
#1
May1011, 04:38 PM

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Let's say you're mass is 100kg and you're traveling at 0.9C. The relativistic effect on your mass is roughly 2.3 so you now have a mass of 230kg.
If you went whipping past me, would you appear to have a mass of 230kg or 100kg? Seems like a nobrainer. But it keeps tripping me up! TIA! CG 



#2
May1011, 04:42 PM

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#3
May1011, 04:45 PM

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#4
May1011, 05:43 PM

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Is increase in mass relative?The current state of the art in GR is that it is not "mass" that causes gravity, but the "stressenergy tensor", a more complex mathematical object with more than one component. You might take a look at http://ajp.aapt.org/resource/1/ajpia...sAuthorized=no A more advanced paper that talks about the "field" of an ultrarelativistic moving mass that you might find interesting (if it isn't too advanced) is http://arxiv.org/abs/grqc/0110032. However, it talks about what one can actually measure, which is not a "force" at all, but the Riemann tensor, which can very very roughly be thought of as the rate of change of the Newtonian field. In fact, one can equate parts of the Riemann tensor to Newtonian "tidal gravity". You can see from the text that the resulting "field" of a moving mass isn't spherically symmetric at all, and that any attempt to think of it as a spherically symmetric Newtonian field is doomed. It's interesting and educational to note that the electric field (which is much easier to define!) of an ultrarelativistic moving charge isn't spherically symmetric at all either. If you're really interested in understanding all the details, and have the necessary background in at least special relativity, I'd suggest studying the ultrarelativistic moving charge first, and coming back to the ultrarelativistic moving mass when you've understood the previous case. 



#5
May1011, 06:41 PM

P: 93

In a way a similar question to the mass in different frames of reference  but even more vexing!
Consider a black hole basking in the light of nearby stars which are moving slowly enough relative to the BH so that thier speed can be ignored. First of all, how does one calculate the blue shift of the starlight at say the EH, and then further into the hole? As the BH absorbs the light it aquires mass. But how much mass? If instead of a hole it were a lightwieght asteroid of the same size, the ambient starlight recieved over a year might have mass equivilent of say 1 gram. But the BH blueshifts the light to a much higher energy and therefore the mass of the infalling light is increased. Does the BH gain 1 gram or does it gain the increased amount? Also I hear that at the EH time stops. I question this but assuming this is right, then surely incoming light is blueshifted to infinity? 



#6
May1011, 07:03 PM

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Anyway, on the subject of blueshift, an observer falling through a black hole will only see a finite blueshift as he crosses the horizon, he won't see the rate of clocks in the outside universe sped up to infinity. However a rotating black hole also has an inner "Cauchy horizon" beyond the outer event horizon (see the third diagram on this page), and there the blueshift is predicted to go to infinity in general relativity, which is the main reason physicists don't really trust general relativity's predictions about what goes on at the center of a rotating black hole, and tend to think we'll need a theory of quantum gravity to model what goes on beyond the Cauchy horizon. 



#7
May1011, 07:53 PM

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It's rather interesting to note that the amount of mass added to a black hole by dropping one kilogram into it can be made arbitrarily low, and approaches zero as one approaches the event horizon. If you drop a kilogram into the black hole from infinity, the black hole gains 1kg. If you drop a kilogram into the black hole from near the event horizon, the black hole basically gains nothing  to be more precise, an arbitrarily small amount that approaches zero as one approaches the event horizon. Let's consider that we have a 1kg mass at inifinity, and a very long, very strong, rope. If you lower the 1kg further and further into the black hole on a rope, where does the energy go? Well, as the object is lowered into the black hole, it does work, "at infinity". In Newtonian terms, one would say that the mass loses energy due to the "gravitational binding energy" as its lowered into the field, and that this extracted energy is what does the work at infinity. However, there's no solid definition in GR of "gravitational binding energy", except in the Newtonian limit. While we can usually (though not always) compute the mass of a system, we can't say exactly how it's distributed, or where it is. There are several possible ways of doing the computation, all of which give different answers as to "where" the mass is, though they all give the same end result. 



#8
May1011, 08:14 PM

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Whoa!
Do you guys think, from the way the OP presented his/her question, that he/she could understand all this? Really? Zz. 



#9
May1011, 09:05 PM

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#10
May1011, 09:07 PM

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Zz. 



#11
May1111, 06:27 AM

P: 3,178

Indeed, as JesseM clarified, the increase in relativistic mass is relative.
(Now let's see if the OP still has issues!) Note: although offtopic, it sounds as if JesseM made a mistake in the force calculation. The equation for acceleration along x (the same direction as the motion): F = gamma^{2}*m*a = gamma^{3}*m_{0}*a, with m = relativistic mass and m_{0} = rest mass. However, for acceleration in the normal direction (perpendicular on the motion): F = m*a = gamma*m_{0}*a 



#12
May1111, 07:08 AM

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#13
May1111, 02:49 PM

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I am having a more imeadiate problem with the EH. I would love to know how the formula for Rs was derived because there is a very serious problem with meerly evaluating the newtonian formula for escape velocity with v set to c. It only works for the proper speed (disance/proper time). Rs is the point at which an infalling object reaches a proper speed of c. The velocity reached at EH is less than c when measured by any other means. Photons still overtake the infalling object and the photons are not blue shifted to infinity at this point. To say things like 'all paths inside the EH point to the singularity' are in my view wrong, particularly if you make that all lightlike paths. Light always travels at c and therefore has infinite proper speed  and can escape from within the EH as long as it's path is along a very narrow cone. If it is a finite blueshift on the way in (except maybe at cauchy), it will be a finite redshift on the way out. Only if there is a zero sized point at the center, will photons face an infinite red shift and not be able to escape. 



#14
May1111, 03:19 PM

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#15
May1111, 06:02 PM

P: 93

The other day I posted a question in another thread but did not get a response. I considered the eliptical orbit of a planet around a star. The orbit continually converts PE to KE and back again. Ignoring the star's radiation, the total massenergy of the system should be a constant. Your answer appears to confirm this statement but only sort of. The total gravity of the system should be that given by the actual mass of the star and planet, plus thier PE and KE. I am told that a hot object has more mass than an identical cold one, due to the vibrational energy, which is just KE  so I think the KE has to be added to the mass, and to be constant, so must the PE. The trouble is, I can't for the life of me, figure out the mechanism by which PE contributes to massenergy and thus to gravity. I am further baffled that 1Kg is added only if droped from infinity. I would have thought that while the closer to the hole the less was added, but would not have thought it possible that less than 1Kg could be added. It seems to me that if you have say one solar mass in one place, the total massenergy is one solar mass. If you have twp solar masses in one place the total massenergy is two solar masses. But if you have two solar masses in two places you have more than two solar masses worth of massenergy because there is PE. This must be connected to energy of free space (in GR terms rather than exotic stuff like zeropoint energy). Or have I completly lost the plot? 



#16
May1111, 07:31 PM

P: 93

Proper speed was mentioned because as far as I can make out, a proper speed of c would be reached by an object falling from infinity to the EH. This falls well short of a real speed of c.
In the case of matter, I proposed in an earlier post, that there may be a discrepency between velocity attained by an infalling object and that needed to escape. Normally these are considered the same at all points along the trajectory. However for dense stars where the speeds are such that relitivistic effects become significant, matter falling in does not have it's acceleration impeaded by mass aquisition due to speed while outgoing matter would be impeaded by this. This proposition may be invalid but I was trying to find a way in which it was valid to say matter could not escape not matter how fast it went. In the case of light, it makes perfect sense for a photon to stay on the boundary of a future light cone but what I don't get is the idea that the EH can be compared to a light cone. In what sense is the EH moving out (at c or any other speed)? It moves out only when the mass is added to as is clear from the equation. In any case if time no longer progresses at EH is it not better simply to state the photons stay where they are because they are frozen in time? It is irrelevent how fast something is going if it is in a locale where time dilation has become infinite and time has stopped? In any case why would outgoing light stop at the EH? The curvature gets more severe with r < Rs so an outgoing photon would never get to the EH. I shall now answer your question concerning my 'view' that all physicists are wrong about this. I don't take that view because I am not qualified to do so but I do pose the question. I pose the question because I know that where humans are concerned it is due dillegence to do so  and I am qualified to say that because I have studied the psychology! I am currently studying the math of GR but it is very difficult to square the math with the numerous written descriptions of objects falling into a black hole etc. I am not going to pretend I am any good at math but I never had this problem with any other aspect of physics. The math and the descriptions were for the most part, complimentry. I do need to see a full derivation of the Rs equation. Hopfully in that will be meanings I have so far failed to garner. 



#17
May1111, 08:06 PM

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Keep in mind that the "size" of any surface, whether the event horizon or the surface of the Earth, is totally dependent on the type of coordinate system you use, and in general relativity there are no global "preferred" coordinate systems like inertial frames in special relativity (although as noted above you can still have locally inertial frames in SR), you can use any coordinate system you want and the equations of general relativity will work equally way in each one as long as you define the metric correctly in that coordinate system. See this article on "diffeomorphism invariance" for more on the complete freedom you have to define coordinate systems however you like in general relativity. It might seem counterintuitive that just by switching coordinate systems a spherical surface can go from having a fixed radius to an expanding one, but in general relativity neither version is more "correct" than the other. And another diagram which explicitly shows the worldlines of light rays bounding the light cones, both light rays which were aimed "inward" (the straight lines at 45 degrees) and light rays which were aimed "outward" (the curved lines, which you can see continue to approach the singularity inside the horizon): 



#18
May1211, 06:04 PM

P: 93

JesseM, thany you for this very detailed response. Clearly I have much reading to do. You are certainly right to say that it is counterintuative to switch between a coordinate system that does not expand to one that does. You can switch coordinate systems in other areas of math and physics, not just GR  except that in so doing there are more consequences perhaps! My key problem does appear to be linked directly to this issue. In short I have not yet found a coordinate system that produces the widely accepted result describing an EH.
I note that you make the point that inside the horizon there no outgoing light because a light beam sent in any spatial direction will always have a decreasing Schwarzschild radius. I am not sure what is meant by a decreasing radius or why it is nessesary. But it was this point about no outgoing light, that a lecturer at a public lecture I attended many years ago, completely screwed up. When I said that it did not make sense that a micron outside the EH light could be outbound while a micron inside it the light was bent in effect by 180 degrees, he huffed and puffed and ended up admitting he hadn't thought of that! All he needed was an explanation of how it was valid for the radius to expand at c and the 180 degrees charge would have been debunked. I can see it to a certain extent but not completely. If a number of objects fall towards the hole, regardless of thier projected entry point or angle, each would see the others as receeding and the further away they were, the faster would be the recession  very much like the galaxies in the universe and this would serve to create the 'illusion' that space was expanding. I take it that the math suggests the expansion is real rather then an illusion. I also would have thought this effect would simply continue once inside the EH. Are you saying the objects cannot see objects that are nearer the singularity because no light can head outwards? This differs from other accounts in which photons do travel outwards to the EH but cannot get beyond it because of the expandsion at c (niether account I can as yet concur with). Does this expansion concept have anything to do with the assertion that once the Rs of a dense object exceeds it's physical radius, the object imeadiately begins its collapse to a point? As I understand it, the assertion has nothing to do with limits of nuetron degeneracy pressure as these limits are completely unknown, but is a direct consequence of the math of GR. I ask this because I am wondering what happens when a small mass lands on one side of a neutron star and pushes Rs that last trillionth of a micron so that it now matches the physical radius. Would the collapse be in any way lopsided? Also, presumably nearby orbiting objects (such as another nuetron star), would extend the 3 solar mass limit, preventing a collapse. I am intrigued as to what might happen with a 1 solar mass nuetron star in an eliplical orbit around a 3 solar mass nuetron star. During part of the orbit, the larger body would become a black hole only to be repeatedly denuded. Don't ever let it be said that black hole geeks have no sense of humor! 


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