GR: "Proper Distance" Meaning and Usage

[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:

<br /> \frac{dr}{ds} = \sqrt{1+\frac{1}{r0} - \frac{1}{r}}<br />

(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:

<br /> ds = \frac{dr}{\sqrt{1+\frac{1}{R} - \frac{1}{r}}}<br />

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:

<br /> \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}<br />

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.

 

[PAllen]

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.

The definition of proper distance doesn't use anything related to the NNT condition or any coordinates at all. Check the definition section again to see this. The NNT idea comes in with an attempt to say something limited about the extremal properties spacetime geodesics. I establish that without restrictions in nearby paths, no max/min statement is possible at all, even in SR. So I am only looking here to propose a restricted sense in which you can declare a max/min property.

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.

I am not referring to these geodesics at all. Much later, in talking about physical meaning of proper distance I talk about 3-geodesics in a carefully chosen simultaneity slice. But in this section I am only talking about spacetime geodesics and spacetime paths.

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.

This observation gets at a small omission in the extremal conjecture section, where an additional issue I dealt with later (in the "meaning of proper distance" section) I forgot to include in the geodesic extremal conjecture. I thought of this early today, but haven't had a chance to fix it. That, indeed, is related to what you say here.

I meant to specify that a spacelike geodesic is a local minimum related to NNT paths in a coordinate system that includes that geodesic as an NNT path. This is they way I did it later.

Then I think there is nothing wrong with my conjecture. I will see if I can still edit the original post, otherwise I will post a corrected version of the extremal conjecture.

 

[PAllen]

In reference to my long post #51, which I can no longer edit, I describe a limited statement about in what sense a spacelike geodesic can be considered minimal. The statement in that post is correct but worded in a coordinate dependent way. I realize a somewhat stronger, more coordinate independent statement is implicit, as follows (you need to read this in conjunction with post #51 for terminology):

Suppose there is a path p between two events with spacelike separation,
and interval(p) < interval(nearby geodesic between the same events). Then in every coordinate
system in which the geodesic is NNT, p will violate NNT.

 

[pervect]

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.

The method you describe for distance is due to Born, it's the concept of using Born rigid motion to define length, though it's one I also like and have espoused in the past.

A more detailed calculation confirms to my satisfaction that the fermi-normal distance will give the same result as the measurement of distance via a Born rigid ruler.

(I felt I needed to confirm that the timelike worldlines of the accelerating observers at constant fermi-normal distance were orthogonal to the space-like geodesics along which the distance was measured. This will be true if the metric in these coordinates is diagonal away from the origin, which appears to be the case. Using the same series approach but including higher order terms, I convinced myself that metric was in fact diagonal.)

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.)

It is a proper distance in the sense that it's a distance measured along a specific curve. Passionflower's distance will also be a proper distance - if he ever gets around to specifying the curve along which it's measured, that is :-)

However, I don't believe that passionflower's distance will be equivalent to the distance measured by dropping a Born rigid ruler.

 

[yuiop]

Just a quick note to acknowledge that Passionflower pointed out a major mistake in my calculations for the motion of a falling body with an initial velocity at infinity in post #69. I have now edited that post and hopefully the equations are now correct. Thanks for checking Passionflower. It is good to know that someone is checking the posted equations so that they might be a useful and accurate reference to someone in the future .

 

[yuiop]

It is a proper distance in the sense that it's a distance measured along a specific curve. Passionflower's distance will also be a proper distance - if he ever gets around to specifying the curve along which it's measured, that is :-)

However, I don't believe that passionflower's distance will be equivalent to the distance measured by dropping a Born rigid ruler.

As far as I can tell, Pf's distance is the distance measured by a series of free falling observers each with their own clock and very short Born rigid ruler. The observers would have to synchronise clocks at infinity and then jump at regular intervals of say one second. At the time the leading observer arrives at the event horizon, each observer notes their spatial separation from their nearest neighbours at that time (according to their own local clock) and the distance to the event horizon according to a given observer at "that time" is the sum of all the individual measurements. The observers have to be very close together and the Born rigid rulers very short. Unlike Fermi normal observers and clocks, the Pf free fall observers are not physically connected in any way and do not experience proper acceleration. They will notice that their separation from their immediate neighbours is continually changing according to measurements using their Born rigid rulers or radar measurements. The Pf distance should also agree with the Schwarzschild coordinate distance and with what is measured by a free falling observer using a rolling wheel meter like the odometer of a car, as also pointed out by Pf. I am pretty sure the Pf measurement is closely related to the PG coordinate system. It would be interesting to check that out.

 

[Passionflower]

You are welcome yuiop, I definitely can use some practice in checking mistakes in formulas as the formulas I write myself are often full of mistakes.

I attempted to generalize the formulas to obtain the proper velocity of a free falling observer wrt a local stationary observer, in the attempt to generalize the, what Pervect calls, 'the passionflower distance'. Then perhaps we can do the same thing but then for the 'Fermi distance'.

Apparently the velocity function can be parametrized, if I am not mistaken such thing was developed by Hartle (or perhaps he copied it before, I don't know that).

Basically the proper velocity wrt a local stationary observer of a radially free falling observer can be described as:

<br /> v=\sqrt {{\frac {{A}^{2}-g_{{{\it tt}}}}{{A}^{2}}}}<br />

where:

<br /> g_{{{\it tt}}}=1-2\,{\frac {m}{r}}<br />

In the case the free fall is from infinity A simply becomes 1.
In the case the free fall has an initial velocity then A is calculated as:

<br /> A={\frac {1}{\sqrt {1-{v}^{2}}}}<br />

Now, if we assume the formula below as given by yuiop is both correct and not an approximation:

<br /> \frac{dr}{dtau} = \sqrt{\frac{2m}{r} - \frac{2m}{R}} <br />

then there is a catch.

For A smaller than 1 the free fall starts from a given height. I was trying to obtain the correct formula in order to plug in the right A in such a case and here is where I got some surprises (I assume for the greater minds on the forum this is yet another demonstration of my lack of understanding). I did not succeed in expressing this in terms of r only. Initially I was trying to reason that if in some way I could 'subtract' the escape velocity at the given R and convert that into A I would get the expected results. But that did not work, it turns out that A is no longer constant during the free fall from a given height, we could express it in the following way to get the desired results, but it is ugly:

<br /> A = -{\frac {\sqrt { \left( -rR+2\,mR-2\,rm \right) R \left( -r+2\,m<br /> \right) }}{-rR+2\,mR-2\,rm}}<br />

This 'monstrosity' is not even real valued!

Now the interesting question is why, assuming again the formula based on this is exact, does the parameter depend on r? Perhaps I made a mistake or is perhaps the formula to obtain the proper velocity from a given R only an approximation?

For instance it is rather tempting to calculate A for a drop at h independently of r by using:

<br /> A=\sqrt {1-2\,{\frac {m}{h}}}<br />

The result is close to yuiop's formula but not identical. The 'shape' of the slabs look rather similar.

 

[JesseM]

Is it true that each hovering observer's local surface of simultaneity in their local inertial rest frame is parallel to a line of constant Schwarzschild time and varying radius? If so, maybe Passionflower's distance would be the proper distance along a single spacelike curve where the tangent at each point along the curve is parallel to the local surface of simultaneity of a freefalling observer at that point?

 

[Passionflower]

Mathpages gives a parameter K defined as

K = \sqrt{1-V^2}

or alternatively as:

K = 1/\sqrt{1-2m/R}

It is obvious that the Hartle parameter A is related to the mathpages parameter by A = 1/K.

This suggests you should end up with:

A = \sqrt{1-2m/R}

for the constant parameter in terms of the apogee height R.

Right, I think I did that, but then the results do not seem identical to the formula you quoted (see the graph). Could you confirm? It looks like it is off by some linear factor. I probably made some mistake.

Hopefully we can work out the kinks because then we have a generic velocity formula for a free falling observer. Then it is simply a matter of finding the Lorentz factor based on the obtained velocity v(r) and divide (because it is length contraction! ) the integrand by the Lorentz factor to get the correct integrand to obtain a proper distance in all free falling radial scenarios (except for acceleration that is, but I think it is better to get the kinks out free falling motion first). By the way, the parametric approach has the advantage that we can extend things to a solution with a rotating black hole by applying some adjustments. As far as I am concerned that would then cover most areas where the notion of distance makes some sense, as the concept of proper (e.g. physical) distance in non-stationary spacetimes do not make much sense to me (except perhaps in things like FLRW models where distances are 'pumped up' in time). To me, but that is likely because I am simple minded, the generic distance discussion is rather academic for arbitrary non-stationary spacetimes as there is really not much to calculate.

By the way once this is sorted out it would be really interesting to lay out (in a separate topic) all the Doppler shifts (including a decomposition into kinematical and gravitational components) for all radial motion wrt a stationary local observer, an observer from infinity and of course wrt other free falling observers. As far as I know, no such generic formula is available.

 

[yuiop]

Apparently the velocity function can be parametrized, if I am not mistaken such thing was developed by Hartle (or perhaps he copied it before, I don't know that).

Basically the proper velocity wrt a local stationary observer of a radially free falling observer can be described as:

<br /> v=\sqrt {{\frac {{A}^{2}-g_{{{\it tt}}}}{{A}^{2}}}}<br />

where:

<br /> g_{{{\it tt}}}=1-2\,{\frac {m}{r}}<br />

O.K. I have figured it now. The equation given by Hartle is not proper velocity (dr/ddtau), but the coordinate velocity (dr'/dt') according to a local stationary observer using his local stationary rulers and clocks. Proper velocity is the velocity using the proper time of the falling clock which is a different thing.

Starting with the Schwarzschild falling coordinate velocity I gave in #69:

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

and using dr'/dr = 1/sqrt(1-2m/r) and dt/dt' = 1/sqrt(1-2m/r) to relate local measurements of the stationary observer to the observer at infinity such that dr/dt = dr'/dt' (1-2m/r), the local coordinate velocity is:

<br /> \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m}{r}(1-V^2) +V^2} <br />

This can be written using A = 1/sqrt(1-v^2) as:

<br /> \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m/r +V^2A^2}{A^2} } <br />

\Rightarrow \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m/r + A^2 - A^2 +V^2A^2}{A^2} }

(because A^2-A^2 =0)

\Rightarrow \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m/r + A^2 - A^2(1-V^2)}{A^2} }

\Rightarrow \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m/r + A^2 - 1 }{A^2}

\Rightarrow \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{ A^2 - (1 -2m/r) }{A^2}

\Rightarrow \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{ A^2 - g_{\it tt} }{A^2}

 

[Passionflower]

O.K. I have figured it now. The equation given by Hartle is not proper velocity (dr/ddtau), but the coordinate velocity (dr'/dt') according to a local stationary observer using his local stationary rulers and clocks.

Just to verify we are on the same page, you do not mean coordinate velocity in terms of Schw. coordinates right? Because clearly a local stationary observer would not measure such coordinates with local clocks and rulers. Also the free falling observer would measure the same velocity right, since this is simply the principle of relativity at work, right (e.g. time dilation, length contraction, locally Minkowskian)? But if so, then what is proper velocity in this context if you call this one coordinate velocity? Celerity?

Since:

<br /> {dt \over d\tau} = {A \over g_{tt} }<br />

It should be staightforward to convert this coordinate velocity to proper velocity.

By the way it would be helpful is someone could attach the relevant page from Hartle, unfortunately I do not have it: http://books.google.com/books?id=azZmQgAACAAJ&dq=inauthor:%22James+B.+Hartle%22&hl=en&ei=uWHCTLO0I5CCsQP1xrjZCw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CC0Q6AEwAA

 

[yuiop]

Just to verify we are on the same page, you do not mean coordinate velocity in terms of Schw. coordinates right? Because clearly a local stationary observer would not measure such coordinates with local clocks and rulers. Also the free falling observer would measure the same velocity right, since this is simply the principle of relativity at work, right (e.g. time dilation, length contraction)? But if so, then what is proper velocity in this context if you call this one coordinate velocity? Celerity?

I was just trying to make clear the distinction between proper velocity and coordinate velocity. Proper velocity (or celerity) is coordinate distance (which depends on the observer) divided by the proper time of the moving object. In post #69 I gave the proper velocity of the falling object according to the Schwarzschild observer at infinity as:

<br /> \frac{dr}{dtau} = \sqrt{\frac{2m}{r} + \frac{ V^2}{(1-V^2)}} <br />

where tau is the proper time of the falling clock. This can be converted to the proper velocity as measured by a local observer at coordinate radius r by using dr'/dr = sqrt(1-2m/r) so that: <br /> \frac{dr&#039;}{dtau} = \frac{1}{\sqrt{1-2m/r}} \sqrt{\frac{2m}{r} + \frac{ V^2}{(1-V^2)}} <br />

and this can be simplified using A = 1/sqrt(1-V^2) and g_{tt} = (1-2m/r) to:

<br /> \frac{dr&#039;}{dtau} = \sqrt{\frac{A^2 - g_{tt}}{g_{tt}}}<br />

The equation for the local velocity of the falling object given by Hartle is obviously not the proper velocity.

Since:

<br /> {dt \over d\tau} = {A \over g_{tt} }<br />

It should be staightforward to convert this coordinate to proper velocity.

Yes we can do it that way too. Starting with the Hartle equation for local velocity:

<br /> \frac{dr&#039;}{dt&#039;} =\sqrt {{\frac {{A}^{2}-g_{{{\it tt}}}}{{A}^{2}}}}<br />

the local proper velocity is:

\frac{dr&#039;}{dtau} = \frac{dr&#039;}{dt&#039;}*\frac{dt&#039;}{dt}*\frac{dt}{dtau} = \frac{A}{\sqrt{g_{tt}}} \sqrt {\frac {{A}^{2}-g_{tt}}{A^2}} = \sqrt{\frac{A^2 - g_{tt}}{g_{tt}}}

 

[Passionflower]

I was just trying to make clear the distinction between proper velocity and coordinate velocity. Proper velocity (or celerity) is coordinate distance (which depends on the observer) divided by the proper time of the moving object. In post #69 I gave the proper velocity of the falling object according to the Schwarzschild observer at infinity as:

Ok, it looks we are on the same page.

After some tinkering, the simplest generic formula I came up with is not the 'Hartle' approach but this:

<br /> v_{(r)} = \sqrt {2m \left( {r}^{-1}-{{\it r_0}}^{-1}-{\frac { \left( -r+1<br /> \right) {{\it v_0}}^{2}}{r}} \right) }<br />

I worked out a position-velocity equivalent, and it turned out it is much simpler to express an initial velocity as a (virtual) position than vice versa.

Lorentz factoring the equation gives:

<br /> {1 \over \gamma} =\sqrt {1-2\,m \left( {r}^{-1}-{r_{{0}}}^{-1}-{\frac { \left( -r+1<br /> \right) {v_{{0}}}^{2}}{r}} \right) }<br />

Multiplying 1/gamma by the integrand used by a stationary observer to obtain the distance for an arbitrary radial free falling observer gives:

<br /> \int _{{\it r1}}^{{\it r2}}\!\sqrt {1-2\,m \left( {r}^{-1}-{{\it r_0}}^<br /> {-1}-{\frac { \left( -r+1 \right) {{\it v_0}}^{2}}{r}} \right) }{\frac <br /> {1}{\sqrt {1-2\,{\frac {m}{r}}}}}{dr}<br />

A few plots:

 

[yuiop]

Ok, it looks we are on the same page.

After some tinkering, the simplest generic formula I came up with is not the 'Hartle' approach but this:

<br /> v_{(r)} = \sqrt {2m \left( {r}^{-1}-{{\it r_0}}^{-1}-{\frac { \left( -r+1<br /> \right) {{\it v_0}}^{2}}{r}} \right) }<br />

Before I check this out in detail, can you confirm that to use your formula, that if r0 is not infinite then v0 must be set to zero and if v0 is not zero then r0 has to be set to infinite?

 

[Passionflower]

Before I check this out in detail, can you confirm that to use your formula, that if r0 is not infinite then v0 must be set to zero and if v0 is not zero then r0 has to be set to infinite?

Yes, as far as I understand both parts 'on' would lead to something unphysical.

 

[yuiop]

After some tinkering, the simplest generic formula I came up with is not the 'Hartle' approach but this:

<br /> v_{(r)} = \sqrt {2m \left( {r}^{-1}-{{\it r_0}}^{-1}-{\frac { \left( -r+1<br /> \right) {{\it v_0}}^{2}}{r}} \right) }<br />

If I set v0 = 0 then the above equation reduces to:

<br /> v_{(r)} = \sqrt {2m \left( {r}^{-1}-{{\it r_0}}^{-1}\right) }<br />

which agrees with the mathpages equation I gave in #69 for the proper velocity dr/dtau:

... 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}}

Now if I set r0 = infinite in your equation, it reduces to:

<br /> \frac{dr}{dt} = \sqrt{\frac{2m}{r}(1-v_0^2) +2mv_0^2} <br />

which does not agree with either the mathpages coordinate velocity or proper velocity:

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 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)}}

although it is similar to the coordinate velocity. More tinkering required I think

You did not specify if v_{(r)} is intended to be a proper or coordinate velocity or if the observer that makes the measurement is local or at infinity.

[EDIT] Actually looking back I am guilty of the same thing in #69 and did not make it clear that all measurements in that post are according to the Schwarzschild observer at infinity.

 

[Passionflower]

Now if I set r0 = infinite in your equation, it reduces to:

<br /> \frac{dr}{dt} = \sqrt{\frac{2m}{r}(1-v_0^2) +2mv_0^2} <br />

which does not agree with either the mathpages coordinate velocity or proper velocity:


although it is similar to the coordinate velocity. More tinkering required I think

There are different kind of velocities, pure coordinate velocity for an observer at infinity will obviously be zero at the EH.

You did not specify if is intended to be a proper or coordinate velocity or if the observer that makes the measurement is local or at infinity.

The velocity this formula is supposed to track is the velocity as measured by a local stationary observer. I am not using the observer at infinity in any way, this observer is by the way physically speaking rather uninteresting.

Perhaps that is the source of some of the confusion we had communicating this. At all times I talked about the velocity as measured by a local stationary observer.

So far I do not see anything wrong with the formula, and the plots look right as well.

Perhaps you could present an alternative and then we can plot those out as well. I would not have been wrong the first time. :)

 

[pervect]

As far as I can tell, Pf's distance is the distance measured by a series of free falling observers each with their own clock and very short Born rigid ruler. The observers would have to synchronise clocks at infinity and then jump at regular intervals of say one second. At the time the leading observer arrives at the event horizon, each observer notes their spatial separation from their nearest neighbours at that time (according to their own local clock) and the distance to the event horizon according to a given observer at "that time" is the sum of all the individual measurements. The observers have to be very close together and the Born rigid rulers very short. Unlike Fermi normal observers and clocks, the Pf free fall observers are not physically connected in any way and do not experience proper acceleration. They will notice that their separation from their immediate neighbours is continually changing according to measurements using their Born rigid rulers or radar measurements. The Pf distance should also agree with the Schwarzschild coordinate distance and with what is measured by a free falling observer using a rolling wheel meter like the odometer of a car, as also pointed out by Pf. I am pretty sure the Pf measurement is closely related to the PG coordinate system. It would be interesting to check that out.

You've gotten a key point - because PF's ensemble of observers doesn't maintain a constant distance from each other, they don't define a Born rigid distance.

The motion of the observers can't be ignored and is important to specify, because different observers have different notions of space and time.

As far as using small, born-rigid rulers goes - if you are in an area of space-time small enough that the effects of curvature are small, radars and rulers agree. This is why the SI folks were able to switch from a physical, rigid rod, to a radar-like method of counting the wavelengths of a particular frequency of a cesium standard without a major upheaval. When you consider how atoms maintain their distance, it's not surprising, the forces between nearby atoms are based on the same electromagnetic fundamentals that the radar is based on.

One of the advantages of the radar concept for measuring distance is that it makes the implied concept of simultaneity more manifest than using rigid bars.

Let one wolrdline hold the radar. The other worldline reflects the radar signal. Then the midpoint between the time of signal emission and its reception on the radar world-line is simultaneous with the reflection of the radar signal on the reflection worldline.

This defines a short segment of a curve on the radar world-line, that propagates in a spatial direction. It's an alternate way of defining "orthogonal" without the math, though the 4-vector approach remains more convenient in my opinion. And if you imagine repeating the process, by putting a bunch of radars on a bunch of worldlines, and adjusting the timing so that the short line-segments all connect, you'll generate the entire curve, the curve of simultaneity, the one that PF's distance is being measured along.

 

[Roshanjha14]

I want to know that is it possible that we will get all our lost things when the big CRUNCH occur?

 

[yuiop]

The velocity this formula is supposed to track is the velocity as measured by a local stationary observer. I am not using the observer at infinity in any way, this observer is by the way physically speaking rather uninteresting.

In that case you formula still does not reduce to the correct equations for the local velocity when r0 = infinite or v0 = 0. These are the correct equations:

The local velocity of a falling particle with initial velocity v0 at infinity, according to an observer at r, was derived in #79 and is:

<br /> \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m}{r}(1-v0^2) +v0^2} <br />

The local velocity of a falling particle with apogee r0 according to an observer at r is:

<br /> \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m(r^{-1} - r0^{-1})}{(1-2m/r0)}} <br />

Perhaps that is the source of some of the confusion we had communicating this. At all times I talked about the velocity as measured by a local stationary observer.

So far I do not see anything wrong with the formula, and the plots look right as well.

Perhaps you could present an alternative and then we can plot those out as well. I would not have been wrong the first time. :)

A general formula that combines both the above equations should be something like (if I have got it right):

\frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m(r^{-1}(1-v0^2) - r0^{-1})}{(1-2m/r0)}+v0^2}

Just to be clear, I am not talking about proper velocity and I assume you are not either.

 

[yuiop]

I want to know that is it possible that we will get all our lost things when the big CRUNCH occur?

Yes, all your mysteriously lost odd socks will be right there next to you, but they will still be rather difficult to find because everyone elses missing odd socks and the rest of the universe will be right there next to you too.

 

[Passionflower]

Now if I set r0 = infinite in your equation, it reduces to:

<br /> \frac{dr}{dt} = \sqrt{\frac{2m}{r}(1-v_0^2) +2mv_0^2} <br />

which does not agree with either the mathpages coordinate velocity or proper velocity:

You might want to double check that.

I get:

<br /> \sqrt {2m \left( {r}^{-1}-{\frac { \left( -r+1 \right) {{\it <br /> v_0}}^{2}}{r}} \right) }<br />

The local velocity of a falling particle with initial velocity v0 at infinity, according to an observer at r, was derived in #79 and is:

<br /> \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m}{r}(1-v_0^2) +v_0^2} <br />

This formula is identical to mine if we take r0 = infinity.

This is identical:

<br /> \sqrt {{v_0}^{2}+{\frac {1-{v_0}^{2}}{r}}}=\sqrt {{r}^{-1}-{\frac {<br /> \left( -r+1 \right) {v_0}^{2}}{r}}}<br />

<br /> \frac{dr single-quote}{dt single-quote} = \sqrt{\frac{2m(r^{-1} - r_0^{-1})}{(1-2m/r_0)}} <br />

It looks like you are right on that one.

 

[yuiop]

Now if I set r0 = infinite in your equation, it reduces to:

<br /> \frac{dr}{dt} = \sqrt{\frac{2m}{r}(1-v_0^2) +2mv_0^2} <br />

You might want to double check that.

I get:

<br /> \sqrt {2m \left( {r}^{-1}-{\frac { \left( -r+1 \right) {{\it <br /> v_0}}^{2}}{r}} \right) }<br />

Starting with your equation:

<br /> \sqrt {2m \left( {r}^{-1}-{\frac { \left( -r+1 \right) {{\it <br /> v_0}}^{2}}{r}} \right) }<br />

\Rightarrow \sqrt {\frac{2m}{r}-\frac { 2m\left( -r+1 \right) <br /> v_0^2}{r}} <br />

\Rightarrow \sqrt {\frac{2m}{r} +2mv_0^2-\frac { 2mv_0^2}{r}} <br />

\Rightarrow \sqrt {\frac{2m}{r}(1-v_0^2) +2mv_0^2}<br />

So we both get the same result for the reduction of your equation when r0 = infinite, and this does not agree exactly with the equation I gave in #89:

The local velocity of a falling particle with initial velocity v0 at infinity, according to an observer at r, was derived in #79 and is:

<br /> \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m}{r}(1-v0^2) +v0^2} <br />

Do you now at least agree that the equations I gave in #89 are correct?

I note that sometimes you are explicitly using rs = 2m and other times using rs = 1 which might lead to errors.
Your generic equation is close for the r0 = infinity case if we only use rs =1 but it is miles out for the v0 = 0 and r0<infinite case.

 

[Passionflower]

Ok, let's take a look at the formulas you presented.

The local velocity of a falling particle with initial velocity v0 at infinity, according to an observer at r, was derived in #79 and is:

<br /> \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m}{r}(1-v0^2) +v0^2} <br />

When I enter m=1/2, v0=1/2 and r=1 I get a v>1 this does not seem right. In fact v reaches 1 already at r=4/3 using your formula. Unless I am mistaken v=1 should only happen at the EH. My formula gives v=1 at r=1.

I note that sometimes you explicitly using rs = 2m and other times using rs = 1 which might lead to errors.

Yes, I apologize for that, I changed many formulas around to express them in terms of m instead of rs but some have not been updated.

 

[yuiop]

Ok, let's take a look at the formulas you presented.

The local velocity of a falling particle with initial velocity v0 at infinity, according to an observer at r, was derived in #79 and is:

<br /> \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m}{r}(1-v0^2) +v0^2} <br />

When I enter m=1/2, v0=1/2 and r=1 I get a v>1 this does not seem right. In fact v reaches 1 already at r=4/3 using your formula. Unless I am mistaken v=1 should only happen at the EH. My formula gives v=1 at r=1.

I am not sure how you get that result. I get:

<br /> \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m}{r}(1-v0^2) +v0^2} = \sqrt{\frac{1}{1}(1-(0.5)^2) +(0.5)^2} = \sqrt{1-0.25 + 0.25} = 1<br />

Try your formula:

<br /> <br /> \frac(dr&#039;}{dt&#039;} = \sqrt {2m \left( {r}^{-1}-{\frac { \left( -r+1 \right) {{\it <br /> v_0}}^{2}}{r}} \right) }<br /> <br />

using m=7, r=2m and v0 =1/2 and you will see where the problem is. (My (mathpages) equation works for any arbitrary value of m.)

 

[Passionflower]

I am not sure how you get that result. I get:

<br /> \frac{dr&#039;}{dt&#039;} = \sqrt{\frac{2m}{r}(1-v0^2) +v0^2} = \sqrt{\frac{1}{1}(1-(0.5)^2) +(0.5)^2} = \sqrt{1-0.25 + 0.25} = 1<br />

Try your formula:

<br /> <br /> \sqrt {2m \left( {r}^{-1}-{\frac { \left( -r+1 \right) {{\it <br /> v_0}}^{2}}{r}} \right) }<br /> <br />

using m=7, r=2m and v0 =1/2 and you will see where the problem is. (My equation works for any arbitrary value of m.)

Sorry yuiop I am definitely not having my day, must be my jet lag due to moving from China to the USA.

Ok, now I think your formulas look good, I am going to play with them a little. Would it be possible to get a generic "Fermi distance formula"? I am going to take your formulas to get a generic distance formula as well.

Ok, here are the plots for the formulas you presented, and they look good. One thing is very different, the 'dropped from a r value' observers. In my plot the distance decrease is definitely not linear while using your formulas it appears linear. But I think yours are correct and mine were wrong.

In the last graph the stationary observers are obviously not in motion so each point on the plot represents a different observer at a different r value.

 

[pervect]

The paramaterized equations for a general spacelike geodesic will be given by the functions r(s) and t(s) that satisfy

<br /> \frac{dt}{ds} = \frac{C}{1-2m/r}<br />

<br /> \left(\frac{dr}{ds}\right)^2 = 1 + C^2 - 2m/r<br />

where C is an arbitrary constant. To get the fermi-normal distance associated with the worldline of some particular time-like observer, you'll have to choose the correct value of C so that your geodesic is orthogonal to the worldline of your time-like obsever at the point where they cross.

 

[Roshanjha14]

Can anyone suggest me any book to start with general theory of relativity and cosmology?

 

[Roshanjha14]

Can anyone describe me simplest way that what the general theory relativity acutually tell.