Reversal of proper time for a falling object

[PAllen]

Hi DrGreg, I appreciate what you are getting at, but my argument would be this. When we want to know what happens in a region we have never tested, we can only take the laws we have tested and confirmed in our region and extend them to the unknown region. If we says the laws change in the unknown region, we can say anything we like about the unknown region. By way of example let us say we had a hypothetical universe where proper time τ of a moving clock related to coordinate time t by the relation dτ = dt(1-v^2/c^2) and have never actually tested what happens when v>c. We could predict using the laws we have tested that for v>c that dτ runs in the opposite direction to dt. Now if we are uncomfortable with proper time running in reverse, we can say that v>c that are different laws and the relation becomes dτ = -dt(1-v^2/c^2) and that it is the dt that changes it nature rather than the proper time of the moving clock. Let's say sometime later we actually (in the hypothetical universe) get the clock exceed c and observe that the moving clock is now running backwards relative to all our clocks and natural processes. Should we still cling to the notion that the proper time of the moving clock cannot run backwards and therefore all our clocks and natural processes must have become spacelike instead of timelike? Why should our clocks and rulers change their nature, when they have not undergone any acceleration or other measurable change, just because one clock in the (hypothetical) universe is exceeding c? Of course if there was an observer riding with the accelerated clock, they would say the clock is still advancing in time while we say it is going backwards.

It really seems like you are not reading what people are writing. We keep saying clocks and rulers don't change their nature across the horizon, neither do any laws don't change. Neither do any intervals change nature. Further, as I noted, in some coordinate systems (e.g. Kruskal), no coordinates change their nature. All that is true is that using some particular schemes for labeling events, the label called 't' has one meaning in one region and another meaning in another region. What determines its meaning? The metric. The metric is the only thing in GR that gives meaning to coordinate labels.

Let me ask you this: try to phrase your argument in Kruskal coordinates. If you are saying something physical, it should not matter what coordinates you use. Only, there you will find that along a free fall trajectory that crosses the horizon dτ/dV remains positive. Why? Because in these coordinates V is timelike coordinate everywhere.

 

[PeterDonis]

For r<2GM/c^2 the proper time runs in the reverse direction relative to Schwarzschild coordinate time.

Let me go back to the OP and ask the following question: what do you think the above quoted observation means, physically?

 

[yuiop]

Let me go back to the OP and ask the following question: what do you think the above quoted observation means, physically?

This is a big question and I will come back to it later and there are many other replies here that deserve to be digested and responded to as time allows. Thanks all. For now I would like to leave you all with an analogy and ask what you make of it.

Let us say we have a rectangular piece of tracing paper with x-axis along the long edge of the rectangle and y-axis on a short edge. On a wall we mark off a similar coordinate system with the x' axis horizontal and y' axis vertical. We lay the tracing paper on the wall and ensure the x-axis of the tracing paper is parallel with the x' axis of the wall grid and the y and y' axes are similarly parallel. We glue the left edge of the paper to the wall. Now we put a diagonal fold about a third of the way along the tracing paper so that that the last two thirds of the long rectangle are now vertical instead of horizontal. On the paper that has been moved position, the x lines are now parallel to the y' lines on the wall and the y lines are now parallel to the x' lines. Can we now say the x coordinates of the paper have become ylike or should we say that that the y' coordinates of the wall have become x'like? We now put another diagonal fold in the last third of the paper so that in this section x is antiparallel to x' so that increasing x on the paper is related to decreasing x' on the wall. Can we argue that nothing has changed on the paper and that x' coordinates on the wall have now changed their nature and no longer measure horizontal distances on the wall correctly? Imagine an a two dimensional creature restricted to living on the tracing paper. As far as it is concerned, the paper is flat and the wall must be somehow distorted in space. Who is correct. The 2D creature on the paper who says the paper is flat and the wall is folded in space or another observer who says the wall is flat and the paper is folded? As far as I can see, the x' coordinates on the wall always measure an x'like or horizontal property of the wall whatever we do with the paper. Folding the paper changes the relation of the paper to the wall, but does not change anything about the quantity that x' coordinates are defined to measure, namely horizontal distances along the wall. Why is it that in Schwarzschild coordinates that we seem to take the position that when the paper folds, it is actually the wall that folds and refuse to acknowledge the possibility that maybe the paper folded?

 

[PeterDonis]

I would like to leave you all with an analogy and ask what you make of it.

I'm not sure I understand the analogy you are trying to draw. Which coordinates on a black hole spacetime do you see as corresponding to the paper coordinates? Which ones do you see as corresponding to the wall coordinates?

 

[A.T.]

What about this from an old thread:

Note: The negative proper time for clock D when R=2.13 is not a typo. The proper time of a clock at the centre of a gravitational mass is negative always negative for radii less than 2.25m or less than $$\frac{9}{8} R_{s}$$

If true, you would have proper time reversal without any event horizon.

Question to yuiop, if you still remeber: What interior Schwarzshild solution did you use to derive this? Was is the exact one, or just a weak field approximation? The latter might explain the strange result because R = 9/8 R_s is definitely not a weak field.

 

[yuiop]

What about this from an old thread:

If true, you would have proper time reversal without any event horizon.

Question to yuiop, if you still remeber: What interior Schwarzshild solution did you use to derive this? Was is the exact one, or just a weak field approximation? The latter might explain the strange result because R = 9/8 R_s is definitely not a weak field.

At Rs < R < 9/8 Rs there is an apparent horizon where gravitational time dilation goes to zero located below Rs. The result came from an exact solution. This was discussed in a recent thread and we agreed there are certain limitations to the validity of the result. See https://www.physicsforums.com/showpost.php?p=3857054&postcount=24 and following posts.

The interior solution assumes that the massive body has even density and is static. The negative proper time of the clock at the centre probably suggests that a massive body cannot have even density or be static when the surface radius is less than 9/8 Rs.

 

[yuiop]

Let me go back to the OP and ask the following question: what do you think the above quoted observation means, physically?

The observation that the free falling clock goes in the reverse direction to coordinate time suggests to me that we may be wrong in some of our assumptions about what happens inside black holes such as the assumption that an object passes straight through the event horizon and continues to the singularity at the centre. See my previous post which is related. Reversal of proper time implies that the thermodynamic arrow of time is reversed which is unlikely so if we obtain a negative time result we should revisit our assumptions. I understand that most people here take the view that the free falling clock is not measuring time any more below the event horizon.

I'm not sure I understand the analogy you are trying to draw. Which coordinates on a black hole spacetime do you see as corresponding to the paper coordinates? Which ones do you see as corresponding to the wall coordinates?

To me the wall represents Schwarzschild coordinates, because that solution is static and unchanging and the falling observer does not materially change anything about the gravitational field is the massive body is large relative to the observer. The paper coordinates represent the reference frame of the free falling observer. The location of the free falling observer is continually changing relative to the gravitational field and he can probably even notice changes in his own coordinate system over time due to tidal effects and limitations of the equivalence principle, if his coordinate system is extended over a large distance. The rate of the clock that the falling observer uses as a reference, is continually changing over time as he falls, relative to the clock of the Schwarzschild master clock. In SR we can argue about whose clock is really going slower, but in GR it is indisputable that the falling clock is ticking slower than the stationary SC clock and continues to get even slower as it falls.

For example in these forums it is often stated that the speed of light near a black hole is the same as it is at infinity and that local measurements trump distant measurements. Let's examine these beliefs more closely. Let us say that a observer, (call him Adam) near the black hole but at constant radius claims that he measures the speed of light to be c locally. Eve, who is far away from the black hole also measures the local speed of light to be c but additionally claims that that the speed of light near Adam is slower than the speed of light at her radius. Adam counter claims that he measures the local speed of light to be c and the same as the local speed of light measured by Eve so the speed of light does not vary with distance from the black hole, because local measurements trump distant measurements. Eve points out Adam measures the speed of light using a clock that is ticking slower than her own clock so his measurements are distorted. Adam says his his clock is ticking at a rate of one second per second so it must be running at the same rate as Eve's clock. Eve then gets Adam to agree to a series of experiments transporting clocks and bringing them back together and eventually gets Adam to agree that clocks lower down do actually physically run slower than clocks higher up. He also agrees that the time dilation affects are not reciprocal as in SR. Once he concedes those points, he has to agree that the speed of light lower down must be relatively slower than the speed of light higher up and that local measurements can be distorted by the effects of gravity on his measuring tools.

If we time a race and find that the everyone in the race has broken the world record by a significant margin and later find the the race clock had a fault and was running slow, do we still make a new entry in the record books or declare that result invalid? At the very least we should try and determine by how much the race clock was running slow and correct for that error.

 

[Dale]

Hi Dale. It would not be the first time I learned something from you so I will play along and give the obvious answer, that in that case +dx² would appear to be the time coordinate.

I am very aware that it is generally accepted that timelike becomes spacelike and vice versa below the event horizon in Schwarzschild coordinates, but I have yet to see a convincing proof as to why that should be so. Maybe you can do it.

So, the rule is that the time coordinate is the one with the + sign in the metric, irrespective of the symbol used. Therefore, we simply apply this rule to the Schwarzschild metric. Let's just use units where c=G=1 and M=1/2 and let's compute the metric at r=2 and r=1/2 with θ=90º.

At r=2 we have:
dτ² = +0.5dt² -2dr² -4dθ² -4dφ² (t is the time coordinate)

At r=1/2 we have:
dτ² = -dt² +dr² -0.25dθ² -0.25dφ² (r is the time coordinate)

It follows directly from the rule for identifying the time coordinate. Do you disagree with the rule for identifying the time coordinate? If so, then what rule would you propose instead?

 

[PAllen]

The observation that the free falling clock goes in the reverse direction to coordinate time suggests to me that we may be wrong in some of our assumptions about what happens inside black holes such as the assumption that an object passes straight through the event horizon and continues to the singularity at the centre. See my previous post which is related. Reversal of proper time implies that the thermodynamic arrow of time is reversed which is unlikely so if we obtain a negative time result we should revisit our assumptions. I understand that most people here take the view that the free falling clock is not measuring time any more below the event horizon.

How can you you so persistently misread statements that have been repeated and clarified dozens of times? Everyone here (except possibly you) has stated multiple times that the free fall clock continues to measure time, perfectly normally, in the forward direction. Yet another way to state the flaw in your interpretation is the dτ/dt is a coordinate dependent quantity, and per universal interpretation of GR cannot be taken to mean anything observable without a computation based on invariant values. There is the rub: dτ/dτ is obviously 1; dτ/dt referred to the local frame of the free fall world line is also 1 straight through. So this is what I call the 'fundamental mistake of newbie GR interpretation' - why I rail against relating observables to coordinate quantities without analysis in terms of invariants.

Note something observable that is quite easy to compute: as the free faller passes through and beyond the EH, they can continue to get signals from outside the EH (it is only the reverse that is prevented by the EH). Suppose these signals are clock readings from an exterior clock. Then the free faller notes that the exterior clock continues to advance in the same direction as theirs, with no discontinuity or behavior change at the EH, right down to the singularity. (Note: the relative rates of the two clocks will vary depending on the particular free fall trajectory and the location of the exterior clock; but they will both be advancing steadily, in the same direction, as observed by the free faller).

[Edit: The only thing we say changes at the EH in this discussion is the nature of dt in one particular set of coordinates; and that this coordinate feature has no particular physical significance, because it doesn't affect observables such as the scenario described in the prior paragraph.]

 

[PAllen]

To me the wall represents Schwarzschild coordinates, because that solution is static and unchanging and the falling observer does not materially change anything about the gravitational field is the massive body is large relative to the observer.

Here is yet another key misunderstanding. The SC metric inside the EH is not static at all. This would be clear if you analyzed in terms of tensors and invariants rather than coordinates. In particular, there are no timelike killing vectors inside the EH, so it is not static. This, of course, is related to your error of insisting that (P, P+Δt) has a nature determined by the letter t rather than by observing the sign of its contraction with the metric.

 

[yuiop]

...
At r=2 we have:
dτ² = +0.5dt² -2dr² -4dθ² -4dφ² (t is the time coordinate)

At r=1/2 we have:
dτ² = -dt² +dr² -0.25dθ² -0.25dφ² (r is the time coordinate)

It follows directly from the rule for identifying the time coordinate. Do you disagree with the rule for identifying the time coordinate? If so, then what rule would you propose instead?

Two alternative proposals:

1)Time coordinate is that measurement made using a clock.
2)Time coordinate is that unique coordinate that can only increase while all other coordinates can increase or decrease.

I am trying to get at an understanding of the actual physical meaning of time coordinate changing from being timelike to being spacelike. I am sure it is not the most simplistic casual conclusion that clocks become rulers and vice versa.

Using your rule for identifying the time coordinate of the timelike coordinate as being the coordinate with the opposite sign to all the others in the metric we can note that above the event horizon the coordinate with the + sign can only increase for increasing dτ. By logical extension, below the event horizon we can conclude that dr can only increase and dt can run in either direction for increasing dτ.

 

[PAllen]

Two alternative proposals:

1)Time coordinate is that measurement made using a clock.
2)Time coordinate is that unique coordinate that can only increase while all other coordinates can increase or decrease.

I am trying to get at an understanding of the actual physical meaning of time coordinate changing from being timelike to being spacelike. I am sure it is not the most simplistic casual conclusion that clocks become rulers and vice versa.

Using your rule for identifying the time coordinate of the timelike coordinate as being the coordinate with the opposite sign to all the others in the metric we can note that above the event horizon the coordinate with the + sign can only increase for increasing dτ. By logical extension, below the event horizon we can conclude that dr can only increase and dt can run in either direction for increasing dτ.

Well let's apply your alternatives to interior SC coordinates:

1) A curve of t constant, r varying, inside the EH can be computed to be a timelike curve, and the the proper time along it will be a monotonic function of r. Thus, this criterion says r is timelike, not t.

2) For this criterion, we must clarify 'along a timelike path'. For, trivially, a spacelike path can go back in time. Then, using the interior metric it is easy to see that any timelike path must have r changing monotonically towards 0, while t can vary in either direction along a timelike path.

[The only peculiarity is that r decreasing is 'forward in time' along free fall trajectory continued across the EH.]

So, your own criteria agree with what we've all been saying.

Your last summary paragraph is correct except for the inverted convention on r. The metric only contains squared differentials, thus the timelike coordinate is constrained to change monotonically, but can follow either directional convention.

Noting that the exterior and interiors SC solutions are actually two separate coordinate patches, following a free fall world line through the EH, the coordinate description of the path (within the problematic SC coordinates) has the following character:

coordinate value of timelike coordinate just above EH goes to infinity. Then, the time along the world line just below the EH resumes with time like coordinate r=Rs-ε (coordinate discontinuity at EH itself), and continues to r=0 at time of reaching singularity.

 

[Dale]

1)Time coordinate is that measurement made using a clock.

This rule doesn't work. A clock measures proper time, not coordinate time. In the metric, that is the quantity on the left hand side, not any of the quantities on the right hand side.

2)Time coordinate is that unique coordinate that can only increase while all other coordinates can increase or decrease.

This rule might work, but it needs a little bit of clarification. First, if you are just considering a chart on the manifold then all coordinates can vary over their complete range. So, you must be restricting it somehow.

As much as possible, we don't want to rely on other theories, like thermodynamics, so I would think that the best way to do that would be to identify some event and then to identify all nearby events that are inside the future light cone from that event. If we do that we will usually find that for three of the coordinates the "inside" events will have larger and smaller coordinate values compared to the reference event and for one there will only be coordinates which are larger than the reference event.

I think this clarifies your intention without distorting it in any way.

Using your rule for identifying the time coordinate of the timelike coordinate as being the coordinate with the opposite sign to all the others in the metric we can note that above the event horizon the coordinate with the + sign can only increase for increasing dτ. By logical extension, below the event horizon we can conclude that dr can only increase and dt can run in either direction for increasing dτ.

Yes. Although for a black hole r can only decrease, so the time coordinate would be -r.

 

[PeterDonis]

The observation that the free falling clock goes in the reverse direction to coordinate time suggests to me that we may be wrong in some of our assumptions about what happens inside black holes such as the assumption that an object passes straight through the event horizon and continues to the singularity at the centre.

So you're interpreting the reversal to imply that *coordinate* time is the "fixed point" whose direction is unchanging, and *proper* time is what has "reversed direction". That clarifies things. It's also backwards; actually, Schwarzschild coordinate time is more like the "paper", and the underlying spacetime geometry is the "wall", as we'll see below.

I understand that most people here take the view that the free falling clock is not measuring time any more below the event horizon.

As PAllen said, this is a serious misunderstanding of what all of us have said. You yourself showed that the worldline of a freely falling object remains timelike all the way down to r = 0. That means that a clock falling along such a worldline continues to "measure time". We all agree that it does.

To me the wall represents Schwarzschild coordinates, because that solution is static and unchanging

Only outside the horizon! This may be a key source of the misunderstanding. Inside the horizon, Schwarzschild spacetime is *not* static. That means that you cannot use the "static" faraway observer as a point of reference for events inside the horizon.

Try this exercise: look at a diagram of Schwarzschild spacetime in Kruskal coordinates. Look at what the curves r = constant and t = constant look like. Outside the horizon, in "region I", the r = constant curves are hyperbolas running up the diagram that gradually get flatter as you move outward. The t = constant curves are straight lines whose slope varies with t (the t = 0 line is the "X-axis" of the diagram, t < 0 lines slope down and to the right, t > 0 lines slope up and to the right). So the r = constant curves are timelike and the t = constant curves are spacelike. (Note that the horizon, r = 2M, the 45 degree line up and to the right, is not included in the Schwarzschild exterior chart, except for the "center point" at the origin of the diagram, where t = plus infinity; this is where all of the t = constant lines *cross*, which is why there is a coordinate singularity in the Schwarzschild chart.)

What does this tell us about the character of the coordinates? (Remember that this diagram leaves out the angular coordinates altogether, so we are only considering r and t.) Since the r = constant lines are timelike, that means any small line element with dr = 0 and dt nonzero will be timelike. So t is a timelike coordinate. Similarly, since the t = constant lines are spacelike, any small line element with dt = 0 and dr nonzero will be spacelike. So r is a spacelike coordinate.

Now look at "region II", inside the horizon. Here the r = constant curves are hyperbolas running across the diagram, and the t = constant curves are lines running up the diagram from the "center point". The t = 0 line goes straight up; t > 0 lines go up and to the right, t < 0 lines go up and to the left. So here, r = constant lines are spacelike, and t = constant lines are timelike. This tells us, using the same reasoning as above, that inside the horizon, r is timelike and t is spacelike. This also tells us why the spacetime inside the horizon is not static: since the r = constant lines are now spacelike, there cannot be any static observers that stay at the same r.

But it also tells us something else. What will the worldline of a freely falling observer look like in this diagram? It will start out somewhere in the lower right corner, at very large r and very large negative t. It will gradually move up the diagram, and to the left (but staying at less than 45 degrees to the vertical, since it is always timelike), until it reaches the horizon line. At that point, it is at r = 2M, t = plus infinity. Now continue following the worldline into region II, the interior. It will keep moving up, and somewhat to the left. That means it will have to continue decreasing its r coordinate, but it will also now be *decreasing* its t coordinate! This should be obvious from looking at how the t = constant lines are laid out in region II; the ones closer to the horizon line have higher values of t, so as the freely falling worldline moves away from the horizon, it moves to lower values of t.

So what has actually happened? What has happened is that the Schwarzschild *coordinate* time is like the "paper" in your scenario; it flips around inside the horizon so that it runs in the opposite direction, relative to the "wall", the actual underlying spacetime geometry as diagrammed in the Kruskal diagram. (Actually, it would be more correct to say that the "paper" has been flipped by 45 degrees, sort of like the horizon line being the "fold".)

For example in these forums it is often stated that the speed of light near a black hole is the same as it is at infinity and that local measurements trump distant measurements.

No, that's not what is often stated. What is often stated is that an observer anywhere in spacetime will measure the *local* speed of light to be c. That is *not* the same as saying the speed of light is "the same as it is at infinity" without qualification--in a curved spacetime there is not a unique way to compare speeds between distant locations. You discuss different ways the comparison can be made, and rightly point out that they give different answers; but there is no way of showing that any of those answers is the "right" one. One observer says that his local measurements stay the same, so the speed of light stays the same; another says that gravity distorts the measuring tools, so the speed of light changes with location. In a sense, both are right, because they are talking about different interpretations of the term "speed of light".

 

[PeterDonis]

(Actually, it would be more correct to say that the "paper" has been flipped by 45 degrees, sort of like the horizon line being the "fold".)

Correction, I should have said "flipped by 90 degrees".