Another Schwarzschild question

[snoopies622]

Recently I read that from the perspective of a distant observer
(r \gg r_s =\frac{2GM}{c^2})
the speed of a beam of light moving directly towards the center of a spherical (non-rotating, non-charged) object decreases because if we set ds^2=d\theta ^2 =d\phi ^2 = 0 then
\frac {dr}{dt}=c(1-\frac {r_s}{r}).

I was wondering why this interpretation is favored over the following one: in the Schwarzschild metric, the speed of light stays the same everywhere, but the distance between successive r coordinates increases asympotically as r \rightarrow r_s. This would also be consistent with
\frac {dr}{dt} \rightarrow 0 for light as r \rightarrow r_s.

Does this interpretation fail somehow?

 

[Mentz114]

If both positions explain the observations equally well, we can't say which one is 'actually correct'. And to me they do just that.

 

[George Jones]

Recently I read that from the perspective of a distant observer
(r \gg r_s =\frac{2GM}{c^2})
the speed of a beam of light moving directly towards the center of a spherical (non-rotating, non-charged) object decreases because if we set ds^2=d\theta ^2 =d\phi ^2 = 0 then
\frac {dr}{dt}=c(1-\frac {r_s}{r}).

This is a coordinate speed. Any observer, freely falling or acccelerated, at any r that the light whizzes by measures the speed of light to be c.

I was wondering why this interpretation is favored over the following one: in the Schwarzschild metric, the speed of light stays the same everywhere, but the distance between successive r coordinates increases asympotically as r \rightarrow r_s. This would also be consistent with
\frac {dr}{dt} \rightarrow 0 for light as r \rightarrow r_s.
Does this interpretation fail somehow?

I don't know what "distance" means. I do know what "coordinate speed" and "measured speed" mean.

 

[yuiop]

Recently I read that from the perspective of a distant observer
(r \gg r_s =\frac{2GM}{c^2})
the speed of a beam of light moving directly towards the center of a spherical (non-rotating, non-charged) object decreases because if we set ds^2=d\theta ^2 =d\phi ^2 = 0 then
\frac {dr}{dt}=c(1-\frac {r_s}{r}).

I was wondering why this interpretation is favored over the following one: in the Schwarzschild metric, the speed of light stays the same everywhere, but the distance between successive r coordinates increases asympotically as r \rightarrow r_s. This would also be consistent with
\frac {dr}{dt} \rightarrow 0 for light as r \rightarrow r_s.

Does this interpretation fail somehow?

Hi Snoopies,
As you know there is usually more than one physical interpretation of a given situation in relativity. The speed of light is always c as measured by a local observer and according to an observer at infinity the speed of light is:

c' = \frac {dr}{dt}=c(1-\frac {r_s}{r})

The two observations can be combined into one more general equation:

c' = \frac {dr}{dt}=\frac{c(1- r_s/r)}{(1 - r_s/R)}

where R is the location of the observer and r is the location of the measurement. It is easy to see when R=r the local speed of light c' = c.

At the event horizon the speed of light is zero according to an observer at infinity and c according to a stationary observer located at the event horizon. It is a mute point whether you can actually have a statinary observer at the event horizon. In fact if you look at the general equation for the speed of light that I gave above, the local speed of light at the event horizon is not zero, but 0/0 which means undetermined. Anyway, the event horizon is mathematically tricky. Further down into the black hole below the event horizon things get more interesting. While a local observer (if one could actually exist there) always sees the local speed of a falling photon as positive and always heading from the event horizon to the assumed singularity at the centre, while the observer at infinity see the photon as heading in the opposite direction away from the singularity. Although this by itself does not prove that everything asymptotically "gravitates" towards the event horizon, close inspection of the equations for the acceleration and velocity of particles with rest mass support this view and further suggest there is no mass at the singularity (IMHO).

 

[snoopies622]

Thanks all. I have a follow-up if I may.

Suppose I have a series of ropes of lengths x, x+\Delta x, x+2\Delta x, x+3\Delta x....etc. and I bend each one into a circle and place these circles concentrically around our spherical mass. Since in Schwarzschild coordinates r= C/2 \pi by definition of r, each of these ropes may be regarded as a coordinate marker for r. If the space were flat, the radial distance between each rope would be \Delta x / 2 \pi. Am I correct in my suspicion that in the non-flat case with the massive sphere at their center, the radial distance between successive ropes will increase as we approach (but not reach) the event horizon? If it is necessary to define “distance”, I hastily offer, ‘what one measures by lining meter sticks end-to-end and then counting them’.

 

[yuiop]

Thanks all. I have a follow-up if I may.

Suppose I have a series of ropes of lengths x, x+\Delta x, x+2\Delta x, x+3\Delta x....etc. and I bend each one into a circle and place these circles concentrically around our spherical mass. Since in Schwarzschild coordinates r= C/2 \pi by definition of r, each of these ropes may be regarded as a coordinate marker for r. If the space were flat, the radial distance between each rope would be \Delta x / 2 \pi. Am I correct in my belief that in the non-flat case with the massive sphere at their center, the radial distance between successive ropes will increase as we approach (but not reach) the event horizon? If it is necessary to define “distance”, I hastily offer, ‘what one measures by lining meter sticks end-to-end and then counting them’.

Call the value \Delta x / 2 \pi the coordinate distance.The vertical "distance" as measured using physical meter sticks will be greater than the coordinate distance. The distance as measured by halving the time it takes for a radar signal to travel down to a lower concentric circle and back up again and multiplying by the standard speed of light will be greater than the ruler distance. The distance as measured by halving the time it takes for a radar signal to travel up to a higher concentric circle and back down again and multiplying by the standard speed of light will be less than the ruler distance but still greater than the coordinate distance. The ruler distance might seem a simple and unarguable distance but it gets complicated near an event horizon because it takes an infinite number of meter sticks to extend from the lowest concentric circle to the event horixon. Just for clarity it should be added that the rulers do not length contract horizontally and both the distant and local observers agree on the length of the circumferance of each horizontal circle whether measured by rulers or orbital periods.

 

[George Jones]

The distance as measured by halving the time it takes for a radar signal to travel down to a lower concentric circle and back up again and multiplying by the standard speed of light

For what it's worth, this time is something measured on *one* clock, and is calculated in

https://www.physicsforums.com/showthread.php?p=928277#post928277

 

[snoopies622]

Wow, there's more to this than I thought. Thank you both for such wonderfully detailed answers!

 

[MeJennifer]

both the distant and local observers agree on the length of the circumferance of each horizontal circle whether measured by rulers or orbital periods.

How do you reason that as it is clear that the clock rates of local and distant observers are not identical?

 

[yuiop]

How do you reason that as it is clear that the clock rates of local and distant observers are not identical?

Derivation of the coordinate horizontal velocity c_H of a photon.

Starting with the Schwarzschild metric:

c^2(dtau)^{2}=(1-{r_s}/{r}) c^2(dt)^{2}- (1-{r_s}/{r})^{-1}(dr)^{2}-r^{2} (d\theta)^{2}-r^{2}\sin^{2}(\theta)(d\phi)^{2}

The proper time rate for a photon is zero so dtau=0.
For horizontal velocity dr=0.
By setting d\phi to zero we are left with:

0=(1-{r_s}/{r}) c^2(dt)^{2}-r^{2} (d\theta)^{2}

which rearranges to:

r (d\theta)/dt = c\sqrt{1-r_s/r} = c_H

which is the coordinate tangential velocity of light according to an observer at infinity.

At the photon orbit the speed of light according to a local observer is c and this corresponds to circumference = orbital period*c = t*c

According to the observer at infinity the circumference is proportional to the coordinate orbital period multiplied by the coordinate tangential velocity of light which is

\frac{t}{\sqrt{1-r_s/r}}*c\sqrt{1-r_s/r} = t*c

which is the same as the photon orbit circumference as measured by the local observer at r.

A similar result can be deduced for other radii by assuming an elaborate system of mirrors to guide the photon path. The circumference is always measured to be the same by the observers at infinity and locally using a light signal.

Although I won't prove it here, the same is true for circumferences measured from the orbital period and velocities of satellites with rest mass. Clocks slow down low in the gravity well but so do orbital velocities by the same amount and the end result is that the circumference is unchanged. Orbital circumference can be thought of as a sort of gravitational invariant.

Perhaps you thought I implied orbital periods where the same no matter who measures them? I can see that what I said might be ambiguous. I meant orbital circumference is the same if calculated from orbital period multiplied by the orbital velocity, no matter whether the measurement is done by a local observer or one at infinity and is the same as the circumference of a Snoopy Ring at the same location measured by local rulers, laid end to end horizontally. Sorry for any confusion.

 

[MeJennifer]

Orbital circumference can be thought of as a sort of gravitational invariant.

In fact they can work as a kind of clock counting the number of full orbits!

The catch is, I think, let me know if you disagree, that in certain spacetimes not all observers might agree on when there is one full circumference.

 

[snoopies622]

I wonder if this invariance of circumference is simply a direct consequence of g_{\theta \theta} being the same in the Schwarzschild metric as it is in flat space.

 

[yuiop]

In fact they can work as a kind of clock counting the number of full orbits!

The catch is, I think, let me know if you disagree, that in certain spacetimes not all observers might agree on when there is one full circumference.

Hi Jennifer,

I only have a few minutes, so I will have to come back to this later when I have more time

There are many kinds of spacetimes (or choices of coordinate systems) so I am sure what you are saying is almost cetainly true. Did you have a particular spacetime in mind when you made that statement?

One that comes to mind is the point of view of an observer orbiting the stationary body. Where would they think the start and finish of a orbit is located relative to a test particle at the same radius orbiting in the opposite direction?

There seems to be a difficulty in defining a location in space to return to. How do we put a "marker" there? Do we define a point in space as the location of an object that no acceleration acting on it? That does not seem to work as a body in orbit and a body falling are clearly not stationary from other points of view. Maybe a non inertial observer has a better point of view of a stationary location. For example am observer on a snoopy ring will observer the greatest proper acceleration downwards on an accelerometer, when he has no orbital motion forward or backwards.

One glimmer of light in the turbulent sea of different spacetimes is that it is not true that "Everything is relative". Rotational motion has an absolute nature. A satellite in geosynchronous orbit with the Earth is NOT the same as a stationary body above a stationary Earth although it might appear to be superficially if the observer in geosynchronous orbit declares himself to be stationary.

 

[snoopies622]

..it takes an infinite number of meter sticks to extend from the lowest concentric circle to the event horizon.

And yet someone falling into a black hole reaches the event horizon in finite proper time. How is this possible?

 

[DrGreg]

Schwarzschild's parabola

it takes an infinite number of meter sticks to extend from the lowest concentric circle to the event horixon.

And yet someone falling into a black hole reaches the event horizon in finite proper time. How is this possible?

Kev has got this wrong. It's true that, for radial motion,

\frac {ds}{dr} = \frac {1} {\sqrt{1 - r_s/r}} \rightarrow \infty (1)​

as r \rightarrow r_s. However the ruler distance to the event horizon is the integral

s = \int_{r_s}^r \frac {ds}{dr} dr (2)​

which is finite.

It turns out that, for a constant t coordinate, the curvature of space outside the event horizon can be visualised as extrinsic curvature in a higher dimensional Euclidean space.

Take the equation for the radial metric (when dt = d\theta = d\phi = 0 and r>r_s),

ds^2 = \frac {dr^2} {1 - r_s/r} (3)​

and substitute

Z = 2 \sqrt{r_s} \sqrt{r - r_s} (4)​

It is not hard to show that

ds^2 = dr^2 + dZ^2 (5)​

Note that (4) is the equation of a parabola in (r, Z) coordinates, and (5) is saying that the ruler distance s is just the Euclidean arc length along the parabola. This arc length is finite, despite dZ/dr becoming infinite at r=r_s.

Z is to be thought of as a "phantom" coordinate in a fourth space dimension, with Schwarzschild space (suppressing time) as a three dimensional surface of revolution of the parabola around the Z axis.

There's a diagram of this to be found at the end of http://www.bun.kyoto-u.ac.jp/~suchii/schwarzschild.html .

Note that as the time dimension has been suppressed, this visualisation does not tell the whole story; nor does it work inside the event horizon.

 

[snoopies622]

Thanks, DrGreg. I was wondering about that integral

<br /> \int_{r_s}^r \frac {dr} {\sqrt{1 - r_s/r}} <br />.

I entered it into this anti-derivative finder

http://integrals.wolfram.com/index.jsp

but the result was something that was both more complex than I expected and which diverged at r=r_s so I assumed that the distance was in fact infinite. Your parabola substitution -- however -- looks very interesting to me.

 

[yuiop]

In fact they can work as a kind of clock counting the number of full orbits! ..

Hi jennifer, Nice observation Orbiting satellites do seem to provide a big easy-to-observe natural clock for comparisons. Only trouble is, there are no natural stable orbits below the photon horizon , maybe slightly higher?

And yet someone falling into a black hole reaches the event horizon in finite proper time. How is this possible?

Hi Snoopies,

Infinity is a tricky subject which is paradoxical to many people including me One example of the uncomfortable coexistence of infinite and finite is the logarithmic spiral that has the peculiar property that "Starting at a point P and moving inwards along the spiral, one has to circle the origin infinitely often before reaching it; yet, the total distance covered on this path is finite. " See http://www.nationmaster.com/encyclopedia/Logarithmic-spiral
I am sure there are many other examples. I do not pretend to understand the exact meaning and significance of infinity and I am still coming to terms with it. I may be wrong about there being an infinite number of physical rulers extending down to the event horizon and DrGreg seams to have raised a valid objection (See below). On the other hand a finite "distance" to the event horizon does not necessarily mean it can be reached in a finite time. I put "distance" in quotes because it has many meanings especially in GR.

Kev has got this wrong. It's true that, for radial motion,

\frac {ds}{dr} = \frac {1} {\sqrt{1 - r_s/r}} \rightarrow \infty (1)​

as r \rightarrow r_s. However the ruler distance to the event horizon is the integral

s = \int_{r_s}^r \frac {ds}{dr} dr (2)​

which is finite.

Hi DrGreg,

I think you have raised a good point here and I might learn something from it :) There are however a number of issues that are puzzling me with this interpretation.

First, I wish to understand the physical meaning of s or ds in this context. From the Schwarzschild metric it would appear to be ds = c*dtau where tau is the proper time, but the proper time according to whom? If we are measuring the distance from coordinate radius r1 to r2 is it the proper time according to:

a) A stationary observer located at r1?
b) A stationary observer located at r2?
c) An observer free falling from r2 to r1?
d) The sum of infinitesimal proper time intervals made by a large number of stationary observers spread out from r1 to r2?

I think case (c) can safely ruled out because we are assuming the coordinate time interval between measurements to be zero and that would require the free falling observer to fall with infinite coordinate velocity. I ask, because the proper time varies with location. It may be that the variation of proper time with radius is already taken into account by the integration, in line with case (d) but I just wanted to be sure.

Next, there seems to be a problem with imaginary values popping up when the "ruler distance" is derived from the Schwarzschild metric.

Starting with the simplified Schwarzschild metric (no rotation):

c^2 {dt}^{2} = <br /> \left(1 - \frac{r_s}{r} \right) c^2 dT^2 - \left(1-\frac{r_s}{r}\right)^{-1}{dr^2}

where dt is a proper time interval and dT is coordinate time interval.

By setting dT to zero:

c^2 {dt}^{2} = - \left(1-\frac{r_s}{r}\right)^{-1}{dr^2}

there is a problem when taking the square root of both sides because one side of the equation always ends up as an imaginary number no matter what side the negative sign is placed. Taking the integral of \frac{c dt}{dr} = \sqrt{\left(\frac{-1}{(1-{r_s}/{r})}\right)} yields values for c*dt that are also imaginary for radii greater than the Schwarzschild radius. How is this issue resolved?

It turns out that, for a constant t coordinate, the curvature of space outside the event horizon can be visualised as extrinsic curvature in a higher dimensional Euclidean space.

Take the equation for the radial metric (when dt = d\theta = d\phi = 0 and r>r_s),

ds^2 = \frac {dr^2} {1 - r_s/r} (3)​

and substitute

Z = 2 \sqrt{r_s} \sqrt{r - r_s} (4)​

It is not hard to show that

ds^2 = dr^2 + dZ^2 (5)​

Note that (4) is the equation of a parabola in (r, Z) coordinates, and (5) is saying that the ruler distance s is just the Euclidean arc length along the parabola. This arc length is finite, despite dZ/dr becoming infinite at r=r_s.

Z is to be thought of as a "phantom" coordinate in a fourth space dimension, with Schwarzschild space (suppressing time) as a three dimensional surface of revolution of the parabola around the Z axis.

There's a diagram of this to be found at the end of http://www.bun.kyoto-u.ac.jp/~suchii/schwarzschild.html .

Note that as the time dimension has been suppressed, this visualisation does not tell the whole story; nor does it work inside the event horizon.

Could you show how equation (5) is obtained?

 

[DrGreg]

Thanks, DrGreg. I was wondering about that integral

<br /> \int_{r_s}^r \frac {dr} {\sqrt{1 - r_s/r}} <br />.

I entered it into this anti-derivative finder

http://integrals.wolfram.com/index.jsp

but the result was something that was both more complex than I expected and which diverged at r=r_s so I assumed that the distance was in fact infinite. Your parabola substitution -- however -- looks very interesting to me.

I confess I never worked out the integral. I knew it had to be finite because of the parabola argument.

But, using the website you quoted, the integral is finite. Did you enter it correctly? If you replace r by x and r_s by a, the site gives formula that works perfectly well at x=a.

Integrate[1/Sqrt[1 - a/x], x] == Sqrt[1 - a/x]*x + (a*Log[-a + 2*(1 + Sqrt[1 - a/x])*x])/2

 

[DrGreg]

First, I wish to understand the physical meaning of s or ds in this context. From the Schwarzschild metric it would appear to be ds = c*dtau where tau is the proper time, but the proper time according to whom? If we are measuring the distance from coordinate radius r1 to r2 is it the proper time according to:

a) A stationary observer located at r1?
b) A stationary observer located at r2?
c) An observer free falling from r2 to r1?
d) The sum of infinitesimal proper time intervals made by a large number of stationary observers spread out from r1 to r2?

I was a bit lax in my notation, and this is indeed a bit of a notational grey area.

To keep things simple, let's just consider flat 2-d spacetime (i.e. 1 space + 1 time coord, with no gravity). Different authors will tell you that the metric is given by one of the following equations.

ds^2 = c^2 dt^2 - dx^2 (I)
ds^2 = dt^2 - dx^2 / c^2 (II)
ds^2 = dx^2 - c^2 dt^2 (III)
ds^2 = dx^2 / c^2 - dt^2 (IV)​

(I) and (II) are referred to as a (+---) metric signature, while (III) and (IV) are referred to as a (-+++) signature. When you get more experienced in the subject, you might even mentally switch from one definition to another without explicitly saying so.

What is s? Well you have to measure it along a curve in spacetime, and it depends what sort of curve it is.

If the curve is a timelike curve, i.e. represents the wordline of a particle, then equation (II) gives you the proper time s = \tau experienced by that particle. (By the way, use "\tau" in tex.) Equivalently equation (I) gives you s = c\tau, and rather more uglier, (III) gives you s = ic\tau and (IV) gives you s = i\tau.

On the other hand, if the curve is spacelike and lies within the surface of simultaneity of an observer, then equation (III) gives you the distance s = d along that curve, according to that observer (with appropriate variants for the other metric equations I-IV). In GR this has to be modified slightly to say that the distance is measured using local rulers, i.e. at each point along the curve you use a local observer at that point to make the infinitesimal measurement ds. The curve has to lie parallel to each local observer's surface of simultaneity at that point, in other words, each local observer is stationary relative to the line in space being measured.

(If the curve is null or lightlike, s = 0, and the curve is the worldline of a photon.)

In my previous post, I was implicitly assuming the (III) version of the metric, because that version is more convenient for measuring distance as opposed to time. By putting dt = 0 I was using the metric to measure distance along a spacelike curve rather than time.

(It might have been less confusing if I'd used a different letter e.g. d instead of s to denote ruler distance. Then, using the metric signature (I) that you had used in previous posts, d = is.)

Does that make more sense now?

Could you show how equation (5) is obtained?

From my original equation (4)

dZ = 2 \sqrt{r_s} \cdot \frac{1}{2}(r - r_s)^{-1/2} dr
dZ^2 = \frac {r_s}{r - r_s} dr^2
dr^2 + dZ^2 = \frac{(r-r_s) + r_s}{r - r_s} dr^2
dr^2 + dZ^2 = \frac{dr^2}{1 - r_s/r}​

(It is also possible to work backwards to obtain (4) from (5), which is what I originally did.)

On the other hand a finite "distance" to the event horizon does not necessarily mean it can be reached in a finite time.

Indeed. It depends which "time" you mean. For the Schwarzschild coordinate time t, or the time of any "stationary" observer, it takes an infinite time. But in the traveller's proper time \tau, it takes a finite time. The difference is due to gravitational time dilation (which isn't included in the "Schwarzschild parabola" space-only visualisation).

 

[snoopies622]

...using the website you quoted, the integral is finite. Did you enter it correctly?

I guess not. I did not know to use "a" to represent a constant, so I just tried different integers instead and the results were always in agreement with each other. For some reason my way produces a different anti-derivative which has \sqrt{\frac{(r-r_s)}{r}}\sqrt{r} for the denominator. I will accept that yours is correct, but for now I don't know why they differ.

 

[yuiop]

I was a bit lax in my notation, and this is indeed a bit of a notational grey area.

To keep things simple, let's just consider flat 2-d spacetime (i.e. 1 space + 1 time coord, with no gravity). Different authors will tell you that the metric is given by one of the following equations.

ds^2 = c^2 dt^2 - dx^2 (I)
ds^2 = dt^2 - dx^2 / c^2 (II)
ds^2 = dx^2 - c^2 dt^2 (III)
ds^2 = dx^2 / c^2 - dt^2 (IV)​

(I) and (II) are referred to as a (+---) metric signature, while (III) and (IV) are referred to as a (-+++) signature. When you get more experienced in the subject, you might even mentally switch from one definition to another without explicitly saying so.

Hi DrGreg,
I was not being critical of your notation and I realize that you are just using the conventional notation as used in textbooks. It is just that ds is indeed a grey area and can mean many different things. It can be seen from the equations you listed that ds can have different units depending on the context and that can be confusing for beginners. Because of the many faces of ds it can even appear to ignore the rules of algebra because when switching from the +--- to -+++ signature the sign of ds does not change which is only true in algebra if ds has the value zero.

IMHO ds should be explicitly stated in terms of other variables to avoid confusion. For example your list could be restated as:

c^2 d\tau^2 = c^2 dt^2 - dx^2 (I)
d\tau^2 = dt^2 - dx^2 / c^2 (II)
-(c^2 d\tau^2) = dx^2 - c^2 dt^2 (III)
-d\tau^2 = dx^2 / c^2 - dt^2 (IV)​

That way it becomes clear that expressions (I) and (III) have units of proper distance and expressions (II) and (IV) have units of proper time. It also becomes clear that the rules of algebra still apply and that changing the metric signature of the right hand side also changes the sign of the left hand side (ds).

(By the way, use "\tau" in tex.)

Sorry for not using tau in the tex. To me it looks too much like lower case R which can also be confusing for beginners who might not be expecting it.

In my previous post, I was implicitly assuming the (III) version of the metric, because that version is more convenient for measuring distance as opposed to time. By putting dt = 0 I was using the metric to measure distance along a spacelike curve rather than time.

OK, starting with version (III) of the radial Schwarzschild metric as you suggest:


-(c^2 {d\tau}^2) = -\left(1 - \frac{r_s}{r} \right) c^2 dt^2 + \frac{dr^2}{\left(1-\frac{r_s}{r}\right)}

and setting dt=0

-(c^2 {d\tau}^2) = \frac{dr^2}{\left(1-\frac{r_s}{r}\right)} (Eq 6)

and taking the square root of both sides and dividing by dr

\frac{\sqrt{-(c^2 {d\tau}^2)}}{dr} = \frac{1}{\sqrt{\left(1-\frac{r_s}{r}\right)}} (Eq 7)

it seems that the expression on the left hand side is an imaginary quantity while the expression on the right hand side is real. This is what was confusing me. I suddenly realized half way through writing this how that apparent mathematical contradiction is solved. The quantity dtau is the imaginary proper time of a particle traveling the distance dr in zero time. In other words it is the motion of an imaginary particle. This can be seen by rearranging (Eq 6) to get:

d\tau = \sqrt{\frac{-dr^2}{c^2\left(1-\frac{r_s}{r}\right)}} (Eq 8)

Eq 8 shows that dtau is imaginary in this case. The square of a imaginary number is a real negative number and so the expression \sqrt{-(c^2 {d\tau}^2)} yields a real value when dtau is imaginary.

I am still a little confused as to how to prove that the value measured by the motion of a virtual particle moving with infinite coordinate velocity, is really the physical ruler distance. Why does this differ from the coordinate distance dr and why is dr time dependent? Intuition suggests the distance dr should not vary with time and should be the same even if the distance is not measured simultaneously at both ends.

From my original equation (4)

dr^2 + dZ^2 = \frac{dr^2}{1 - r_s/r}​

(It is also possible to work backwards to obtain (4) from (5), which is what I originally did.)

Thanks for taking the time to show how you derived it. I now understand how the equation was obtained. However, I am still not certain as to what it actually represents physically. The equations also seems to suggest that there can be no physical static rulers below the event horizon, and that the imaginary proper time of a virtual particle becomes real when it falls below the event horizon and also the opposite: that the proper time of a real particle becomes imaginary below the event horizon.

Either way, thanks for showing me a new tool with which to probe the spacetime in and around the event horizon ;)

 

[yuiop]

I guess not. I did not know to use "a" to represent a constant, so I just tried different integers instead and the results were always in agreement with each other. For some reason my way produces a different anti-derivative which has \sqrt{\frac{(r-r_s)}{r}}\sqrt{r} for the denominator. I will accept that yours is correct, but for now I don't know why they differ.

Hi Snoopies,

I find the quickmath website http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=calculus&s2=integrate&s3=advanced useful for doing online calculus and it has the benifit of allowing you to use variable names of your own choice.

I get the integral of

s = \int_{r_1}^{r_2} \left(\frac {1} {\sqrt{1 - r_s/r}}\right) dr

to evaluate to:

s = \sqrt{r_2(r_2-r_s)}- \sqrt{r_1(r_1-r_s)} + r_s Log\left(\sqrt{r_2} + \sqrt{(r_2-r_s)}\right) - r_s Log\left(\sqrt{r_1} + \sqrt{(r_1-r_s)}\right)

For r_1 = r_s = 1 and using r for r_2 the expression simplifes to:

s = \sqrt{r(r-1)} + Log\left(\sqrt{r} + \sqrt{(r-1)}\right)

where Log is the natural logarithm.

 

[snoopies622]

Thanks for the Quick Math reference.

I must be using it incorrectly. I just asked it for

<br /> <br /> \int \frac{dx}{\sqrt{1-\frac{a}{x}}}<br /> <br />

and it gave me another expression that diverged at x=a. This one had an arctangent in it instead of a natural logarithm. Perhaps I should read the directions...

 

[yuiop]

Thanks for the Quick Math reference.

I must be using it incorrectly. I just asked it for

<br /> <br /> \int \frac{dx}{\sqrt{1-\frac{a}{x}}}<br /> <br />

and it gave me another expression that diverged at x=a. This one had an arctangent in it instead of a natural logarithm. Perhaps I should read the directions...

Hi Snoopies,

I checked out the proplem you are having with the quickmath website. It turns out if you enter 1/sqrt(1-a/r) as the expression to be integrated with respect to r, you get arctangent result and if you enter 1/sqrt(1-s/r) as the expression to be integrated with respect to r, you get the logarithm answer.

The quickmath help page says only the letters i and e are reserved variables so it does not make sense and I have reported the problem as a bug to the website owner. Sorry about that. It was pure luck that I chose s rather than a for the constant.

 

[snoopies622]

Thanks, Kev. I'm getting different answers using different variables and styles (like using "^(-1/2)" instead of "1/sqrt(..)" ) too, but they always diverge at r=r_s. Strange.

 

[George Jones]

Thanks, Kev. I'm getting different answers using different variables and styles (like using "^(-1/2)" instead of "1/sqrt(..)" ) too, but they always diverge at r=r_s. Strange.

Although they look different, the answers are the same.

 

[Mentz114]

Kev,
I used the integration on the quickmath site and got this - see pic.

It has terms that will be imaginary for x>m. Is the program telling me there is no real indefinate integral ?

M

 

[yuiop]

Kev,
I used the integration on the quickmath site and got this - see pic.

It has terms that will be imaginary for x>m. Is the program telling me there is no real indefinate integral ?

M

Nope, its a bug or "feature" in quickmath that gives different presentations which as George points out are the same numerically, but look different. (See the last 3 or 4 posts about the "bug")

To get the log version using a constant that higher in the alphabet than the variable that the integration is with respect to seems to help

For example you could integrate 1/sqrt(1-m/a) with respect to a to get:


\frac{\sqrt{a}*(a - m) +\sqrt{a - m}*m*LN(\sqrt{a} + \sqrt{a - m})}{\sqrt{a}*\sqrt{1 - m/a}}

If you try to use that expression in a spreadsheet it will give up with a division by zero error. By "cheating a little" the expression can be converted to:

\frac{\sqrt{a}*(a - m) +\sqrt{a - m}*m*LN(\sqrt{a} + \sqrt{a - m})}{\sqrt{a - m}}

which makes it obvious where the spreadsheet gets the division by zero error when a=m.

Now divide top and bottom by sqrt(a-m) to get:

\sqrt{a}*\sqrt{a - m} +m*LN(\sqrt{a} + \sqrt{a - m})

My methods may be a little dirty, but the end result is the same numerically as that given by DrGreg and a spreadsheet can handle it ;)

Quickmath does not always come up with the best simplifications and you you have to tune it by hand sometimes. :(

LN is the natural log in the form that most spreadsheets would expect it to be entered.

Just in case you are interested the arctangent is related to the natural logarithm by:

tan^{-1}(z) = i*LN(1-i*z)/2 - i*LN(1+i*z)/2

where i is the square root of -1

which just goes to show the sort of imaginary quantities and infinities that lurk in the equations.

 

[snoopies622]

So when evaluating definite integrals it's OK to "cancel out" zeros? (I realize that this question belongs in the "calculus & analysis" section of PF, but here the context is already set up.)

 

[yuiop]

So when evaluating definite integrals it's OK to "cancel out" zeros? (I realize that this question belongs in the "calculus & analysis" section of PF, but here the context is already set up.)

As far as I know it is OK to cancel out a numerator and denominator that contain exactly the same variables and are of the same form, even if at some point they seem to take on the value 0/0. Be wary of numerators and denominators that contain only constants because if they have the value 0/0 then the result is always indeterminate.

For more complex situations where the expressions for the numerator and denominator differ in form but apppear to give the answer 0/0 when a limit is taken, then L'Hospital's rule can be applied to find out if 0/0 is has a determinate value.

I am no maths expert and if you require an authoritive answer you might be well advised to consult the experts in the calculus and analysis section if one the experts here does not jump in first ;)

 

[yuiop]

Confession time. When I said I was "cheating a little" by substituting

\sqrt{(a - m)} for \sqrt{a}*\sqrt{(1 - m/a)}

the implication that \sqrt{(a - m)} = \sqrt{a}*\sqrt{(1 - m/a)} is not strictly mathematically true because they differ when a is negative. Fortunately gravitational radii are usually taken to be positive, so a is always positive in the context used here and there shouldn't be problem. The indefinite integral I posted earlier should be valid for all radii >=0 but the results are imaginary complex numbers below the event horizon when a < m.

 

[Mentz114]

Kev,

Nope, its a bug or "feature" in quickmath that gives different presentations which as George points out are the same numerically, but look different. (See the last 3 or 4 posts about the "bug")
To get the log version using a constant that higher in the alphabet than the variable that the integration is with respect to seems to help

OK, I did that and get the log solution which is real for a > m. Whoopee.

[edit]The expression below (writing r for a ) assuming r>m>0 can be simplified to

log(\sqrt{r} + \sqrt{r - m}) + r\sqrt{1 - \frac{m}{r}} which is most satisfactory.

I just noticed you've done all this...

M

 

[yuiop]

Hi,

I thought it might be interesting to summerise and compare the ruler distance given by DrGreg with radar distance given by George. Please note I am not claiming that that there is any contradiction or that DrGreg or George made any mistakes. The two measurements are different.

The ruler distance is:

\sqrt{r2*(r2-rS)} - \sqrt{r1*(r1-rS)}<br /> + rS*\left(LN\left(\sqrt{r2} + \sqrt{(r2-rS)}\right)- LN\left(\sqrt{r1} + \sqrt{(r1-rS)}\right)\right)

which is in terms of the definite integral where rS is the event horizon, R1 is the lower coordinate radius and r2 is the higher coordinate radius.

The coordinate radar distance is

c*(rS*LN\left( \frac{r2-rS}{r1-rS}\right) +r2 - r1)

which is the distance measured by halving the time it takes a photon to travel from r1 to r2 and back again and multiplying by c. This distance is the same in either direction in coordinate terms and the distance goes to infinity when r1=rS for any r2>rS.

The radar distance according to a local observer is

c*(rS*LN\left( \frac{r2-rS}{r1-rS}\right) +r2 - r1)*\sqrt{1-\frac{rS}{rX}}

where rX can be r1 or r2 according to which end the observer is located. The observers located at either end of the ruler distance r2-r1 disagree with each other about the radar distance measured by the light travel time.

This allows us to put things into perspective. Imagine an observer attaches a mirror to one end of a meter ruler and goes down close to a black hole event horizon so that the end of the ruler with the attached mirror is just touching the event horizon. The ruler and mirror would have to be massless in order to actually physically do this ;) Now he sends a photon down to the mirror and times how long it takes to come back. Normally in flat space the photon would travel the one meter distance in about 1/(299792458) of a second. Down here, just one physical meter away from the event horizon it takes an infinite amount of time for the light to travel that one meter as measured by the clock of the stationary non inertial observer one meter away from the event horizon.

I wonder if DrGreg or George can provide us with an equation for the time measured by the clock of an observer falling from r2 to r1 and what she would consider the distance r2-r1 to be? For simplicity, consider only the motion of a observer falling from infinity with an inital velocity of zero.

 

[snoopies622]

I was just trying to derive this

c&#039; = \frac {dr}{dt}=\frac{c(1- r_s/r)}{(1 - r_s/R)}

from entry #4 and I realized that I have a more fundamental question: do all observers in this static, spherically-symmetric gravitational field agree on the r coordinate of a particular location, regardless of their own location and state of motion?

 

[DrGreg]

I am still a little confused as to how to prove that the value measured by the motion of a virtual particle moving with infinite coordinate velocity, is really the physical ruler distance. Why does this differ from the coordinate distance dr and why is dr time dependent? Intuition suggests the distance dr should not vary with time and should be the same even if the distance is not measured simultaneously at both ends.

I think you still haven't grasped he point about what ds² means in all cases. Just think again, in flat 2D spacetime when ds^2 = dx^2 - c^2 dt^2, what it means when dt=0. In that case, s is simply distance, s=x, there's no need to think about virtual particles or imaginary times. Then realize that ds is invariant, its value does not change when you change coordinates, e.g. apply a Lorentz transform, so s still represents distance (when ds² > 0, with this sign convention) even when that distance is being measured by someone else other than the observer associated with your choice of coordinates.
You might like to consult some textbooks or online sites for the meaning of ds² (in SR, never mind GR).

To answer your question above, dr does not vary with time. r represents distance according to an observer "at infinity", but for some other "stationary" observer, ds represents the local distance when dt=0. (On the other hand, when dr = d\theta = d\phi = 0, then d\tau = \sqrt{-ds^2/c^2} represents local (i.e. proper) time.)

To summarise (using my metric signature III for the sake of argument -- adapt for other signatures):

if ds² > 0, you have a spacelike curve (by definition) and ds represents distance
if ds² < 0, you have a timelike curve (by definition) and d\tau = \sqrt{-ds^2/c^2} represents proper time
if ds² = 0, you have a lightlike (a.k.a. null) curve (by definition), the worldline of a photon

(Note: I've started a separate thread HERE about the ambiguity of the "ds" notation.)

Thanks for taking the time to show how you derived it. I now understand how the equation was obtained. However, I am still not certain as to what it actually represents physically.

Z doesn't represent anything physically. It's a mathematical trick to help you visualise the distortion of space, and why local distance s is not the same as coordinate "distance" r.

Referring to the http://www.bun.kyoto-u.ac.jp/~suchii/schwarzschild.html , imagine a 2D slice of space (suppressing t and \phi) not as a flat horizontal plane but shaped like the end of a vertically-upwards trumpet, with parabolic vertical cross-section. r represents horizontal radius and Z represents vertical height. Ruler distance along any curve should then be measured within the curved surface of the trumpet. The closer you get the event horizon, where the surface becomes vertical, the larger the discrepancy between the horizontal r distance and the curved s distance (denoted \sigma in the diagram).

Note, however, there is no distortion in the \theta direction -- the circumference of any circle around the centre is still 2 \pi r.

The equations also seems to suggest that there can be no physical static rulers below the event horizon,

Yes, nothing can be static below the event horizon, it has to fall inwards, so "static ruler distance" makes no sense.

 

[DrGreg]

Proper time along a geodesic

I wonder if DrGreg or George can provide us with an equation for the time measured by the clock of an observer falling from r2 to r1 and what she would consider the distance r2-r1 to be? For simplicity, consider only the motion of a observer falling from infinity with an inital velocity of zero.

I don't claim to be an expert in this subject.

According to Woodhouse⁽¹⁾⁽²⁾, the equations of free fall (i.e. geodesics) in the Schwarzschild metric are:

E = \left(1 - \frac{r_s}{r} \right) \frac{dt}{d\tau} = \mbox{constant} ...(1)
J = r^2 \sin^2 \theta \frac{d\phi}{d\tau} = \mbox{constant} ...(2)​

together with the definition of d\tau via the metric. E is energy and J is angular momentum, and both are determined by the initial conditions of motion.

The above is from a reputable textbook so must be true. What follows is my own working, so might not.

In the case of radial motion, d\theta/d\tau = d\phi/d\tau = 0, and the metric simplifies to

\left(1 - \frac{r_s}{r} \right) \left(\frac{dt}{d\tau}\right)^2 - \frac{1}{1 - r_s/r}\left(\frac{dr}{d\tau}\right)^2 = 1 ...(3)​

Eliminating dt/d\tau between (1) and (3) gives

\frac {E^2 - (dr/d\tau)^2}{1 - r_s/r} = 1 ...(4)
\left(\frac{dr}{d\tau}\right)^2 = E^2 - 1 + \frac{r_s}{r} ...(5)
\int_{r_1}^{r_2}\frac{dr}{\sqrt{E^2 - 1 + r_s/r}} = \tau ...(6)​

Feed that into your favourite online integrator. (Good luck, I had problems with one of sites I tried! )

If the particle is released from "rest" (dr/d\tau = 0) at radius r₃, then (5) tells you the value of E, and so (6) becomes

\int_{r_1}^{r_2}\frac{dr}{\sqrt{r_s/r - r_s/r_3}} = \tau ...(7)​

In particular if r_3 = \infty, E = 1 (which makes the integral easy enough to do without computer assistance).

Coordinate time t can be calculated by combining (1) and (5):

t = \int \frac{dt}{d\tau} d\tau = \int \frac{dt/d\tau}{dr/d\tau} dr ...(8)​

As for the distance, the simple answer is zero! The particle passes through both events so the distance between events in the particle's frame is zero. But I guess what you really mean is, what is the distance between two hovering buoys at the start and end points? Off the top of my head, I don't know, I'll have to think about that.

References

1. Woodhouse, N.M.J. (2007), General Relativity, Springer, London, ISBN 978-1-84628-486-1, pages 100, 107, 124.

2. Woodhouse, N.M.J. (2003), http://people.maths.ox.ac.uk/~nwoodh/gr/index.html - online lecture notes on which the book (1) was closely based, pages 54-55 (Lecture 12), p59 (L13), p73 (L15).

Online Integration results

Integrate[1/Sqrt[a^2 - 1 + (2*m)/x], x] ==
(Sqrt[-1 + a^2 + (2*m)/x]*x)/(-1 + a^2) - (m*Log[m + (-1 + a^2 + Sqrt[-1 + a^2]* Sqrt[-1 + a^2 + (2*m)/x])*x])/(-1 + a^2)^(3/2)

Integrate[1/Sqrt[(-2*m)/r3 + (2*m)/x], x] ==
-(r3*(2*Sqrt[m*(-r3^(-1) + x^(-1))]*x + Sqrt[m]*Sqrt[r3]*ArcTan[ (Sqrt[r3]*Sqrt[m*(-r3^(-1) + x^(-1))]* (-r3 + 2*x))/(2*Sqrt[m]*(-r3 + x))]))/ (2*Sqrt[2]*m)

 

[stevebd1]

As you know there is usually more than one physical interpretation of a given situation in relativity. The speed of light is always c as measured by a local observer and according to an observer at infinity the speed of light is:

c&#039; = \frac {dr}{dt}=c(1-\frac {r_s}{r})

The two observations can be combined into one more general equation:

c&#039; = \frac {dr}{dt}=\frac{c(1- r_s/r)}{(1 - r_s/R)}

where R is the location of the observer and r is the location of the measurement. It is easy to see when R=r the local speed of light c' = c.

While gravitational time dilation is expressed as -

t&#039; = t \sqrt{1-r_s/r}

and gravitational redshift is expressed as -

z=\frac{1}{\sqrt{1-r_s/r}}-1

Referring to the quote above from post #4, how come the 2 parts of the equation for velocity of light observed from infinity and locally, that relate to Schwarzschild metric, aren't square rooted to match time dilation and redshift (i.e. as below)?

c&#039; =c\sqrt{(1-\frac {r_s}{r})}

c&#039; =\frac{c\sqrt{(1- r_s/r)}}{\sqrt({1 - r_s/R)}}

In post #10, this seems to have been taken into account-

The proper time rate for a photon is zero so dtau=0.
For horizontal velocity dr=0.
By setting d\phi to zero we are left with:

0=(1-{r_s}/{r}) c^2(dt)^{2}-r^{2} (d\theta)^{2}

which rearranges to:

r (d\theta)/dt = c\sqrt{1-r_s/r} = c_H

which is the coordinate tangential velocity of light according to an observer at infinity.

Is there a difference between coordinate tangential velocity of light and simply velocity of light?

Great thread by the way

 

[yuiop]

Is there a difference between coordinate tangential velocity of light and simply velocity of light C?

Great thread by the way

Hi Steve,

"coordinate tangential velocity of light" is just my long winded way of saying the horizontal speed of light according to an observer at infinity which I said equates to:

c_{h} = c_{o}\sqrt{1-r_s/r}

where c_{o} is "simply velocity of light" as measured locally in a vacuum when there is no gravitational field present or in this case the speed of light as it measured locally in the gravitational field.

The horizontal speed of light according to the observer at infinity can be calculated from the gravitational time dilation equation as:

c_{h} = \frac{dx_{h} }{dt } = \frac{dx_{h} &#039;}{(dt &#039; / \sqrt{1-r_s/r})} = c_{o}\sqrt{1-r_s/r}

I also said the vertical coordinate speed of light is:

c_{v} = c_{o} (1- r_s/r)

which does not seem to agree with the gravitational time dilation factor. This is because the length contraction has not been taken into account. While local horizontal rulers do not length contract, local vertical rulers do and the equation is:

dx_{v} &#039; = \frac{dx_{v}}{ \sqrt{1-r_s/r}}

The vertical speed of light according to the observer at infinity can be calculated from the gravitational time dilation equation in the same way as:

c_{v} = \frac{dx_{v} }{dt } = \frac{(dx&#039;_{v} \sqrt{1-r_s/r} )}{(dt&#039; / \sqrt{1-r_s/r})} = c_{o}(1-r_s/r)

The local observer always measures the speed of light as 299,792,458 meters/second in a vacuum.

Note that only the component of length that is vertical is subject to gravitational length contraction which has an analogy in SR. In SR, only the component of length parallel to the motion, contracts.

 

[yuiop]

I was just trying to derive this

c&#039; = \frac {dr}{dt}=\frac{c(1- r_s/r)}{(1 - r_s/R)}

from entry #4 and I realized that I have a more fundamental question: do all observers in this static, spherically-symmetric gravitational field agree on the r coordinate of a particular location, regardless of their own location and state of motion?

Hi Snoopies,

When the observer is at R = infinity the above equation reduces to the more familiar

c&#039; = \frac {dr}{dt}=c(1- r_s/r)

As for the agreement on r by all observers the answer is generally no.

It all depends on what methods you use to make the measurements and how the results are interpreted. It is true that if a hovering local observer measures the velocity and orbital period of a local satellite at his location and calculates the circumference of the orbit and divides by 2pi he will get a value for r that agrees with the coordinate value for r. Because there is no horizontal length contraction he will get a similar result if he physically measures a ring co-located with the orbit using rulers. An observer orbiting at the same radius would get a different value for the circumference and possibly for r depending on how he interprets the results. If the orbiting observer considers himself to be stationary there will be no apparent attraction to the main gravitational body and he would have a very strange view of how physics works because from his point of view gravity appears to be working on the hovering observer who appears to be orbiting but has no effect on himself. He will also discover that orbital velocity is directional. If he launches a test satellite in the opposite direction to the the apparently orbiting hovering observer with the same speed he will see it spiral off into space. If he launches another test satellite in the same direction as the hovering "orbiting" observer but with the twice the speed he will see that it too stays in stable orbit. In other word he has a very asymmetrical view of the gravity and how the universe works if he insists he is stationary but he is perfectly entitled to his point of view according to relativity.

If observers measure r using physical vertical rulers they will get a different measurement from the coordinate radius and if the use light radar signals to measure distances they will get yet another measurement which will vary again depending on where the observer is. However, all observers could agree (if the desire and cooperation was there) to convert all measurements to standard measurements that they could plot on one universal map so that they can meaningfully compare measurements. That map would be called the Schwarzschild metric. That would require the orbiting observer to agree he is in fact orbiting and not stationary, but if he comes around to that point of view he would see that the physics of the universe is lot simpler and more symmetrical than he imagined when he insisted he was stationary.