Pete

But why do you think that definitions must be mathematical to be rigorous? E.g. define "time" mathematically. Pete

I never learned QFT so I'll have to take your word for this until I learn it myself. Why did you qualify it with perturbative? Pete

On the contrary. Negative gravitational mass most definitely implies gravitational repulsion. The weak energy condition does not contradict negative mass. The energy density could be positive and yet leave the mass density as negative. In the weak field limit (which applies to our accelerating universe) the following relation holds del^2 Phi = rhogravitational mass = 4pi(u + 3p)/c2 where u = energy density and p = pressure. When the pressure has the value p = -u then rhogravitational mass < 0 which results in gravitational repulsion. One version of the equivalence principle states that all matter accelerates at a rate which is independant of its mass. This still holds for gravitational repulsion. Pete

"might not"? I guess so. But the same could be said of the inflationary universe model. As it is currently explained and understood it is very much a gravitational effect. Its description is postulated in two ways (1) by the presents of tension (negatgive pressure) or (2) by a positive cosmological constant. Alan Guth has this to say about it. From http://www.edge.org/documents/day/day_guth.html If it wasn't gravitational in nature then what other possibilities exist? It is gravity that drives the expansion of the universe is it not? After all the acceleration of matter which is independant of its mass is basically the definition of gravity itself. And the accelerating expansion of the galaxies apart from each other is clearly indendant of the mass of the galaxies. I disagree. That is a postulate based on the assumption that negative energy density. However mass density is what determines whether there is a gravitational repulsion and the two are not proportional. In the case of negative pressure scenrio its the negative pressure can overcome the positive energy density. That would lead to gravitational repulsion. I recall that the same arguement holds for a positive cosmological constant. I disagree here too for the reason stated above. What are you basing that on? Pete

That's news to me. The article which postulted their existance did so using both relativity and quantum mechanics. Well .. as you said ... I was being lazy. I went back and corrected them. Pete

I disagree. The optical path length is the same. The photons merely take rest stops along the way. To be precise, there is a finite time between photon absorption and photon emission. To be precise a photon's proper mass has never been measured to be exactly zero. There is only an upper bound on it. Similarly with its speed in that it has never been proven to be exactly the same value in all inertial frames because all measurements to date have a non-zero margin of error.

The total inertial mass of such a process is conserved even when the total rest mass is not. The inertial mass shows up in terms of things like kinetic energy (a photon's energy is considered to be all kinetic energy). And its inertial mass that equals gravitational mass, not rest mass. So there will be no change in total mass when a star is formed. Pete

I disagree with respect to given very heuristically. Something is called heuristic when it involves or servers as an aid to learning. The definition given by MTW is not of that nature. Perhaps you were expecting something more mathematical? If so then note that the definition given by MTW is defined in terms of mathematical objects. Quite often when something physically basic is defined it can't be given in terms of math since it is that which must be given in order to provide a mathematical description. I disagree here too. Falling off the spacetime is different than terminating. Something that terminates has a finite endpoint whereas something that falls off the spacetime might not be as such. What were you expecting in terms of a definition for it to be rigorous rather than what you call "inuitive"? Pete

Actually that's wrong. There is gravitational repulsion. Dark energy is one example. It is responsible for the inflationary model of the universe as well as the observed accelerating expantion of the universe. This only becomes apparent in Einstein's general theory of relativity where the cosmological constant is used to provide an equivalent negative gravitational mass density. In cosmology there is an object called a vacuum domain wall. Objects placed near the wall are gravitationally repelled by the wall. Pete

Okey dokey - From Gravitation by Misner, Thorne and Wheeler, page 934 Pete

Good question. I guess I'd have to say that its what Einstein's theory of relativity predicts. But it was 2000N, not 4000N. And you have to keep in mind that the scale is at rest in the planet-bound observer's frame. Consider this from another standpoint. Let us use the Equivalence Principle to analyze this. Consider a scale which is uniformly accelerating in the inertial frame S where the weighing surface has its normal vector in the direction of the scales acceleration. A body is sliding across the scales weighing surface. According to an observer in S who is momentarily at rest with respect to the rest frame of the scale the force on the body due to the scale will have the value F = dp/dt. Since the acceleration is perpendicular to motion it can be shown that F = mta where mt = the transverse mass of the particle. It so happens that mt = m = relativistic mass of particle. Therefore the force on the bodydue to the scale, which we call the weight of the body, is W = F = ma. Using the equivalence principle we then get W = mg where g = a = acceleration due to gravity and the field is a uniform gravitational field. Pete

I should also point out that Einstein defined the mass density of radiation to be proportional to the energy density. This was later seen to be an error in the general case. Pete

You're most welcome. Misner, Thorne and Wheeler define the term singularity in their text Gravitation. I'll post that definition later, perhaps tommorow. Pete

In SR there are no preferred frames. And I didn't say there were in GR. What I did said is that in GR one can at times choose a frame which is at rest with the source of gravity and that frame will have unique properties. I.e. that although there is no meaning for "the" rest frame one can at times (as in the example I gave) define such a frame which is unique, e.g. one in which the source of gravity is not moving. The analogy in EM would be a frame of reference at rest with respect to a static charge distribution. One can then say that they are at rest with respect to a unique frame of refererence. Unique does not mean absolute. As a concrete example consider a uniform gravitiational field in frame S'. In that frame there is no g_01 cross term in the metric and thus no frame-dragging. If the observer transforms to a frame of reference S' which is moving at constant speed in a direction which is perpendicular to the field then in that frame there will be a cross term g_0'1' in the metric and thus in that frame there will be a frame-dragging effect. That means that if you launched a particle straight up it would be inertially deflected by the field. This wouldn't happen in the frame S. Thus one can determine a rest frame which is different than a rest frame in S'. I didn't say that the mass of the planet didn't change. Sorry if you got that impression. I was using an example in which the observer was at rest with respect to the source of gravity. The mass does indeed change if the planet moves. Pete

No. Its the body that is moving, not the planet. However, if the planet was moving then g would change. Pete

Let us consult Wald's text on this topic. On page 214 Wald writes One needs to distinguish between the mathematical term and the physical term. The mathematical term for singularity is used to refer to points where a quantity goes to infinity as one approaches that point. A physical singularity is a mathematical singularity if it exists in all possible coordinate systems. If there is no physical singularity at a point then the singluarity is purely mathematical and is referred to as a "coordinate singularity." It seems that the expert isn't Wald. Pete Let us consult Wald's text on this topic. On page 214 Wald writes One needs to distinguish between the mathematical term and the physical term. The mathematical term for singularity is used to refer to points where a quantity goes to infinity as one approaches that point. A physical singularity is a mathematical singularity if it exists in all possible coordinate systems. If there is no physical singularity at a point then the singluarity is purely mathematical and is referred to as a "coordinate singularity." Pete

I disagree. The observer can tell if he is at rest in a gravitational field. If the observer is moving at constant speed perpedicular to the field lines (i.e. sliding along the ground) then there will be a frame-dragging effect in the frame which is moving which is not there in the rest frame. If the scale is at rest in the gravitational field then the scale will read 2000N because W = mg. W_0 = m_0*g, W = gamma*m_0*g = 2m_0*g = 2W_0 = 2000N. This means that a moving body weighs more than the same body at rest. Pete

That information is available in the article I referenced above. If you read the article you'd see that in the years 2000-2005 is the most recent data in Fig. 1 and it shows that 8 out of 12 relativity texts use it. The article The inertia of stress, Rodrigo Medina, Am. J. Phys. 74(11), November 2006 is more recent. The concept of the inertia of stress is one that comes from the concept or relativistic mass. The current definition of mass is given at Argonne National Laboratory http://www.neutron.anl.gov/hyper-physics/inertia.html and refers to relativistic mass rather than rest mass. Same thing with Lawrence Berkeley National Laboratory http://aether.lbl.gov/www/classes/p139/animation/sr.html and CERN http://teachers.web.cern.ch/teachers/archiv/HST2001/accelerators/teachers%20notes/cyclotron.htm Then there are articles such as this http://www2.fpm.wisc.edu/safety/Radiation/2000%20Manual/chapter12.pdf Also for the sake of example, I constructed a list of online university lecture notes on the use of rel-mass. Its on my web site at - http://www.geocities.com/physics_world/ref/relativistic_mass.htm. See University Lecture Notes - Online Examples Pete

I've done more research into this concept than most people so I'm sure I'm diffferent in that respect. I've come to learn more general applications of the concept of mass that most people. In fact I've come to learn that the concept of mass as rest mass cannot be made to apply in general and therefore cannot serve as a valid definition of mass in general. Also, I don't understand what it is that every physicist here disagrees with. What is it that they are disagreeing with? The number of texts/books etc. in which the concept is used? The number of journal articles which employ the notion? If so then take a look at figure 1 in the article I quoted. It shows that there are more relativity textbooks utilize the concept of relativistic mass than those that don't (with the exclusion of the years 1980-1984) First off let us recall exactly what I said. I.e. in response to the comment By mass we mean .. I wrote Do you beliece that is untrue? Now recall the statement I was questioning the use of the term seldom when I wrote Do you think the source I provided for this data is wrong? If so then why? If you have a source which gives other data I would like to read it. Thanks. Pete

Its called dark matter because it isn't luminous. It is detectable through its gravitational effects. Then it would be better in the future to say "physics" rather than "science" when it is physics that you are talking about. Until we can determine what all forms of dark energy are it cannot be said where they are. Dark matter might consist of elementary exotic particles or it may consist of objects like black holes, cold stars, large gas giants, brown dwarfs etc. Not yet. Unknown as of yet. The existance of galaxies doesn't depend on dark matter. It is the rotation curves which indicates its existance. The curves show that there is more matter there than can be accounted for my stars. Pete

I see no reason to assume that's true in all areas of science in all situations. Some areas of science are non-mathematical in its very nature. There may be certain experiments which are analyzed with statistics but that is for verification purposes and not for predictions. Pete

I assume you're joking? Pete

That depends on the definition. E.g. http://www.merriam-webster.com/dictionary/singularity In any case the singularity of a black hole is a point. Pete

Different observers will observe different things. Observers outside the black hole will observer the matter to approach the event horizon but never cross it. However these outside observers will eventually be unable to observe the matter due to an ever increasing red shift. An observer who is falling into the black hole will observe the matter to be torn apart by tidal forces. After that it gets nasty and eventually falls into the singularity and compressed into a point. Yuk! Pete

If you are moving perpendicular to a gravitational field then your weight would be W = mg = m0g/sqrt(1 - v2/c2) The derivation of this is found on my website here http://www.geocities.com/physics_world/gr/weight_moving_body.htm I derived this in two ways. The first one is heuristic and thus easy to follow. The second is a rigorous derivation using general relativity. Pete