we can have a little coordinate independent numerical example and plug in the numbers and see where the differences are.

We take M=1/2 and we have a space station at a proper distance of 1 from the EH. A free falling (from infinity) probe is passing by and just when this probe is at the same r value as the space station the captain of the probe reports his observed distance. The captain of the space station reports the observed velocity of the probe.

So we have:
M = 1/2
Proper Distance Space Station: 1

I get:

Schwarzschild radial coordinate of the space station: 1.232270555
Observed distance to the EH by the probe: 0.232270555
Velocity as observed by the space station: 0.9008385532
Lorentz Factor: 2.303328900

So do you get the same if you calculate the problem by using Fermi normal coordinates? If not, what do you get?

By the way if you have an objection to measuring the distance to the EH, an objection which I do not share by the way, then feel free to calculate it up to the EH+epsilon.

Edited: update the probe distance.

 

[yuiop]

I think it might help the discussion if we have a quick review of what proper distance means in SR first before applying the concept to GR. Well, it might help me if no one else.

Let us say the distance between Earth (E) and Mars (M) is one light year. O.K. it isn't really, but let's say it is to make the maths easier. By defining the E-M distance as one light year I mean a radar signal sent from Earth to Mars takes two years and similarly the radar distance from Mars to Earth and back to Mars takes two years. Now to an observer traveling between Earth and Mars at 0.8c the apparent distance is 0.6 lyr and this distance appears real to the traveller. A rod of proper length 0.6 lyr at rest in the reference frame of the traveller spans the E-M distance and the ends of the rods appear to be level with Earth and Mars simultaneously in the travellers rest frame. In other words, in all respects, the proper distance from Earth to Mars appears to be 0.6 lyr to the traveller and this is consistent with the reduced proper travel time of the traveller, while in the rest frame of Earth the proper distance E_M appears to be 1 lyr. So is the proper distance from Earth to Mars observer dependent or is there a single definition of the E_M distance that is observer independent. Presumable 1 lyr? If the distance between Earth and Mars is observer dependent, then presumably there is no such thing as "THE" proper distance distance between Earth and Mars.

The above discussion is about the proper distance between two physical objects at rest with respect to each other. There is also the issue of the proper distance between two events. When we talk about the the proper distance between two events, I think it almost universally agreed that this is the distance measured in the reference frame where the two events occur simultaneously. When referring to proper distances, is it implicit that we are talking about the space like separation of two simultaneous events and not the distance between two physical locations or the length of a physical object. I suspect in the preceding discussions, people are using proper distance to mean different things so it might help to come to an agreement of what it means in SR and how it can be adapted to GR.

In another thread we seemed to come to an uneasy agreement that that if a short object of unit proper length dS is dropped from infinity that its coordinate length would be dr = dS*(1-rS/r). This seems to be at odds with the claim by passionflower (supported by a book) that dS = dr for a short object falling from infinity. This also appears to be supported by the definition of the Fermi normal length which is:

$$\frac{dr}{ds} = \sqrt{1+\frac{1}{r0} - \frac{1}{r}}$$

(see https://www.physicsforums.com/showthread.php?t=435999)

The above indicates that when r0 = r, then it is indeed true that dr = ds. So which is it? dr = ds or dr = ds*(1-rS/r) for an object falling from infinity?

 

[PAllen]

Unique Proper Distance and Its Physical Interpretation
--------------------------------------------------

This essay aims to clarify the titled question (in the
affirmative) in response to one of the two variants of proper time
that pervect originally asked about:

"I am aware of two meanings of the term "proper distance" in GR. The
first is when you have points in flat space-time, or space-time that's
locally "flat enough", in which case it is defined as it is in SR, as
the Lorentz interval between the two points. This usage of the term
implies that one is considering short distances, or is working in a
flat space-time. "

I claim to give a unique, physically meaningful, answer to this
without any limitations (assuming some conjectures are true). Of
course, most observers won't measure this proper distance, any more
than they would measure proper time. All would calculate this proper
distance the same, but for most observers it wouldn't 'look like a
distance' because the events are not simultaneous.

Along the way, I will argue that one criteria suffices to be able to
state an extremal property for geodesics, both timelike and
spacelike.

A specific motivation for all of this is a way to give a robust global
definition to J. L. Synge's 'world function' idea from his 1960 GR
book. Basically, WF(p1,p2) is +(interval squared) if p1,p2 have
timelike relation, -(interval squared) if spacelike, and 0 if they could
be on a null geodesic. (Synge deliberately glosses over the issues
below, but explicitly admits he is doing so).

Definitions
-----------

spacelike / timelike separation between events - obvious, but just for
completeness: if one event is within the forward or backward pointing
light cones of the other, the relation is timelike; if not, it is
spacelike (well, of course, if it is on the light cone, we could say
light like, but we are not interested in this case; everyone would
perceive and measure this interval as zero).

A spacelike path has ds**2 with (+++-) metric non-negative everywhere.
A timelike path has ds**2 with (+---) metric non-negative everywhere.
A null geodesic (light path) can be considered a degenerate case of either.

Interval along a path is undefined for a path that is neither timelike
nor spacelike. Otherwise, it is obviously integral of ds of
appropriate sign signature.

Geodesic - May be defined either of two ways: a curve that parallel
transports its tangent (formalization of 'straight as possible')
(affine definition); or vary a distance function. For variational
definition, you must vary among all timelike curves or all spacelike
curves (else you get imaginary contributions) (variational
definition). If you mechanically do the variation, getting the Euler
necessary condition, the result is the same as the affine
definition. See end of the essay for a comment on the significance of
this.

NNT path - (non-negative time): Given a coordinate patch, any path in
that patch which is non-decreasing in t coordinate. Obviously, which
paths in spacetime are NNT is strongly coordinate and frame dependent
(for spacelike NNT paths; I believe NNT, not NNT character of timelike
paths is invariant). For both timelike and spacelike geodesics, this
will be key to defining in what limited sense they have local extremal
properties (generally, they have no such properties, even in SR, as I
shall show).

ST foliation (for a given spacelike geodesic): A foliation in which
all points of the given spacelike geodesic are simultaneous. It seems
obvious that this is always possible (in an infinite number of ways).

Geodesic Extremal Conjecture
----------------------------
A geodesic represented in some coordinate system has the following
global property:

1) If timelike, it is a local maximum interval of timelike NNT paths for
that coordinate patch.
2) If spacelike,[EDIT: and if the geodesic is NNT in this coordinate system,] it is a local
minimum interval of spacelike NNT paths for
that coordinate patch. [EDIT: the additional clause about how the spacelike geodesic
is embedded is not needed for the timelike case, because no coordinate system
can change the time ordering of events on a timelike path]

I can easily argue that you can't say more than this. I can't prove
this conjecture, but believe it is true, and would be very interested
if someone came up with a counterexample.

To see that you can't say more than this, only SR is needed. Consider
the timelike geodesic from (x,t) = (0,0) to (0,1) in some inertial
frame with minkowski metric and c=1. Now consider the NNT violating
path (0,0) to (0,.5) to (.01,.1) to (0,1). This has much longer
interval than the so called proper time maximum for a timelike
geodesic. Such reverse timelike deviations can be made as small as
possible, so this violates the possibility of making even a local
maximum claim unless you rule out non-NNT paths, as I propose.

For a spacelike geodesic, consider (0,0) to (1,0) in this
frame. Consider the path (0,0) to (.5,.49) to (1,0). It is all
spacelike and has much lower interval than the geodesic. However, it
is not an NNT path.

Unique Proper Distance Definition
-----------------------------

Proper distance between spacelike p1 and p2 is the greatest lower
bound of the intervals along all spacelike geodesics between them.

We can also define proper time between timelike p1 and p2 as least
upper bound of the intervals along all timelike geodesics between
them.

Normally, one might think there is a unique geodesic between two
points. In many non-flat geometries, this is false. I believe the
worst case is exemplified by as simple a case as the 4-sphere. If you
pick a pair of polar points, there is a 3 continuous parameter family
of geodesics between them (all with the same length).

However, I would like to boldly conjecture that the set of geodesic
intervals is never open from above for timelike points, nor open from
below for spacelike points. Then one may claim there is a possibly
non-unique extremal geodesic. All I can say to justify this is that I
can easily construct cases where the set of interval distances for all
paths between points is open, after much effort, I can't conceive of a
case for the set of geodesic intervals.

Accepting this conjecture, the physical meaning of proper time between
two events is easy to state: the proper time along a maximal geodesic
between them (the most 'pure time' path).

Meaning of Unique Proper Distance
---------------------------------

Given any minimal geodesic between two spacelike points, and any ST
foliation for it, then the proper distance (already defined) will also
be the 3-space distance using the positive definite spatial submetric
of the ST foliation. This follows from my geodesic conjecture, because
all NNT paths for for this foliation will be in the 3 surface of
simultaneity containing the minimal geodesic.

Put succinctly, the physical meaning of unique proper distance is
the 3-distance in a 3-surface of simultaneity containing a minimal
geodesic between the points.

My claims are that even though we defined proper distance using only
spacetime invariants, and thus computable in any coordinate patch
containing the points, it has a simple interpretation as the 'best
3-distance' between the events. I conjecture that for any minimal
geodesic, and any ST foliation for it, you would find that the unique
proper distance is the 3-distance for that foliation.

-------------------------------------------------------------
Affine vs. variational definition (more)
---------------------------------

In Physics, when one does a variation, you typically stop with what a
mathematician would call the first (of several) necessary (but not
sufficient) conditions for a local minimum or maximum. In doing so, I
think all one has established (since it comes out the same as the
affine definition) is that 'being as locally straight as possible' is
the first necessary condition for a local extremum. In other words, a
circuitous route to the affine definition. To me, the affine
definition is more meaningful than attaching meaning to a 'saddle
point'. In particular, physics is generally local, with infinitesimal
changes of state. It makes much more sense to think of light seeking
local straightness than global shortest distance; when they are the
same, it is only because local straightness is the first necessary
condition for global extremal properties. Similarly, for a more
abstract system, using Lagrangian analysis, it makes much for sense to
think of the system as trying to preserve its 'trend' or 'abstract
momentum' than magically seeking a global minimum (that it usually
fails to find anyway).

 

[yuiop]

In another thread we seemed to come to an uneasy agreement that that if a short object of unit proper length dS is dropped from infinity that its coordinate length would be dr = dS*(1-rS/r). This seems to be at odds with the claim by passionflower (supported by a book) that dS = dr for a short object falling from infinity. This also appears to be supported by the definition of the Fermi normal length which is:

$$\frac{dr}{ds} = \sqrt{1+\frac{1}{r0} - \frac{1}{r}}$$

(see https://www.physicsforums.com/showthread.php?t=435999)

The above indicates that when r0 = r, then it is indeed true that dr = ds. So which is it? dr = ds or dr = ds*(1-rS/r) for an object falling from infinity?

O.K. I have given this a little more thought (and I probably still need to give it a lot more thought.). Let's call the simultaneous distance between two stationary shell coordinates as measured by a falling observer ds'. Then the equation dr = ds' applies to this measurement. The other equation dr = ds*(1-rS/r) applies to the length of the falling object as measured simultaneously by the Schwarzschild observer at infinity. It seems on further thought that these two equations are not necessarily in conflict because we are talking about different measurements by different observers.

Now dr = ds' only applies over a short distance because it applies to a free falling observer and over longer distances a string of free falling observers or clocks will be moving relative to each other and the equation becomes invalid. I believe the Fermi normal length takes this into account with the equation:

$$ds = \frac{dr}{\sqrt{1+\frac{1}{R} - \frac{1}{r}}}$$

For extended distances the integrated distance has to be used and the Fermi normal distance to the event horizon (using rS=1) according to the free falling observer at R, is:

$$\Delta s = \int_{rs}^{R} \frac{dr}{\sqrt{1+\frac{1}{R} - \frac{1}{r}}} = \frac{1}{2}\left(\frac{R}{R+1}\right)^{3/2}\ln\left(\frac{R^2(1+\sqrt{1+1/R})}{1+\sqrt{1+R}}\right) +\frac{R^2-\sqrt{R}}{R+1}$$

This, I believe represents the distance to the event horizon according to a free falling observer at R using a set of clocks that are accelerated in just the right way, that they maintain constant proper separation from the point of view of the free falling observer, similar to the clocks in SR Born Rigid accelerating motion.

 

[Passionflower]

Now dr = ds' only applies over a short distance because it applies to a free falling observer and over longer distances a string of free falling observers or clocks will be moving relative to each other and the equation becomes invalid.

For a free falling observer (free falling from infinity) the proper radial distance to the EH (or for those who have an objection, which I do not share btw, in measuring up to the EH, just use EH+epsilon) is simply r-2M.

I try to find a reason why this would only be true for smaller r values but I fail to see any reason why (which of course does not mean you are wrong). Could you explain why you think rho = r-2M would only be valid in case the distance is small?

Here is a plot where we can compare three distances (for now I just called the distance you calculated "Fermi Distance"):

 

[yuiop]

For a free falling observer (free falling from infinity) the proper radial distance to the EH (or for those who have an objection, which I do not share btw, in measuring up to the EH, just use EH+epsilon) is simply r-2M.

I try to find a reason why this would only be true for smaller r values but I fail to see any reason why (which of course does not mean you are wrong). Could you explain why you think rho = r-2M would only be valid in case the distance is small?

I look at it this way. Imagine a series of observers jump from a great height. Each observer waits until the preceding observer has fallen a distance of one light-second before jumping. Each of these observers measure dr = ds locally but his measurement is fairly meaningless as a measure of proper distance, because these observers would notice that they do not maintain a constant separation of one light second wrt each other as they fall due to tidal effects. In order to maintain a constant proper separation as they fall, only one observer can truly be in freefall and the other observers would have to experience proper acceleration. The observers experiencing proper acceleration will not have the same falling velocity as free falling observer, so they do not measure dr = ds but instead measure dr = ds*sqrt(1+1/R-1/r) where R is the height of the special free falling observer. In order to measure dr = ds their falling velocity has to be exactly the escape velocity at that height which is only true for a free falling observer. In an earlier thread you mentioned having a falling rod that was "rigid" as a measure of proper distance and you acknowledged that such a rod would have to have rockets or some other physical means to counteract tidal effects and maintain a proper length in a manner analogous to Born rigid motion, so I think you should easily see what I am getting at here.

 

[Passionflower]

I look at it this way. Imagine a series of observers jump from a great height. Each observer waits until the preceding observer has fallen a distance of of one light-second before jumping. Each of these observers measure dr = ds locally but his measurement is fairly meaningless as a measure of proper distance, because these observers would notice that they do not maintain a constant separation of one light second wrt each other as they fall due to tidal effects. In order to maintain a constant proper separation as they fall, only one observer can truly be in freefall and the other observers would have to experience proper acceleration. The observers experiencing proper acceleration will not have the same falling velocity as free falling observer, so they do not measure dr = ds but instead measure dr = ds*sqrt(1+1/R-1/r) where R is the height of the special free falling observer. In order to measure dr = ds their falling velocity has to be exactly the escape velocity at that height which is only true for a free falling observer. In an earlier thread you mentioned having a falling rod that was "rigid" as a measure of proper distance and you acknowledged that such a rod would have to have rockets or some other physical means to counteract tidal effects and maintain a proper length in a manner analogous to Born rigid motion, so I think you should easily see what I am getting at here.

I need to digest this a little, but this spotted my immediate attention: "a distance of of one light-second ". Mixing radar and proper distance is very much going to complicate matters. Is it possible to restate you explanation or do you think I am wrong in thinking that adding radar distance very much complicates things?

 

[yuiop]

I need to digest this a little, but this spotted my immediate attention: "a distance of of one light-second ". Mixing radar and proper distance is very much going to complicate matters. Is it possible to restate you explanation or do you think I am wrong in thinking that adding radar distance very much complicates things?

Radar distance complicates things a little, but if the radar distance remains constant over time from the point of view of a single observer using a single clock, then it is not too complicated. Physical rods used as rulers can not be trusted because they are subject to tidal stress and strain and they have to be calibrated using radar distance anyway. I guess we could in principle get the clocks to maintain constant proper distance by calculating what proper acceleration they should have, but I think the radar system would be simpler. It would be nice if someone could calculate the required proper acceleration of the free falling clocks . What method do you propose for ensuring that the proper separation remains constant in the free falling frame?

 

[pervect]

Well it should be very simple to resolve this issue as we can have a little coordinate independent numerical example and plug in the numbers and see where the differences are.

We take M=1/2 and we have a space station at a proper distance of 1 from the EH. A free falling (from infinity) probe is passing by and just when this probe is at the same r value as the space station the captain of the probe reports his observed distance. The captain of the space station reports the observed velocity of the probe.

So we have:
M = 1/2
"Proper" Distance Space Station: 1

I get:

Schwarzschild radial coordinate of the space station: 1.232270555
Observed distance to the EH by the probe: 1.232270555
Velocity as observed by the space station: 0.9008385532
Lorentz Factor: 2.303328900

So do you get the same if you calculate the problem by using Fermi normal coordinates? If not, what do you get?

Are you sure that the "passionflower distance" to the EH is 1.232270555? I would think it would only be .23227, i.e. 1.2322 - 1

I agree that the fermi-normal distance from the space station at r=1.2322 to the event horizon is equal to 1.

The fermi-normal distance from the probe to the event horizon when it passes the space station is somewhere between .243 and .244

This is clearly not equal to your number (or your number -1).

This is found by solving numerically the differential equation for r(s), i.e.

dr/ds = -1/sqrt(1+1/1.2322 -1/r) with r(0) = 1.2322, and finding the value of s for which r(s) = 1.

I agree that the velocity of the probe relative to the space station is 1/sqrt(r) = .9, approx, and that the gamma factor is 2.3.

So, the numerical example confirms that passionflower distance is not equal to the Fermi-normal distance.

 

[Passionflower]

Radar distance complicates things a little, but if the radar distance remains constant over time from the point of view of a single observer using a single clock, then it is not too complicated.

Ok.

What method do you propose for ensuring that the proper separation remains constant in the free falling frame?

I wonder why you think we need to ensure that. After all we are not measuring the distance between two free falling points here, instead we measure the distance between a free falling point and a stationary r location, a situation where I do not readily see any 'ruler sturm und drang'.

A Schwarzschild spacetime is static and that means that distances do not change over time. Stationary space stations will forever measure the same distance to a fixed r location. The only question here is how does that distance change if something moves wrt that fixed r location. The simple answer seems to be to include the Lorentz transformation.

To enrich the matter a little, think of a free falling from infinity observer having a frictionless wheel on a road consisting of stationary (and thus accelerating differently at each r value) asphalt with fixed meter markers painted on it. Connected to that wheel is a distance meter. If we follow the meter markers we know we have to Lorentz contract the distance and then we find out this corresponds with r but if we are guided by the distance meter of the wheel it appears that we would not measure the Lorentz contracted distance but instead we would measure the meter markers which is in fact the same distance as the distance obtained by the stationary observer.

I am probably wrong, and that would not matter to me in the least, the last thing I want to do is to give an impression that "I know it all'. Actually this Fermi distance is interesting but I am still not convinced this is the correct way to measure the distance. Do you have references to the literature I can consult?

What would really help this discussion though is that we are not going to mix in radar distance. I think radar distance is very useful but it is something different than ruler distance.

Are you sure that the "passionflower distance" to the EH is 1.232270555? I would think it would only be .23227, i.e. 1.2322 - 1

Oops and of course! I updated the prior posting accordingly.

So, the numerical example confirms that passionflower distance is not equal to the Fermi-walker distance.

Indeed, I calculate: 0.2435040558

 

[Ich]

Let me clarify this. I believe it will be a 3-distance in any frame in which the end events are simultaneous and the minimal interval geodesic is an amissable path (meaning has dt = zero along the path

IOW, any spacetime geodesic is also a geodesic of a hyperplane in which it is fully contained.
That seems to be true, but why do you bother? The distance is already defined by the geodesic, without reference to frames or foliation.

1) Attach a master clock to the centre of gravity of the falling object.
2) Attach further clocks above and below the master clock at intervals defined in (3).
3) Arrange the clocks so that the radar distance between any two clocks is the same and the clocks are spaced close enough to each other so that the radar distance measured from either end of the unit gap is approximately the same to an agreed accuracy.
4) Attach a mechanism to each clock that adjusts its spatial separation from its neighbouring clocks so that the unit gap length as defined in (3) is actively maintained to provide continuous active calibration of the ruler.

This is actually the https://www.physicsforums.com/showpost.php?p=2920004&postcount=303".
I'm not entirely sure - but confident - that you get a geodesic with this definition, though.

 

[yuiop]

This is actually the https://www.physicsforums.com/showpost.php?p=2920004&postcount=303".
I'm not entirely sure - but confident - that you get a geodesic with this definition, though.

The method I described in #40 was based on a definition for proper distance given by pervect in another thread. My only contribution was the addition of active maintenance of the ruler in changing tidal situations.

I wonder why you think we need to ensure that. After all we are not measuring the distance between two free falling points here, instead we measure the distance between a free falling point and a stationary r location, a situation where I do not readily see any 'ruler sturm und drang'.

A Schwarzschild spacetime is static and that means that distances do not change over time. Stationary space stations will forever measure the same distance to a fixed r location. The only question here is how does that distance change if something moves wrt that fixed r location. The simple answer seems to be to include the Lorentz transformation.

To enrich the matter a little, think of a free falling from infinity observer having a frictionless wheel on a road consisting of stationary (and thus accelerating differently at each r value) asphalt with fixed meter markers painted on it. Connected to that wheel is a distance meter. If we follow the meter markers we know we have to Lorentz contract the distance and then we find out this corresponds with r but if we are guided by the distance meter of the wheel it appears that we would not measure the Lorentz contracted distance but instead we would measure the meter markers which is in fact the same distance as the distance obtained by the stationary observer.

OK the introduction of a rolling wheel as a distance measuring device is interesting. In SR, a wheel with a circumefrence of one meter will measure out one Kilometer on a road after 1000 revolutions are registered an on a counter attached to the wheel, if the wheel is rolled very slowly. If the wheel is traveling at 0.8c then the meter attached to the wheel will register 600 revolutions after traveling 1 km along the same road. The wheel device therefore gives a nice method to measure coordinate distances. I think you are also right that the wheel meter effectively gives an integrated coordinate distance even for an accelerating "vehicle" that the wheel meter is attached to. However, the wheel device does not measure the distance between two points simultaneously so we have to question whether what the wheel measures is in fact a proper distance and so we are back to semantics. The Fermi normal distance is an attempt to define simultaneity for a free falling observer and measure the distance simultaneously.

I would however agree with you, that if an observer falls from infinity with your wheel device, the meter attached to the wheel as it falls/rolls will agree with Schwarzschild coordinate distance and not with the proper distance measured by rulers that are stationary in the metric. If we have a metric based on Schwarzschild distance and proper time of a falling observer I think we should end up with GP coordinates. See http://en.wikipedia.org/wiki/Gullstrand–Painlevé_coordinates.

 

[Ich]

The method I described in #40 was based on a definition for proper distance given by pervect in another thread.

Well, and the method I described was based on a definition for Rindler distance given by pervect in https://www.physicsforums.com/showthread.php?p=1194653#post1194653" - four years ago. Good to see that people still - or, yet again - learn from him.
The method is fairly standard, I think. I used it (together with the appropriate simultaneity convention) mainly to set up "as static as possible" coordinates in cosmology. You find that the Big Bang is now some 25 GLY away, among other interesting things.

 

[pervect]

Unique Proper Distance and Its Physical Interpretation
--------------------------------------------------

This essay aims to clarify the titled question (in the
affirmative) in response to one of the two variants of proper time
that pervect originally asked about:

"I am aware of two meanings of the term "proper distance" in GR. The
first is when you have points in flat space-time, or space-time that's
locally "flat enough", in which case it is defined as it is in SR, as
the Lorentz interval between the two points. This usage of the term
implies that one is considering short distances, or is working in a
flat space-time. "

I claim to give a unique, physically meaningful, answer to this
without any limitations (assuming some conjectures are true). Of
course, most observers won't measure this proper distance, any more
than they would measure proper time. All would calculate this proper
distance the same, but for most observers it wouldn't 'look like a
distance' because the events are not simultaneous.

Here's the problem I see. Your NNT (Non Negative time) condition isn't as innocent as you think. You've already noted that it's not coordinate independent, and I believe it's going to introduce coordinate dependency into your definition.

You start out by making some space-time slice in order to be able to distinguish out negative time from positive time, which means you have defined space-like slices of simultaneity.

Now, the interesting part comes in just when the geodesics lying entirely in this space-like slice are not the same as the geodesics in space-time.

Clearly, if they are different, and if we take our initial point and final point as being "at the same time", some section of the geodesic in space-time is going to violate your NNT condition.

So what's going to be the net result? Well, if the end time is the same as the start time, not even a small section of the geodesic can ever advance forward in time. So the entire geodesic must lie within the surface of simultaneity.

As a consequence, your approach doesn't seem to be any different than just specifying your space-like slice by fiat.

 

[Ich]

As a consequence, your approach doesn't seem to be any different than just specifying your space-like slice by fiat.

With the requirement that a spacetime geodesic be in the slice, you can forget about the slice and use the geodesic instead to define distance. The slice is completely irrelevant then.

 

[yuiop]

Actually this Fermi distance is interesting but I am still not convinced this is the correct way to measure the distance. Do you have references to the literature I can consult?

What would really help this discussion though is that we are not going to mix in radar distance. I think radar distance is very useful but it is something different than ruler distance.

If you look carefully at post #54 (the post you were responding to), you will see that I never actually mentioned radar distance. You simply assumed that when I mentioned "a distance of one light-second" that I must be talking about the radar distance but that is not necessarily true.

Anyway, let us assume we have an "infinitely rigid" ruler that is two light seconds long in flat space. (We can calibrate it's proper length in flat space using radar distance or your slow meter wheel device.) We place a clock cT and observer at the top of the rod, a second clock cM and observer at the middle of the rod and a third clock CB and observer at the bottom of the ruler and synchronise the clocks in flat space using the Einstein clock synchronisation convention. The observer at the middle is the primary observer. We drop the rod and observers and ask the observer at the bottom of the rod to note the time tZ on his local clock cB as he passes the event horizon. (How this observer knows he is at the event horizon is a topic in itself ). We ask the primary observer where he was when his local clock cM read tZ. This distance is the Fermi normal length to the event horizon according to the primary free falling observer. Note that the observers at the top or bottom of the rigid falling rod are not free falling and not inertial.

Finally (and this question is open to anyone) would it useful to reserve the term "proper length" to refer exclusively to the length of a physical object as measured in the rest frame of the object and reserve the term "proper distance" or "proper spacelike separation" to refer exclusively to the spatial separation between two events as measured in a reference frame where the two events are simultaneous?

 

[Passionflower]

Finally (and this question is open to anyone) would it useful to reserve the term "proper length" to refer exclusively to the length of a physical object as measured in the rest frame of the object and reserve the term "proper distance" or "proper spacelike separation" to refer exclusively to the spatial separation between two events as measured in a reference frame where the two events are simultaneous?

Proper distance or proper radial distance is often used in the Schwarzschild solution to mean the physical radial distance between two coordinate values (as opposed to the coordinate distance which is not really a distance).

 

[yuiop]

Proper distance or proper radial distance is often used in the Schwarzschild solution to mean the physical radial distance between two coordinate values (as opposed to the coordinate distance which is not really a distance).

As you pointed out, a rolling meter wheel device would measure coordinate distance and if the wheel was rolling along a radial "road" at free fall velocity, its measurement would agree with the Schwarzschild radial distance. Are you saying the distance measured by the meter wheel device is not really a distance?

I assume you meant the coordinate distance is not a proper distance, no?

 

[PAllen]

Finally (and this question is open to anyone) would it useful to reserve the term "proper length" to refer exclusively to the length of a physical object as measured in the rest frame of the object and reserve the term "proper distance" or "proper spacelike separation" to refer exclusively to the spatial separation between two events as measured in a reference frame where the two events are simultaneous?

Yes, I think it would be good to distinguish proper distance from proper length in this way. Note that in GR, having the ends simultaneous isn't very meaningful. You could still come different lengths in different coordinate systems. My argument is you would say something more like: the spatial separation between two events in a coordinate system containing a minimal geodesic between them within a hypersurface of simultaneity.

 

[Passionflower]

I assume you meant the coordinate distance is not a proper distance, no?

Right, the Schwarzschild r coordinate is not a measure of physical distance, in fact it is a measure of curvature. To obtain a physical distance from these coordinates we typically have to integrate, however there is one exception and that is for the physical distance (at least one way of measuring distance) observed by a free falling (from infinity) observer. For this observer the physical distance is equal to the Schwarzschild coordinate difference. But as Pervect demonstrated there are multiple ways of defining distance. What would be useful I think is to be able to express all radial motion using these Fermi coordinates as well.

E.g. we can identify three simple conditions:

1. Free fall from infinity with a given initial velocity (including v=0)
2. Free fall from a given r value.
3. An linearly accelerating observer falling in the field (e.g. the magnitude of his acceleration is smaller than the inertial acceleration of the field).

By the way the free falling observer from infinity is also interesting wrt PG coordinates, or sometimes called "The River Model".

 

[yuiop]

What would be useful I think is to be able to express all radial motion using these Fermi coordinates as well.

E.g. we can identify three simple conditions:

1. Free fall from infinity with a given initial velocity (including v=0)
2. Free fall from a given r value.
3. An linearly accelerating observer falling in the field (e.g. the magnitude of his acceleration is smaller than the inertial acceleration of the field).

To start you off here are the equations for radial motion in Schwarzschild coordinates based on information from the mathpages website

http://www.mathpages.com/rr/s6-07/6-07.htm

1. Free fall from infinity with a given initial velocity V (including V=0)

The coordinate velocity of a free falling particle at r, with initial velocity of V at infinity is:

$$\frac{dr}{dt} = (1-2m/r) \sqrt{\frac{2m}{r}(1-V^2) +V^2}$$

The time dilation factor for the falling particle is:

$$\frac{dt}{dtau} = \frac{1}{(1-2m/r) \sqrt{1 - V^2}}$$

It follows that the proper velocity of a free falling particle in terms of the proper time of the particle is:

$$\frac{dr}{dtau} = \sqrt{\frac{2m}{r} + \frac{ V^2}{(1-V^2)}}$$

2. Free fall from a given r value.

The coordinate velocity of a free falling particle at r that was initially at rest at R is:

$$\frac{dr}{dt} = \frac{(1-2m/r)}{ \sqrt{1-2m/R} } \sqrt{\frac{2m}{r} - \frac{2m}{R}}$$

The time dilation factor for the particle dropped from R is:

$$\frac{dt}{dtau} = \frac{ \sqrt{1 - 2m/R}}{ (1-2m/r) }$$

It follows that the proper velocity of a free falling particle dropped from R in terms of the proper time of the particle is:

$$\frac{dr}{dtau} = \sqrt{\frac{2m}{r} - \frac{2m}{R}}$$

3. An linearly accelerating observer falling in the field (e.g. the magnitude of his acceleration is smaller than the inertial acceleration of the field).

This is more complicated. Some equations for this situation were given in #345 of this old thread: https://www.physicsforums.com/showthread.php?p=2747788#post2747788

[EDIT] The equations for a particle with initial velocity V have been edited to correct a major mistake in the calculations pointed out by Passionflower. [/EDIT]

 

[PAllen]

IOW, any spacetime geodesic is also a geodesic of a hyperplane in which it is fully contained.
That seems to be true, but why do you bother? The distance is already defined by the geodesic, without reference to frames or foliation.

The definition of proper distance doesn't employ coordinate dependent quantities. However, this was in the section 'physical meaning'. In SR, one often talks about proper distance between two events being the distance you would measure in a frame in which the events are simultaneous. I was seeking to make an equivalent statement in GR, where the SR statement is totally inadequate (actually, it isn't adequate in SR if you allow accelerated frames, but you can trivially fix it by restricting to inertial) For GR and events on either side of e.g. a black hole, you need more of a fix. So my attempt at the best equivalent statement in GR is:

The proper distance between two events is equal to the 3-distance measured in a coordinate system containing a minimal geodesic in a hypersurface of simultaneity.