- #71
Dale
Mentor
- 35,332
- 13,562
In a non-stationary spacetime the term "at rest" has no clear meaning.Anamitra said:The observer is at rest at the initial point of motion.He is trying to investigate the motion from a laboratory.
In a non-stationary spacetime the term "at rest" has no clear meaning.Anamitra said:The observer is at rest at the initial point of motion.He is trying to investigate the motion from a laboratory.
The spatial distance between a pair of spatial points may change due to changes in g(11),g(22) and g(33) with time. But the coordinate labels[spatial] should not change. The physical distance between a pair of laboratories[along some spatial curve] may change with time but each laboratory should stand on the same coordinate labels relating to x,y and z. The laboratory is at rest refers to these unchanging coordinate labels.DaleSpam said:In a non-stationary spacetime the term "at rest" has no clear meaning.
Anamitra said:The spatial distance between a pair of spatial points may change due to changes in g(11),g(22) and g(33) with time. But the coordinate labels[spatial] should not change. The physical distance between a pair of laboratories[along some spatial curve] may change with time but each laboratory should stand on the same coordinate labels relating to x,y and z. The laboratory is at rest refers to these unchanging coordinate labels.
This acknowledges the fact that the physical distance from some other spatial point[given by coordinate labels--x,y,z] may change with time.Calculation along some spatial curve is implied.
PAllen said:I will describe a possibly relevant confusion I had that was clarified by Bcrowell and Dalespam in another thread a couple of months ago.
It is simply wrong to attach any physical meaning to the coordinate labels of an *arbitrary* coordinate system. They could be called a,b,c,d instead of z,y,t,z. Further, it is wrong to assume that the basis vector along some coordinate label at some spacetime point has the same character (timelike, spacelike, null) as at a different spacetime point. Bcrowell provided an example of a frequently used coordinate system for SR that has two lightlike basis vectors and two spacelike basis vectors and *no* timelike basis vectors. Yet it is not only well defined, it was particularly favored by Dirac (I think).
JesseM said:Why do you say this path is "timelike"? In most coordinate systems typically used in physics (Minkowski coordinates, for example), if dt=0 along a path then the path is spacelike, not timelike.
Anamitra said:I would request the audience to consider the following link in regard of the general nature of the ongoing conversation::
https://www.physicsforums.com/showpost.php?p=3062372&postcount=59
Anamitra said:The time component of the four velocity[As suggested by PAllen] is zero. Further differentiation wrt to propertime[ds] yields time component of the momentum vector[multiplication by rest mass is required] which is again zero. It is zero energy particle![Assuming a particle is capable of moving along a timelike curve/path]
yuiop said:Note that defining timelike paths as ones that are contained within past and future light cones is a tautology, because the orientation of light cones depends on how you define timelike and spacelike in the first place.
In most situations a physicist would just use the coordinate speed I think, but there are other options, for example in cosmology the "velocity" in the Hubble formula is based on the rate at which proper distance (measured along a path confined to a single surface of simultaneity in cosmological coordinates) changes with coordinate time, and as I mentioned above "proper velocity" measures the rate at which coordinate distance changes with the moving object's own proper time. If you want a specific formula you have to specify what notion of "speed" you want to measure.Anamitra said:1) What formula should he use to calculate the speed or to define the speed of the particle at some point of the path?
What do you mean by "time interval"? The coordinate time in his frame? The particle's own proper time? In the case of proper time I already gave you the formula for that...Anamitra said:2) What formula should he use to calculate the time interval or to define the time interval of the particle for the entire motion?
That is just the point. The coordinate labels may very well change. Simply because a coordinate is labeled "t" does not imply that it is timelike. There are many such examples.Anamitra said:The spatial distance between a pair of spatial points may change due to changes in g(11),g(22) and g(33) with time. But the coordinate labels[spatial] should not change.
yuiop said:Could you make it clear what you mean by the "time component of the four velocity". Let us define the four velocity as:
[tex]\frac{c^2dt^2}{d\tau^2} - \frac{dx^2}{d\tau^2} - \frac{dy^2}{d\tau^2} - \frac{dz^2}{d\tau^2} = (x_0, x_1, x_2, x_3) [/tex]
This quantity is always equal to the speed of light squared.
I would normally take the time component to mean [itex]x_0[/itex]. Is that what you mean? When this component has the value zero, the four velocity is still the speed of light (squared). Further differentiation would give the four acceleration and a value of zero here just means that the particle is inertial with constant velocity, which does not by itself imply zero momentum or energy.
.
JesseM said:In most situations a physicist would just use the coordinate speed I think, but there are other options, for example in cosmology the "velocity" in the Hubble formula is based on the rate at which proper distance (measured along a path confined to a single surface of simultaneity in cosmological coordinates) changes with coordinate time, and as I mentioned above "proper velocity" measures the rate at which coordinate distance changes with the moving object's own proper time. If you want a specific formula you have to specify what notion of "speed" you want to measure.
JesseM said:What do you mean by "time interval"? The coordinate time in his frame? The particle's own proper time? In the case of proper time I already gave you the formula for that...
PAllen said:I think the speed of a particle measured by a 'scientist' can be given in a coordinate independent way. The particle has some 4 velocity at some event on its world line; assume the 'scientist' has some different 4 velocity at the same event. The scientist would define the time axis to be their 4 velocity; they could define a spacelike unit vector normal (in the spacetime sense) to their 4 velocity in the direction of the particle's spatial motion. The the particle's 4 velocity dot product this spatial vector, divided by the particle's 4 velocity dot the scientist's 4 velocity, would be the particle's speed as perceived by the scientist.
Note that even for as simple a case Swarzschild coordinates, you wouldn't directly use the coordinates to define intuitive speed of anything - two of them are angles..
JesseM said:Coordinate time is completely irrelevant, only proper time matters if you want to know what physical clocks are doing. If you think it somehow makes a difference that dt=0, I think you're misunderstanding something basic about the physical meaning of proper time (and the fact that it's independent of what coordinate system you choose--we could easily describe the exact same physical path in a different coordinate system where dt was not zero, and the proper time along the path would necessarily be exactly the same).
Anamitra said:In my system --the traditional [t,x,y,z]rectangular system I am getting the time component of the momentum four-velocity--as zero[in some particular situation]. Consequently energy works out to be zero.
If you get the value of energy as a non zero quantity in your frame/system of coordinates, how do you interpret the situation----as something physically different from what I am getting?
Anamitra said:In relation to my previous post:
Energy is a conserved quantity but not an invariant.But the rest energy part should not change.
Anamitra said:Let us first come back to the basic issue---whether there there can be an intercoversion between a spacelike path and a timelike/null path by the effect of gravity
[tex]{ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}[/tex]
This is a strange way to word it. A given spacetime path is, by definition, fixed in time and character. I think what you must mean is that a similar (e.g same coordinate slope) coordinate path in different regions of spacetime can be spacelike in one region and timelike in another. That is obviously true, and whenever it is true, a trivial coordinate transform can convert it to the simpler case of a coordinate axis having different character in different regions of spacetime.Anamitra said:Let us first come back to the basic issue---whether there there can be an intercoversion between a spacelike path and a timelike/null path by the effect of gravity[Due to changes in the values/expressions representing the metric coefficients in a time dependent field]
[tex]{ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}[/tex]
Correct. Change in coordinate system will never change an invariant like the spacelike/timelike character of a path.Anamitra said:The value of the metric coefficients can always change by the action of gravity---this is a known fact.The sign of ds^2 can also change[I am referring to such a possibility here] . By choosing a different coordinate system you can never change the physical consequences.
Of course. No one disagreed with this. From my point of view, I was simply proposing the simplest example of this.Anamitra said:Even if you consider some weird geometry with unusual signs for the metric coefficients[in case such an action is possible]or even if you apply any other type of contrivance, the huge number of cases[in relation to the interconversion] indicated in the last paragraph cannot be ruled out.
Anamitra said:The light cone mechanism has been clearly depicted in the following link:
https://www.physicsforums.com/showpost.php?p=3061481&postcount=38
Many other side issues have come up in this thread.I will definitely address them[and I have been addressing them]
The answer to that is very clearly and definitively, "no".Anamitra said:Let us first come back to the basic issue---whether there there can be an intercoversion between a spacelike path and a timelike/null path by the effect of gravity
If you let y1=y2=0 and z1=z2=0 then this is exactly the example I gave previously. Note that the spacelike and timelike paths are different paths because the t coordinate differs by a few minutes.Anamitra said:A simple illustration:
I am in a laboratory at A(x1,y1,z1) . Pallen and yuiop are at different one B(x2,y2,z2). A few minutes ago we had spacelike paths between the two labs. Now we have timelike paths thaks to the gravitational effects.
Anamitra said:Regarding Time dependence:I can always choose a frame of reference where the spatial coordinate labels [x,y,z] do not change with time for my inferences.The values/expressions for the metrics do change due to gravitational effects.PAllen and others can always choose a frame where the coordinates are changing. The physical nature of the conclusions should not change.
A simple illustration:
I am in a laboratory at A(x1,y1,z1) . Pallen and yuiop are at different one B(x2,y2,z2). A few minutes ago we had spacelike paths between the two labs. Now we have timelike paths thaks to the gravitational effects.
Would the physical nature of the conclusions change if by some suitable transformation the spatial coordinates are made to vary with time?
Anamitra said:Let us first come back to the basic issue---whether there there can be an intercoversion between a spacelike path and a timelike/null path by the effect of gravity[Due to changes in the values/expressions representing the metric coefficients in a time dependent field]
[tex]{ds}^{2}{=}{g}_{00}{dt}^{2}{-}{g}_{11}{dx1}^{2}{-}{g}_{22}{dx2}^{2}{-}{g}_{33}{dx3}^{2}[/tex]
A spacetime path remains spacetime in time independent gravitational fields. Their nature remains invariant wrt coordinate transformation[in all types of fields time dependent or independent].In my illustration the coordinate system is not being changed or transformed.We are considering changes in the metric coefficients in the same coordinate system(t,x,y,z). Conclusions should remain unchanged for all other frames once the change [in tne metric coefficients]has taken place.PAllen said:This is a strange way to word it. A given spacetime path is, by definition, fixed in time and character. I think what you must mean is that a similar (e.g same coordinate slope) coordinate path in different regions of spacetime can be spacelike in one region and timelike in another. That is obviously true, and whenever it is true, a trivial coordinate transform can convert it to the simpler case of a coordinate axis having different character in different regions of spacetime.
This does not come in the way of my argumentsPAllen said:Correct. Change in coordinate system will never change an invariant like the spacelike/timelike character of a path.
yuiop said:Let's try a particular example in Schwarzschild metric. Assume dx2 and dx3 are zero so we are considering the two dimensional radial case so:
[tex]ds^2 = (1-2M/r)dt^2 - (1-2M/r)^{-1} dr^2 [/tex]
For a timelike path, ds^2 is positive. Below the event horizon (say r=M) a path with dt=0 is a valid timelike path because:
[tex]ds^2 = - (1-2M/M)^{-1} dr^2 = +dr^2 [/tex]
Can we calculate a permissible velocity, momentum and energy of such a particle?
PAllen said:This is getting at your confusion. You can speak of a space *time* path being spacelike. What you mean si that the path between:
(t1,x1,y1,z1) and (t1,x2,y2,z2) maintaining t=t1 is spacelike,
while the path:
(t2,x1,y1,z1) and (t2,x2,y2,z2) maintaining t=t2 is timelike.
These are two completely independent paths through spacetime, and the situation implies, mostly, that the meaning of the coordinates *has* changed. There is, presumably, a family of spacelike paths connecting the world lines of the two labs at different proper time points along the world lines. The situation above simply means the the coordinate representation of these paths looks very different at different points along the worldlines.
Anamitra said:This part is absolutely OK. Thanks for that PAllen!
Now we are thinking of two spacetime paths A and B where B is a subset of A.t for initial point of A is greater than t for initial point of B. The part between initial point of A and initial point of B is time like and the rest is spacelike when the observer is at A.When the observer reaches the initial point of B he is amazed to find that the rest of the journey can be carried out since the path has become timelike!
The example in the following link is in tune with what you are saying!
https://www.physicsforums.com/showpost.php?p=3061386&postcount=36
Anamitra said:This part is absolutely OK. Thanks for that PAllen!
Now we are thinking of two spacetime paths A and B where B is a subset of A.t for initial point of A is greater than t for initial point of B. The part between initial point of A and initial point of B is time like and the rest is spacelike when the observer is at A.When the observer reaches the initial point of B he is amazed to find that the rest of the journey can be carried out since the remailing path has become timelike due to gravity!
The example in the following link is in tune with what you are saying!
https://www.physicsforums.com/showpost.php?p=3061386&postcount=36
Anamitra said:PAllen has clearly misread/misinterpreted the thought experiment described. Let me help him in getting the matter clarified.
WE consider a spacetime curve running from X[t1,x1,y1,z1] to Z[t3,x3,y3,z3] via Y[t2,x2,y2,z2]. The curve between X and Y is time like and the curve between Y and Z is spacelike . t1<t2
In the time t1 to t2 the observer reaches from X to Y along the timelike curve with the expectation that he will find a space like curve between Y and Z.[or may be observers at the spatial position of Y standing for a long time before his advent will inform him about the nature of the path ahead of him--and how it has changed]. The apparently unreachable spacetime point is now reachable!
Anamitra said:You can always take it in this way:Observers have been standing at the spatial position of y from time<t2.They knew very well that the spacetime point Z had a spacelike separation for time<t2.This information was transmitted to the observer at the spacetime point X at some suitable time<t2[or may be when the observer [initialy at X ] is on the way to Y!
The observer ,when he arrives at Y is amazed to find that the path ahead of him has become timelike!
Observers standing at the same spatial point have the same notion at time=t2
[Spacetime point Y is the same for all observers instantaneously,when the moving observer arrives there]