You are both right. The clocks will keep time with each other, the observers will grow old and die, entropy rules etc.
Nice one : 'I tick therefore i am?' [ +10 points ]
As for 1000 years : I'm just emphasising that when we say 'observer' we mean 'measurement system'. Humans don't have to be there at the time. We can look at artifacts - things left behind - of events and still deduce what went on eg. the clocks still agree. Indeed processes have occurred in the universe long before humans either turned up or started to make clocks. What are clocks but human readable regularities in some conveniently handled physical system ? Ditto for length measuring devices. In our minds then clocks and rulers are useful pictures/icons for 'what is happening to time?' and 'what is happening to distance?'. When we test a prediction of GR we still must use actual clocks and rulers ( LIGO being the best ever made by humans so far ) but we may still think of them for discussion in a virtual sense. The absolutes here are the speed of light ( c ) and Newton's constant ( G ) in this universe. If you want different values then go to another reality. In The Relativities we relate or compare numbers obtained from different measurement systems.
We are in a scenario which is spherically symmetric ie. by assumption this is Karl Schwarzschild's analysis. This means that you can arrange a series of concentric shells ( centred around the black hole ) and each shell will have the same two-ness about it's surface. If you like you might for a given shell mark out a latitude/longitude grid upon it. You can talk of 'areas' ( measured in square degrees ) on the sphere bounded by angles eg. the area within 53 to 54 degrees latitude and 107 to 108 degrees longitude.
Questions 2 : Take a particular shell well outside of the event horizon. We cannot ( even in theory ) go from a point on that shell straight through to the centre of the black hole and out to a point on the shell on the other side*.
Given that we can't measure the diameter/radius in that traditional sense, how would we distinguish/measure the 'width' of a shell ?
{ HINT : LowGuy continues to avoid his contract. What a wimp. We paid him in advance! But we have managed to convince him to maybe scoot around while going no closer to the hole as HighGal. We may get some use out of him before we kill him .... :-) }
Cheers, Mike.
* This is code for a traverse using a 'proper length' ie. a ruler carried by an observer who measures along some line by placing 'end-to-end' and keeping 'straight'. By keeping track of how many end-to-ends there were, we could calculate the 'proper distance' from the center of the sphere to its surface.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
What are clocks but human readable regularities in some conveniently handled physical system ? Ditto for length measuring devices.
I wondered if clocks and rulers are equivalent for each observer. The speed of light is a constant, so we only need really good clock to measure a distance? [*]
Quote:
Questions 2 : Given that we can't measure the diameter/radius in that traditional sense, how would we distinguish/measure the 'width' of a shell ?
I feel we had a hint in the "square degrees" so i'm going to say tile it, and count the tiles. We have a survived a thousand years so tiling a black hole would seem reasonable to keep with the bathroom meme. Of course there is some mathematics involve pi and square roots, but these are trivial details for HighGal.
I wondered if clocks and rulers are equivalent for each observer.
Unto themselves, yes. To be exact that is an a priori assumption. Or if you like it would be a pretty perverse universe otherwise and what would we then do ? There is more depth to that but not fruitful for us at present ( cue metaphysics ).
Quote:
The speed of light is a constant, so we only need really good clock to measure a distance? [*]
Bravo ! Indeed ! One way of 'calibrating' one's thinking is to simply define the speed of light to have the value of 1 ( one ). One what ? One light-speed of course ! There's a couple of good reasons :
- it's easier to draw diagrams with 45 degrees as the light world lines. As opposed to the angle whose tangent is 1/(3 * 10^8) ! :-)
- it's more natural in that you tend to think of matters without reference to human-centric concerns. So going cosmic is less of a jolt.
- you don't have to write the symbol c all the time in equations.
The ( slight ) downside is that when finally emitting a number for real world measurement you have to convert to metres, seconds, parsecs, years, Earth-Sun distances, heartbeats or whatever. Time especially is traditionally measured in a manner that only truly makes sense in this solar system. We actually do alot of this type of calculation daily. We don't realise though eg.
Q. How far is it to Lilydale ?
A. About 20 minutes.
( the underlying transport speed is assumed as per local context ). As you have grasped beautifully, all ( inertial, free space, yada yada ... ) observers when measuring the speed of a specific photon will agree.
So with c = 1 all distances become 'that which takes light to travel across' eg. a light-second of distance ( aka 300,000 kilometres ). You can even flip that over and say things like a 'foot of time' ( aka about a nanosecond ). This manner of re-basing units also flows through ( with other suitable definitions eg. for electric charge ) to mass, energy etc.
For example : the Lorentz analysis ( SR ) has this factor, commonly called 'gamma', which depends on the relative velocity b/w frames
gamma(v) = 1/{SQRT[1 - v^2/c^2]}
and hence simpler to write using SQRT[1 - v^2]. Just convert back to your preferred velocity units when you're done eg. snail-shell-diameters per fortnight ! :-)
Back To The Chase
Quote:
I feel we had a hint in the "square degrees" so i'm going to say tile it, and count the tiles. We have a survived a thousand years so tiling a black hole would seem reasonable to keep with the bathroom meme. Of course there is some mathematics involve pi and square roots, but these are trivial details for HighGal.
Yep. Measure the area of the shell. Divide by four times PI and take the square root. Or you could simply mark and measure a circumference and divide that by two times PI. This radius value is deemed to be that via the measurements made and the subsequent calculation.
Now that is all well and good for a single shell. It avoids death by black hole too. This is Karl's approach and works famously while you are outside of the horizon. How does that help ? Let us think of what happens when there is no black hole there ( slide sideways a few light years to 'empty space' ) :
- do a given shell as above and get the radius. Call it r_1 to be definite.
- move inwards by one metre all over the shell. By definition we have a concentric shell that is slightly smaller. Get the radius, via the same method(s) that never take you to the centre, and call it r_2.
- what is the relationship b/w r_1 and r_2 eg. r_1 minus r_2 ?
- you say : one metre you fool Mike ! What else would it be ? :-)
Therein lies the rub. We've spent a weekend away comparing shells out in the Boondocks. Back to the Black Hole District. Think carefully now :
- two concentric shells as before. ( in fact you ought be objecting to the use of the word concentric by now, but plow on for the moment ).
- let the outer of the two shells have circumference c_1.
- construct a smaller/inner shell such that it's circumference is c_2, but ( careful now ) in a fashion* that makes
c_1 - c_2 = 2 * PI * one metre
Questions 3.
(a) What is the relationship b/w r_1 and r_2 eg. r_1 minus r_2 ?
(b) If I place a one metre ruler, pointing directly inwards towards the black hole with one end just at the outer shell, where will the other end of the ruler be with respect to the inner shell ?
{ HINT : options are
- exactly at the inner shell.
- closer in to the black hole than the inner shell.
- the inner shell is closer to the black hole. }
Quote:
[*] I tick therefore i rule?
Another nice one. Snappy. Flows well, good cadence .... + 15 points. :-)
Cheers, Mike.
* You could have one of those retractable measuring tapes. First wind it out to c_1 then wind a bit in to c_1 - 2 * PI * one metre.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Q. How far is it to Lilydale ?
A. About 20 minutes.
Q. Oh that long?
A. Yeah, it depends on the time.
Quote:
Back To The Chase
- what is the relationship b/w r_1 and r_2
Consenting spheres.
Quote:
Questions 3.
(a) What is the relationship b/w r_1 and r_2 eg. r_1 minus r_2 ?
(b) If I place a one metre ruler, pointing directly inwards towards the black hole with one end just at the outer shell, where will the other end of the ruler be with respect to the inner shell ?
OK i'm stuck without applying some extra knowledge [*] and i'm going to make a guess. I know light red-shifts as it climbs away from a gravity well so assuming HighGal uses the 1m wavelength one tick ruler to measure stuff.
It is going to bluer pointing in. So my initial answer is the ruler will be shorten pointing inwards. So as HighGal measure it, it's more than 1m.
I'm somewhat concerned LowGuy get out his 1m wavelength ruler and it get red-shifted upwards so he thinks the gap is less than a metre. [**]
[*] that this black hole has some gravitational effect on lengths and times.
[**] HighGal points out that LowGuy has a longer tick. *cough*
Well ( using the metric of Karl S ) for a given calculated radial difference the 'actual' spacing b/w the shells is smaller than you find in flat space. Thus for any r_1 - r_2 as deduced by circumferences/areas then the ( proper ) radial distance is larger than that. So the one metre ruler, as placed & oriented in my question, will go further in than the innermost shell. This is one way of expressing the spatial distortion from flat Euclidean ( mass/energy ) in free space. Counter-intuitive, no ?
Another way to think of this is that for a given ingress radially you will pass relatively more shells that are uniformly spaced ( as per their circumference differences ). This is hard to think of !!
Yet another way to say it. Think of the volume ( calculated as per Euclid ) contained b/w two shells. A given volume will lie b/w two shells that are closer together than in flat space. Or a given radial step inwards contains more space. In this sense black holes 'compact' space.
{ Compare this with the case that the metre ruler falls short in my question's construction : it would imply that space has expanded ( has less volume ), which is another scenario entirely. }
{ As for :
Quote:
What is the relationship b/w r_1 and r_2 eg. r_1 minus r_2 ?
This will be one metre. I was just restating the prior discussion slightly differently. To see if you were paying attention .... :-) }
It is perfectly normal to rail against this because it doesn't 'make sense'. Alas it is true, in the sense that when used to predict measurable outcomes, that the Schwarzschild approach works beautifully. The Universe does really work that way. We have the cognitive disadvantage that we have all grown up in a region where that approach is indistinguishable from Euclid ( until recent history ). So the Universe is fine and consistent, but off-Earth we need to adjust our thinking to fit the Cosmic Clockworks. To stop us mere mortals getting confused we have to stick with (a) things that are measurable and (b) be clear about what comparisons are being made. Karl's approach was brilliant in that it separated out the difficult bit ( the special radial character ) from the easy bit ( the spherical symmetry unchanged from free space ).
Notably the language must refer to the approach used, so one cannot be as free as you would normally be in everyday conversation with word usage. This takes some rigour. We don't have a set of objects that are immediately referable on a 'normal' basis say, like an apple could be used to center our thoughts on apple-ness when we talk about it. Intellectually tricky ! :-)
So for :
Quote:
i'm stuck without applying some extra knowledge
that extra knowledge is that Karl's approach works. If you like black holes squash things ! :-)
A Closer Look With Karl
There is a term that recurs in Karl's calculations for the static ( structure not varying with time ) non-rotating black hole case in GR :
2 * G * M / c^2
G and c we know and are constant. The number 2 you also are familiar with from primary school and there's no moves afoot to change it either.
M is the mass of the black hole which in particular means the mass as deduced by distant observation eg. punt some satellites into ( distant ) orbit around the hole, watch for a while, then use Newton to give you the central gravitating mass that would cause said orbit. So yes, far from a black hole Newton's Laws work nicely.
The above term has the units for length. It will be a bigger number for a bigger mass and a smaller number for a smaller mass. In fact you could calculate it for any mass. For every solar mass ( the amount in the Sun ) it is about three kilometers. It is called the Schwarzschild radius ( with symbol R_s used here ). As the two initial components of GW150914 were 36 and 28 solar masses respectively, then their Schwarzschild radii are 36 * 3 = 108 km and 28 * 3 = 84 km. For the Earth it is a centimetre, for me it is ~ 10^[-23]m give or take. For the Sun it is, well, about three kilometres.
{ Some texts call G * M / c^2 the gravitational radius ie. half the Schwarzschild radius. }
Karl came up with a mathematical expression to describe the differences in spacetime positions/events. This is called a metric and is with respect to four dimensions. Essentially it tells one how light will behave in a certain coordinate system. We are at liberty to construct our own coordinate systems. Karl Schwarzschild's is one. But to do physics we need to know how a given system 'works' for the real universe. So each system has a corresponding metric expression that belongs to it, being a converter from coordinate values to real measurable numbers. However over all system/metric combinations the GR principles must be upheld. This is another way of saying that the universe works as it does regardless of how we choose to view, albeit that some ways of viewing are more useful/easier/revealing/interesting than others.
While I Said : "... without delving deeply into the mathematical morass of GR field equations ... " :-))
... we could still examine the Schwarzschild solution to said equations at a simple level.
Firstly : apart from free space it is about the simplest complete solution discovered. The others have extras eg. rotation ( the ( Roy ) Kerr black hole ).
Secondly : the metric depends on a time interval ( ∆t ), a radial distance interval ( ∆r ) and a ( two component ) angular area variable I'll call ∆w.
Thirdly : in this Schwarzschild instance we can look at 'pieces' of the metric independent of other parts. That it is separable was Karl's brilliance in coming up with this.
- if you keep the radius and time constant then the metric describes a spherical shell. I won't even bother writing the expression for this ∆w component. It's the same expression that you would have for Euclidean stuff.
- if you stay on a particular shell point, so ∆r and ∆w are zero then :
[∆τ]^2 = [1 - (R_s/r)] * (∆t)^2[/pre]
.... since only time is varying here, then within the left hand side I have used ∆τ, if only because the units of the right hand side require that to be a time interval. What is this ? ∆τ is the proper time ie. LowGuy's clock rate at a certain physical point if he eventually goes down ....
.... indeed eventually will be the word, no ? Look at the expression surrounded by the square brackets on the right :
[pre]1 - (R_s/r)[/pre]
Questions 4.
(a) What does that expression tend toward as r becomes infinite ?
(b) What does that expression tend toward as r goes toward R_s, while always r > R_s ?
(c) For (a) and (b) what does that mean about the passage of time for LowGuy as viewed by ( well out into Euclidean space ) HighGal ?
(d) BONUS POINTS : What if r = R_s ?
(e) DOUBLE BONUS POINTS : What if r < R_s ? { HINT : we are using the squares of time intervals both sides, so be careful .... :-) }
(f) MEGA BONUS POINTS : Does this work for the Sun, or me, or you too ? :-))
Cheers, Mike.
( edit ) Evidently Karl's choice of spherical polar coordinates was due to the pretty obvious thing that stars are by & large shaped like round balls. It makes good sense that the relevant/interesting dimension is the radial one, as that is true for Newton's formulation also ! :-)
( edit ) Ooops. Consenting plus a longer tick .... hmmmm .... I'll score those together .... with Python Humor Metric perhaps .... OK, SQRT[175^(2/5)] it is then ! :-)
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Or ..... I never know quite how much to write ! :-)
Anyway to be exact the metric is a mathematical device ( a function ) that gives a single number as output if you provide two vectors as input. For our purposes it relays information about the relationship between the coordinate basis vectors. Any 3D vector can be represented as a unique sum of multiples of each basis vector. From Cartesian geometry in three dimension we typically use three unit vectors pointing at mutual right angles eg. i, j and k. The number emitted by the metric is the dot or inner product of those two vectors. With Rene Descarte's system we are somewhat spoiled in that the dot product of a basis vector with itself is one ( say i.i = 1 ) and with any other basis vector is zero ( say i.j = 0 ) ie. it is an orthonormal arrangement with a choice of either right or left handedness. The metric's behaviour over the basis vectors can be summarised in a matrix for all nine possible combinations of two basis vectors inputting said function :
[pre]1 0 0
0 1 0
0 0 1[/pre]
.... thus a 3 by 3 matrix. This matrix isn't the metric. It contains the components of the metric with respect to said basis. You could choose another basis vector set, still in Cartesian space, which has a quite different geometric relationships between themselves. They could be of different lengths, or not mutually orthogonal, for instance.
Even more the basis set could be local ie. not the same set for all points in the 3D space, and this is when it gets really interesting ! I'm sitting in Lilydale right now with my local system of three vectors : one pointing North, another pointing East, with the third pointing straight up above me. Each of you reading this could have a similiarly constructed locally based system. Now step away from the Earth entirely and ask the question : do all these local systems have vectors that point in the same direction ? The answer is no in generality, and although some vectors may coincide two basis sets will be equal ( if and only ) if they arise from the same point on the Earth's surface. Again we have been spoiled by the Cartesian system where we have not given a second ( nor even a first ) thought to the fact that the basis set is uniform across all of 3D space. It was just presented that way during high school without mention.
The Schwarzschild stuff we have been, and will further be, studying has an expression for a spacetime interval. This is a differential scheme in that it examines ( arbitrarily ) small steps in space and time, from one event to a nearby one. Thus the factors multiplying those little steps ( ∆t, ∆r, with ∆θ and ∆φ that make up ∆w ) are rates of change. These rates of change in turn also depend upon where you are ( in 4D space now ).
Quote:
For the cartesian system, the rate of change of the x coordinate with respect to the x coordinate is obviously 1. As the x axis and the y axis are orthogonal, then the rate of change of the x coordinate with respect to the y coordinate is zero. The dot product of one vector with another ( implicitly ) yields how variation along one direction is partly/wholly/not varying along another. Mutatis mutandis.
{ For example, if I am sitting on one of those Schwarzschild spherical shells and move a little bit in angle while staying on the sphere ( so ∆θ and/or ∆φ are not zero ) then a certain actual distance would be traversed, which in turn depends on the sphere's size r. That is, arc length on a circle depends on the radius and the angle included by the arc. Indeed if you go around an entire circle's worth of angle that arc length will be a full circle's worth : the circumference ! :-) }
In summary the metric function, or in full glory the metric tensor, gives us detailed knowledge about how the neighbourhoods of local coordinate bases vary within an entire coordinate system. It is just the ticket for managing all this curvey/wurvey spacey/timey business. It has already informed us that the Schwarzschild setup leads to rather closer packing of spherical shells around a black hole than we would expect in Euclidean space ! :-)
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
[pre]∆τ/∆t = sqrt[1 - (R_s/r)][/pre]
thus ∆τ/∆t is the slowdown of LowGuy's clock as seen by HighGal. If we plot that on a vertical axis with r/R_s on the horizontal :
For instance
(A) r/R_s = 1 means that r = R_s and LowGuy is at the Schwarzschild radius and the ∆τ/∆t = ???
(B) if r/R_s is very big ( well off scale to the right ) then LowGuy is up with HighGal and thus ∆τ/∆t = ????
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Now then Dr Hewson, you pose some tricky questions. Rather than answering them I'll ask you a couple. What does d tau / dt mean? I cannot grok it. Secondly how on earth did Leutnant Schwarzschild manage to find a solution to Albert's field equations in a dugout on the Russian front a hundred years ago?
Now then Dr Hewson, you pose some tricky questions. Rather than answering them I'll ask you a couple. What does d tau / dt mean? I cannot grok it. Secondly how on earth did Leutnant Schwarzschild manage to find a solution to Albert's field equations in a dugout on the Russian front a hundred years ago?
d tau / dt is the rate of LowGuy's clock with respect to HighGal's clock. On the plot this will be one at HighGal's position (B) and zero at the Schwarzchild radius (A).
This is the amazing result of GR flowing from the idea that gravitational fields slow time rates, which in turn is based upon the equivalence of gravity with acceleration, and in turn that is consequent upon the special relativistic slowing of clocks with velocity. A freely falling observer goes through a series of inertial ie. special relativistic frames ( MCRF or momentary co-moving reference frame ) as viewed distantly. That in it's turn relies upon the constancy of light speed .....
Yes Karl did all that when serving in the artillery. He died of an overwhelming skin infection induced by such conditions.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
You are both right. The
)
You are both right. The clocks will keep time with each other, the observers will grow old and die, entropy rules etc.
Nice one : 'I tick therefore i am?' [ +10 points ]
As for 1000 years : I'm just emphasising that when we say 'observer' we mean 'measurement system'. Humans don't have to be there at the time. We can look at artifacts - things left behind - of events and still deduce what went on eg. the clocks still agree. Indeed processes have occurred in the universe long before humans either turned up or started to make clocks. What are clocks but human readable regularities in some conveniently handled physical system ? Ditto for length measuring devices. In our minds then clocks and rulers are useful pictures/icons for 'what is happening to time?' and 'what is happening to distance?'. When we test a prediction of GR we still must use actual clocks and rulers ( LIGO being the best ever made by humans so far ) but we may still think of them for discussion in a virtual sense. The absolutes here are the speed of light ( c ) and Newton's constant ( G ) in this universe. If you want different values then go to another reality. In The Relativities we relate or compare numbers obtained from different measurement systems.
We are in a scenario which is spherically symmetric ie. by assumption this is Karl Schwarzschild's analysis. This means that you can arrange a series of concentric shells ( centred around the black hole ) and each shell will have the same two-ness about it's surface. If you like you might for a given shell mark out a latitude/longitude grid upon it. You can talk of 'areas' ( measured in square degrees ) on the sphere bounded by angles eg. the area within 53 to 54 degrees latitude and 107 to 108 degrees longitude.
Questions 2 : Take a particular shell well outside of the event horizon. We cannot ( even in theory ) go from a point on that shell straight through to the centre of the black hole and out to a point on the shell on the other side*.
Given that we can't measure the diameter/radius in that traditional sense, how would we distinguish/measure the 'width' of a shell ?
{ HINT : LowGuy continues to avoid his contract. What a wimp. We paid him in advance! But we have managed to convince him to maybe scoot around while going no closer to the hole as HighGal. We may get some use out of him before we kill him .... :-) }
Cheers, Mike.
* This is code for a traverse using a 'proper length' ie. a ruler carried by an observer who measures along some line by placing 'end-to-end' and keeping 'straight'. By keeping track of how many end-to-ends there were, we could calculate the 'proper distance' from the center of the sphere to its surface.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
RE: What are clocks but
)
I wondered if clocks and rulers are equivalent for each observer. The speed of light is a constant, so we only need really good clock to measure a distance? [*]
I feel we had a hint in the "square degrees" so i'm going to say tile it, and count the tiles. We have a survived a thousand years so tiling a black hole would seem reasonable to keep with the bathroom meme. Of course there is some mathematics involve pi and square roots, but these are trivial details for HighGal.
[*] I tick therefore i rule?
RE: I wondered if clocks
)
Unto themselves, yes. To be exact that is an a priori assumption. Or if you like it would be a pretty perverse universe otherwise and what would we then do ? There is more depth to that but not fruitful for us at present ( cue metaphysics ).
Bravo ! Indeed ! One way of 'calibrating' one's thinking is to simply define the speed of light to have the value of 1 ( one ). One what ? One light-speed of course ! There's a couple of good reasons :
- it's easier to draw diagrams with 45 degrees as the light world lines. As opposed to the angle whose tangent is 1/(3 * 10^8) ! :-)
- it's more natural in that you tend to think of matters without reference to human-centric concerns. So going cosmic is less of a jolt.
- you don't have to write the symbol c all the time in equations.
The ( slight ) downside is that when finally emitting a number for real world measurement you have to convert to metres, seconds, parsecs, years, Earth-Sun distances, heartbeats or whatever. Time especially is traditionally measured in a manner that only truly makes sense in this solar system. We actually do alot of this type of calculation daily. We don't realise though eg.
Q. How far is it to Lilydale ?
A. About 20 minutes.
( the underlying transport speed is assumed as per local context ). As you have grasped beautifully, all ( inertial, free space, yada yada ... ) observers when measuring the speed of a specific photon will agree.
So with c = 1 all distances become 'that which takes light to travel across' eg. a light-second of distance ( aka 300,000 kilometres ). You can even flip that over and say things like a 'foot of time' ( aka about a nanosecond ). This manner of re-basing units also flows through ( with other suitable definitions eg. for electric charge ) to mass, energy etc.
For example : the Lorentz analysis ( SR ) has this factor, commonly called 'gamma', which depends on the relative velocity b/w frames
gamma(v) = 1/{SQRT[1 - v^2/c^2]}
and hence simpler to write using SQRT[1 - v^2]. Just convert back to your preferred velocity units when you're done eg. snail-shell-diameters per fortnight ! :-)
Back To The Chase
Yep. Measure the area of the shell. Divide by four times PI and take the square root. Or you could simply mark and measure a circumference and divide that by two times PI. This radius value is deemed to be that via the measurements made and the subsequent calculation.
Now that is all well and good for a single shell. It avoids death by black hole too. This is Karl's approach and works famously while you are outside of the horizon. How does that help ? Let us think of what happens when there is no black hole there ( slide sideways a few light years to 'empty space' ) :
- do a given shell as above and get the radius. Call it r_1 to be definite.
- move inwards by one metre all over the shell. By definition we have a concentric shell that is slightly smaller. Get the radius, via the same method(s) that never take you to the centre, and call it r_2.
- what is the relationship b/w r_1 and r_2 eg. r_1 minus r_2 ?
- you say : one metre you fool Mike ! What else would it be ? :-)
Therein lies the rub. We've spent a weekend away comparing shells out in the Boondocks. Back to the Black Hole District. Think carefully now :
- two concentric shells as before. ( in fact you ought be objecting to the use of the word concentric by now, but plow on for the moment ).
- let the outer of the two shells have circumference c_1.
- construct a smaller/inner shell such that it's circumference is c_2, but ( careful now ) in a fashion* that makes
c_1 - c_2 = 2 * PI * one metre
Questions 3.
(a) What is the relationship b/w r_1 and r_2 eg. r_1 minus r_2 ?
(b) If I place a one metre ruler, pointing directly inwards towards the black hole with one end just at the outer shell, where will the other end of the ruler be with respect to the inner shell ?
{ HINT : options are
- exactly at the inner shell.
- closer in to the black hole than the inner shell.
- the inner shell is closer to the black hole. }
Another nice one. Snappy. Flows well, good cadence .... + 15 points. :-)
Cheers, Mike.
* You could have one of those retractable measuring tapes. First wind it out to c_1 then wind a bit in to c_1 - 2 * PI * one metre.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
The measuring stick will be a
)
The measuring stick will be a gnat's cock hair too short.
p.s. A gnat's cock hair is a technical term used in the building trade.
Richard
RE: Q. How far is it to
)
Q. Oh that long?
A. Yeah, it depends on the time.
Consenting spheres.
OK i'm stuck without applying some extra knowledge [*] and i'm going to make a guess. I know light red-shifts as it climbs away from a gravity well so assuming HighGal uses the 1m wavelength one tick ruler to measure stuff.
It is going to bluer pointing in. So my initial answer is the ruler will be shorten pointing inwards. So as HighGal measure it, it's more than 1m.
I'm somewhat concerned LowGuy get out his 1m wavelength ruler and it get red-shifted upwards so he thinks the gap is less than a metre. [**]
[*] that this black hole has some gravitational effect on lengths and times.
[**] HighGal points out that LowGuy has a longer tick. *cough*
Well ( using the metric of
)
Well ( using the metric of Karl S ) for a given calculated radial difference the 'actual' spacing b/w the shells is smaller than you find in flat space. Thus for any r_1 - r_2 as deduced by circumferences/areas then the ( proper ) radial distance is larger than that. So the one metre ruler, as placed & oriented in my question, will go further in than the innermost shell. This is one way of expressing the spatial distortion from flat Euclidean ( mass/energy ) in free space. Counter-intuitive, no ?
Another way to think of this is that for a given ingress radially you will pass relatively more shells that are uniformly spaced ( as per their circumference differences ). This is hard to think of !!
Yet another way to say it. Think of the volume ( calculated as per Euclid ) contained b/w two shells. A given volume will lie b/w two shells that are closer together than in flat space. Or a given radial step inwards contains more space. In this sense black holes 'compact' space.
{ Compare this with the case that the metre ruler falls short in my question's construction : it would imply that space has expanded ( has less volume ), which is another scenario entirely. }
{ As for :
This will be one metre. I was just restating the prior discussion slightly differently. To see if you were paying attention .... :-) }
It is perfectly normal to rail against this because it doesn't 'make sense'. Alas it is true, in the sense that when used to predict measurable outcomes, that the Schwarzschild approach works beautifully. The Universe does really work that way. We have the cognitive disadvantage that we have all grown up in a region where that approach is indistinguishable from Euclid ( until recent history ). So the Universe is fine and consistent, but off-Earth we need to adjust our thinking to fit the Cosmic Clockworks. To stop us mere mortals getting confused we have to stick with (a) things that are measurable and (b) be clear about what comparisons are being made. Karl's approach was brilliant in that it separated out the difficult bit ( the special radial character ) from the easy bit ( the spherical symmetry unchanged from free space ).
Notably the language must refer to the approach used, so one cannot be as free as you would normally be in everyday conversation with word usage. This takes some rigour. We don't have a set of objects that are immediately referable on a 'normal' basis say, like an apple could be used to center our thoughts on apple-ness when we talk about it. Intellectually tricky ! :-)
So for :
that extra knowledge is that Karl's approach works. If you like black holes squash things ! :-)
A Closer Look With Karl
There is a term that recurs in Karl's calculations for the static ( structure not varying with time ) non-rotating black hole case in GR :
2 * G * M / c^2
G and c we know and are constant. The number 2 you also are familiar with from primary school and there's no moves afoot to change it either.
M is the mass of the black hole which in particular means the mass as deduced by distant observation eg. punt some satellites into ( distant ) orbit around the hole, watch for a while, then use Newton to give you the central gravitating mass that would cause said orbit. So yes, far from a black hole Newton's Laws work nicely.
The above term has the units for length. It will be a bigger number for a bigger mass and a smaller number for a smaller mass. In fact you could calculate it for any mass. For every solar mass ( the amount in the Sun ) it is about three kilometers. It is called the Schwarzschild radius ( with symbol R_s used here ). As the two initial components of GW150914 were 36 and 28 solar masses respectively, then their Schwarzschild radii are 36 * 3 = 108 km and 28 * 3 = 84 km. For the Earth it is a centimetre, for me it is ~ 10^[-23]m give or take. For the Sun it is, well, about three kilometres.
{ Some texts call G * M / c^2 the gravitational radius ie. half the Schwarzschild radius. }
Karl came up with a mathematical expression to describe the differences in spacetime positions/events. This is called a metric and is with respect to four dimensions. Essentially it tells one how light will behave in a certain coordinate system. We are at liberty to construct our own coordinate systems. Karl Schwarzschild's is one. But to do physics we need to know how a given system 'works' for the real universe. So each system has a corresponding metric expression that belongs to it, being a converter from coordinate values to real measurable numbers. However over all system/metric combinations the GR principles must be upheld. This is another way of saying that the universe works as it does regardless of how we choose to view, albeit that some ways of viewing are more useful/easier/revealing/interesting than others.
While I Said : "... without delving deeply into the mathematical morass of GR field equations ... " :-))
... we could still examine the Schwarzschild solution to said equations at a simple level.
Firstly : apart from free space it is about the simplest complete solution discovered. The others have extras eg. rotation ( the ( Roy ) Kerr black hole ).
Secondly : the metric depends on a time interval ( ∆t ), a radial distance interval ( ∆r ) and a ( two component ) angular area variable I'll call ∆w.
Thirdly : in this Schwarzschild instance we can look at 'pieces' of the metric independent of other parts. That it is separable was Karl's brilliance in coming up with this.
- if you keep the radius and time constant then the metric describes a spherical shell. I won't even bother writing the expression for this ∆w component. It's the same expression that you would have for Euclidean stuff.
- if you stay on a particular shell point, so ∆r and ∆w are zero then :
[pre][a_spacetime_interval]^2 = [1 - (2 * G * M)/(c^2 *r)] * (∆t)^2
= [1 - (R_s/r)] * (∆t)^2
[∆τ]^2 = [1 - (R_s/r)] * (∆t)^2[/pre]
.... since only time is varying here, then within the left hand side I have used ∆τ, if only because the units of the right hand side require that to be a time interval. What is this ? ∆τ is the proper time ie. LowGuy's clock rate at a certain physical point if he eventually goes down ....
.... indeed eventually will be the word, no ? Look at the expression surrounded by the square brackets on the right :
[pre]1 - (R_s/r)[/pre]
Questions 4.
(a) What does that expression tend toward as r becomes infinite ?
(b) What does that expression tend toward as r goes toward R_s, while always r > R_s ?
(c) For (a) and (b) what does that mean about the passage of time for LowGuy as viewed by ( well out into Euclidean space ) HighGal ?
(d) BONUS POINTS : What if r = R_s ?
(e) DOUBLE BONUS POINTS : What if r < R_s ? { HINT : we are using the squares of time intervals both sides, so be careful .... :-) }
(f) MEGA BONUS POINTS : Does this work for the Sun, or me, or you too ? :-))
Cheers, Mike.
( edit ) Evidently Karl's choice of spherical polar coordinates was due to the pretty obvious thing that stars are by & large shaped like round balls. It makes good sense that the relevant/interesting dimension is the radial one, as that is true for Newton's formulation also ! :-)
( edit ) Ooops. Consenting plus a longer tick .... hmmmm .... I'll score those together .... with Python Humor Metric perhaps .... OK, SQRT[175^(2/5)] it is then ! :-)
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
A Brief-ish Note On The
)
A Brief-ish Note On The Metric ( not essential )
Or ..... I never know quite how much to write ! :-)
Anyway to be exact the metric is a mathematical device ( a function ) that gives a single number as output if you provide two vectors as input. For our purposes it relays information about the relationship between the coordinate basis vectors. Any 3D vector can be represented as a unique sum of multiples of each basis vector. From Cartesian geometry in three dimension we typically use three unit vectors pointing at mutual right angles eg. i, j and k. The number emitted by the metric is the dot or inner product of those two vectors. With Rene Descarte's system we are somewhat spoiled in that the dot product of a basis vector with itself is one ( say i.i = 1 ) and with any other basis vector is zero ( say i.j = 0 ) ie. it is an orthonormal arrangement with a choice of either right or left handedness. The metric's behaviour over the basis vectors can be summarised in a matrix for all nine possible combinations of two basis vectors inputting said function :
[pre]1 0 0
0 1 0
0 0 1[/pre]
.... thus a 3 by 3 matrix. This matrix isn't the metric. It contains the components of the metric with respect to said basis. You could choose another basis vector set, still in Cartesian space, which has a quite different geometric relationships between themselves. They could be of different lengths, or not mutually orthogonal, for instance.
Even more the basis set could be local ie. not the same set for all points in the 3D space, and this is when it gets really interesting ! I'm sitting in Lilydale right now with my local system of three vectors : one pointing North, another pointing East, with the third pointing straight up above me. Each of you reading this could have a similiarly constructed locally based system. Now step away from the Earth entirely and ask the question : do all these local systems have vectors that point in the same direction ? The answer is no in generality, and although some vectors may coincide two basis sets will be equal ( if and only ) if they arise from the same point on the Earth's surface. Again we have been spoiled by the Cartesian system where we have not given a second ( nor even a first ) thought to the fact that the basis set is uniform across all of 3D space. It was just presented that way during high school without mention.
The Schwarzschild stuff we have been, and will further be, studying has an expression for a spacetime interval. This is a differential scheme in that it examines ( arbitrarily ) small steps in space and time, from one event to a nearby one. Thus the factors multiplying those little steps ( ∆t, ∆r, with ∆θ and ∆φ that make up ∆w ) are rates of change. These rates of change in turn also depend upon where you are ( in 4D space now ).
{ For example, if I am sitting on one of those Schwarzschild spherical shells and move a little bit in angle while staying on the sphere ( so ∆θ and/or ∆φ are not zero ) then a certain actual distance would be traversed, which in turn depends on the sphere's size r. That is, arc length on a circle depends on the radius and the angle included by the arc. Indeed if you go around an entire circle's worth of angle that arc length will be a full circle's worth : the circumference ! :-) }
In summary the metric function, or in full glory the metric tensor, gives us detailed knowledge about how the neighbourhoods of local coordinate bases vary within an entire coordinate system. It is just the ticket for managing all this curvey/wurvey spacey/timey business. It has already informed us that the Schwarzschild setup leads to rather closer packing of spherical shells around a black hole than we would expect in Euclidean space ! :-)
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
HINT : re-arranging
)
HINT :
re-arranging
[pre][∆τ]^2 = [1 - (R_s/r)] * (∆t)^2[/pre]
gives
[pre]∆τ/∆t = sqrt[1 - (R_s/r)][/pre]
thus ∆τ/∆t is the slowdown of LowGuy's clock as seen by HighGal. If we plot that on a vertical axis with r/R_s on the horizontal :
For instance
(A) r/R_s = 1 means that r = R_s and LowGuy is at the Schwarzschild radius and the ∆τ/∆t = ???
(B) if r/R_s is very big ( well off scale to the right ) then LowGuy is up with HighGal and thus ∆τ/∆t = ????
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal
Now then Dr Hewson, you pose
)
Now then Dr Hewson, you pose some tricky questions. Rather than answering them I'll ask you a couple. What does d tau / dt mean? I cannot grok it. Secondly how on earth did Leutnant Schwarzschild manage to find a solution to Albert's field equations in a dugout on the Russian front a hundred years ago?
Richard
RE: Now then Dr Hewson, you
)
d tau / dt is the rate of LowGuy's clock with respect to HighGal's clock. On the plot this will be one at HighGal's position (B) and zero at the Schwarzchild radius (A).
This is the amazing result of GR flowing from the idea that gravitational fields slow time rates, which in turn is based upon the equivalence of gravity with acceleration, and in turn that is consequent upon the special relativistic slowing of clocks with velocity. A freely falling observer goes through a series of inertial ie. special relativistic frames ( MCRF or momentary co-moving reference frame ) as viewed distantly. That in it's turn relies upon the constancy of light speed .....
Yes Karl did all that when serving in the artillery. He died of an overwhelming skin infection induced by such conditions.
Cheers, Mike.
I have made this letter longer than usual because I lack the time to make it shorter ...
... and my other CPU is a Ryzen 5950X :-) Blaise Pascal