The metric tensor of general relativity reduces to the metric tensor of special relativity in the absence of gravity. Therefore, both possibilities may exists.
Yes, They coincide locally unless it's a true singularity then nothing is defined there. Locally to go from one frame to another you need a Lorentz transformation which conserve the Lorentz metric i.e the proper time. Globally however, no single coordinate system can permit such transformations: an Atlas is needed which is a collection of charts that must be compatible in the language of differential geometry and which cover a limited portion of the manifold. transforming from one chart to another a Jacobian determinant appears which changes the definition of proper time thus no proper "proper time" can be globally defined.
Dear Barzi,
Thank you for your reply. I want to explain my question. In special relativity, we may define the proper time interval and then we may determine the proper time after integration. In general relativity, we may define the proper time interval only and not the proper time because of the variable components of the metric tensor. Now the question is whether the proper time interval of special relativity is same as the proper time interval of the general relativity? We may have always a coordinates system in general relativity such that the variable components of the metric tensor reduce to the Minkowskian metric tensor in the absence of gravity. In this limit one may think of the reduction of the proper time interval of the general relativity to the proper time interval of the special relativity but if one already assumes that the proper time interval of the general relativity is same as the proper time interval of special relativity no effect of the limit may be seen. I hope that it is sufficient to explain my question but, please feel free to put your views.
Regards
Yes, I think I see what you mean Sir. Well they are the same since without taking the limit you can always diagonalize the metric tensor at a point and by a suitable choice of units get the Minkowskian metric. But your question is more philosophical: which is more fundamental is it the local behavior of the metric or the global one? They way Einstein did it is by assuming they are the same and allowing reference frames to have any relative motion.
Dear Barzi,
Thank you for your reply.I understand your view.Unfortunately,I am unable to write the mathematical expression otherwise, my question of discussion would be more transparent.Any way, please tell me is there any other fault if we assume that the proper time interval of general relativity is different from the proper time interval of the special relativity.
Regards
Good question. Δτ =(1/c) ∫P sqrt(gμν dxμ dxν) where P is a time like world line and the integral is to be interpreted as a line integral. In general relativity gμν is a function of the space time location, choosing a coordinate system where x1,x2,x3 are treated as constants we can write down, Δτ = ∫P sqrt(g00)/c dx0.
This is the generalization of the expression Δτ = ∫ ds/c because sqrt(g00) is a scale factor for curvilinear coordinate systems. When gravity is absent we get back flat space-time; sqrt(g00) is just 1, then the two expressions for proper time intervals are identical.
They are the same when gravity Is absent, but differ when gravity is considered.
Dear J.Decker
Thanks for your reply. This is what I wanted to confirm.In fact I want to derive the equations of motion of particle moving in gravitational field with respected to the proper time of special relativity.Secondly, the most interesting part is, that the proper time of special relativity will automatically comes in the solution of differential equations of motion as the complementary function.Unfortunately,I cannot write the mathematical expression otherwise it would be more easy to explain.
Regards
Well, whatever the metric, the proper time interval is invariant under local reparametrization transformations. However, while it is invariant under global Lorentz transformations, when the metric is flat, it's invariant under local Lorentz transformations, when the metric isn't. This is standard material, cf. for instance, http://www.damtp.cam.ac.uk/user/tong/string/one.pdf
and http://www.damtp.cam.ac.uk/user/tong/string.html for the full course.
Acceleration is equivalent to gravity . Special Relativity is concerned with fixed speed moving references while general relativity deals with accelerated references. In the absence of acceleration, general relativity is transformed into dealing with static references. It means a four dimensional Euclidian space.
No, absence of acceleration does NOT mean four-dimensional Euclidian space. Whether in special or general relativity, spacetime has Lorentzian signature.
Dear Amrit Sorli,
Thank you for your for views about the non existence of proper time and coordinate time.I think there must be an object in your theory which replaces the time.For example,the forth component of vector may replace the time coordinates.But, in Newtonian relativity perhaps it would be difficult to replace the time.
Regards
Coordinate time is dependent on the reference frame and thus not a physical quantity, in special relativity, or general relativity; proper time is Lorentz invariant and, thus, a physical quantity. In general relativity it's not invariant under general coordinate transformations and, thus, not a physical quantity, either All this is known, e.g. http://www.damtp.cam.ac.uk/user/tong/string/one.pdf
Fundamental questions deserve a fundamental answer. A short answer is NO. The reasoning behind it is treated in the link below.
https://en.wikiversity.org/wiki/Hilbert_Book_Model_Project
“A physical quantity is defined by the series of operations and calculations of which it is the result” - A S Eddington
Accepting that, we can say that the physical quantity we call “time” is defined as “that which is measured by a clock”.
Suppose we measure the time elapsed between an event A and an event B, by transporting a clock from A to B. In Newtonian physics it is assumed that the elapsed time will be independent of how the clock is transported from A to B (ie, independent the route taken by the clock - independent of the clock’s trajectory). In Einsteinian physics (Special or General Relativity - makes no conceptual difference) that assumption is not valid. The time elapsed is not merely a property of the pair of events A and B, it is a property of the chosen trajectory between A and B, and is calculable. It is the “proper time” of the trajectory. It is independent of the choice of coordinate system; it is an invariant.
Sir,
My question is about the proper timeintervals and not about the proper time.That is, I want to know the difference between the proper time interval of the special relativity and
Sorry for the incomplete answer.I want to know only the difference between the proper time interval of the special relativity and the proper time interval of the General relativity.
Regards
Hello Ramesh,
I have been discussing some of this question in another thread,
https://www.researchgate.net/post/Is_the_Minkowski_metric_valid_since_it_cannot_explain_Mercurys_perihelion_advance_nor_the_bending_of_light
I don't know if I can answer your question fully and generally, but the difference between the proper time interval for special relativity and Schwarzschild relativity is the presence of two (not equal to one) coefficients in the Schwarzschild expression for the space-time interval.
Thee two coefficients are (1-r_s/r) and (1-r_s/r)^{-1}, and one of them goes in front of the time-component of the differential path element, dt^2, and the other goes in front of the radial component of the differential path element, dr^2.
One of these has the effect of making clocks deeper in the gravitational well go slower. And the other has the effect, (this is my current impression) that it makes the local radial speed of light equal to the universal constant speed of light.
Intro to the Schwarzschild Metric
https://www.youtube.com/edit?o=U&video_id=DMZ4cWyIUhg
Schwarzschild Derivation
https://www.youtube.com/edit?o=U&video_id=lvIOjkQ8N3o
(By the way, I guess the full derivation isn't here. I start to get to a major point in the last two minutes, and was planning to come back to it later, but it looks like I never got around to coming back to it later.)
Once more: the proper time is the integral of the quantity
-mc(g_μν(dx^μ/dλ)(dx^ν/dλ))^(1/2)dλ
over the values of the parameter λ, that parametrizes the trajectory and g_μν is the spacetime metric. This is the proper time of a massive particle (of mass m) and c is the velocity of light.
For special relativity the metric is the Minkowski metric, for general relativity the metric is a solution of Einstein equations different from Minkowski; e.g. Schwarzschild.
All properties of this quantity follow from its mathematical properties, that are the subject of courses, linked to in previous messages.
A difference is that, in special relativity the proper time is invariant under global Lorentz transformations; whereas it is does not have this property in general relativity.
Stam,
Do you know (Except perhaps for the curious presence of m in there, ) both minkowski and Schwarzschild metrics can be expressed in that form?
All one needs to do is set the 16values of g\mu\nu according to the metric of interest.
Also, I wonder if I could hear the reasoning behind your claim that proper time varies under a global Lorentz transformation? I have heard similar claims many times but they always seem to be based on circular or a priori assertion.
To my knowledge Global Lorentz transformation only affects coordinate of what happens. But it doesn't change what happened. Proper time is representing the reading on a clock which cannot change just because observers' locations or velocities are different.
Proper time is given by the expression I wrote for *any* metric, not just Minkowski or Schwarzschild. Its expression is *invariant* under global Lorentz transformations, if the metric is flat, i.e. Minkowski-it suffices to compute its variation. (It helps to actually read what's written!). This is a standard exercise, whose background is provided in the lecture notes linked to.
``A reading on a clock'' is a meaningless expression, by itself.
The equations are different we all know the equation for STR.
And herein the true equations for gravitational time dilation.
https://www.researchgate.net/publication/287213872_Finite_Gravitational_Time_Dilation_In_Black_Holes_Using_Dynamic_Newtonian_Advanced_Gravity_DNAg
Article Finite Gravitational Time Dilation In Black Holes Using Dyna...
Stam,
Why do you say "the reading on a clock, by itself" is meaningless?
Wouldn't you agree that the reading on a clock, generally means "the time that has passed since noon or midnight"?
Better: the amount of time a clock has recorded
It's meaningless, because it depends on the reference frame. It makes sense, without qualification, only in the Newtonian approximation, where time is independent of the reference frame. In special and general relativity this isn't the case.
Only the quantities, that are invariant under the transformations that are relevant, matter (Lorentz invariant quantities for special relativity, general coordinate invariant quantities for general relativity). All other quantities simply are used to construct the invariants. They don't mean anything as such. And this is nothing new, either.
How does the speed of a clock depend on reference frame?
Do you mean that a human choice in reference frame can affect the speed of a clock?
Like the choice in units can affect the numeric value...
Or do you mean that the use of a coordinate system that doesn't consider effects of gravity and motion of the clock will give the wrong answer?
I would agree that different methods could give different units or incorrect predictions but I don't think that makes the quantity meaningless.
The *time* the clock indicates, depends on the frame-in special and general relativity.
So, yes, the choice of a reference frame does affect the clock-but this is a tautology. And this effect isn't meaningful, by itself, because it's a choice of the time coordinate, that's all.
Once more: only invariant quantities have physical meaning. The time coordinate isn't one of them, in special and general relativity, any more than any individual coordinate is. These quantities can define invariant quantities and the latter, only, are physical.
In special relativity, all calculations of invariant quantities give the same answer-and can give different answers for non-invariant quantities.
In general relativity, the invariant quantities are different, that's all.
Comparing quantities that aren't invariant is meaningless. So, whether in special or general relativity, it doesn't make sense to compare the values of individual coordinates in different frames.
And it doesn't make sense to mix up a spacetime description, which simply implies that one time-like direction can always be defined, with a description in which space and time are treated separately and the choice of the time-like direction is fixed.
In the Newtonian approximation the time coordinate is invariant-so, in that approximation, it does make sense to compare clock readings, precisely because they're the same in all frames. This isn't true beyond the Newtonian approximation, however. It's an artifact of the approximation.
SN: Once more: only invariant quantities have physical meaning. The time coordinate isn't one of them, in special and general relativity, any more than any individual coordinate is. These quantities can define invariant quantities and the latter, only, are physical.
But the time on a clock *is* an invariant quantity. If I see the clock read (tick) 12:00 , and then later it read (tick) 12:01, every other observer that sees those two events will agree that the clock has recorded 1 minute of time passed. The clock "ticking 12:00" and clock "ticking 12:01" are not coordinates, They are events. And the measure of the clock between those two events is an invariant. People might disagree on the coordinates of those two events, but everyone will agree on the time the clock indicates between those events.
No, only Newtonian observers will. Events take place in spacetime, in special and general relativity, not, just, time. So providing the time coordinates of the events isn't sufficient to define events.
The ticks of the clock label the time axis of one observer-but this isn't the time axis of all observers, unless one is working in the Newtonian approximation.
In addition, Lorentz transformations don't preserve time intervals; they preserve spacetime intervals. It suffices to do the calculation.
The ``proper time'', mentioned here, is the spacetime interval between a certain class of events, in a given reference frame (that's what ``proper'' means). That it's a spacetime interval implies that it's invariant under global Lorentz transformations-so it has the same value in all inertial frames, that, by construction, are related by global Lorentz transformations. It can, therefore, distinguish frames that are not related by such transformations-that's what this means. That it's a ``time'' refers to the fact that the interval between the events in question is time-like, a notion that makes sense in special and general relativity.
No, only Newtonian observers will.
I think what you mean, here, is that if Newtonian principles of time held in the real universe, then all observers in the universe would find that they "agreed" with the time on any given well-behaved clock. But in a universe where Einsteinian principles of time hold (e.g. the real universe) some observers may disagree that a given clock has correctly measured the amount of time. While the clock has gone from 1pm to 2pm, some observers might say "only 10 minutes has passed for me, while that clock reads 1 hour has passed" while another might say, "Fully 10 hours has passed for me, while that clock has read only 1 hour." So the coordinate time measured by different observers differs. However, there is also an invariant quantity here. All observers in the scenario saw the clock go from 1pm to 2pm. And all would agree that 1 hour has been measured by that clock.
The ``proper time'', mentioned here, is the spacetime interval between a certain class of events,
Consecutive ticks of a given clock (e.g., the clock ticking 1pm, and the clock ticking 2pm) are *in* that class of events.
That it's a spacetime interval implies that it's invariant under global Lorentz transformations
And the proper time between any two events that happen to a real clock would be invariant under global Lorentz Transformations, of course. You will never find a reference frame where the clock does not self-measure 1 hour between its own ticking of 1pm and its own ticking of 2pm. Just as you would never find a 12 inch ruler that did not self-measure itself as 12 inches.
The self-measure of a clock's own time, and the self-measure of a ruler's own length are invariant.
Certainly you can give proper-time between events where there is no clock traveling between them, and you can give proper-distance between events where there are no rulers extending between the events. But the self-measure of time by a clock is an *example* of proper time, and the self-measure of length by a ruler is an *example* of proper distance.
Once more-No. All the statements make sense in one reference frame and no sense in any other, unless it's in the Newtonian approximation, at best. Beyond it they don't, because they don't refer to invariant quantities. That's what matters. So insisting that combinations of time-or space-coordinates alone remain invariant under Lorentz transformations is, simply, wrong. So there's no point in claiming otherwise.
One needs to specify space and time coordinates to define events, but only the spacetime interval, a particular combination of space and time coordinates, makes any sense. Specifying only time or only space coordinates doesn't specify events. But one can use infinitely many combinations of space and time coordinates to specify events. Those that yield the same values for Lorentz invariant combinations of the coordinates are equivalent-or describe the same event. Those that yield different values describe different events. If they yield the same or different values for non-invariant quantities is irrelevant.
In general relativity even that's not enough, since the spacetime interval isn't invariant under general coordinate transformations-these describe what general relativity means. That's the difference between special and general relativity.
And all these statements are the expression of the mathematical properties of the integral of
-mc(gμν(x)(dxμ/dλ)(dxν/dλ)1/2
over the-arbitrary-parametrization of the worldline, xμ(λ). Here μ=0,1,2,3. One of them is time-like and the other three are space-like.
SN: insisting that combinations of time-or space-coordinates alone remain invariant under Lorentz transformations is, simply, wrong. So there's no point in claiming otherwise.
I never claimed that, but I agree it would be incorrect.
SN: One needs to specify space and time coordinates to define events
That is incorrect. I can say that "My clock struck one" without describing the location.
SN: Specifying only time or only space coordinates doesn't specify events.
That is correct. What specifies an event is a description of an interaction, such as ticking of a clock, or emission, reflection, perception of a photon, or a collision of particles.
SN: But one can use infinitely many combinations of space and time coordinates to specify events.
Nuance: One can use infinitely many combinations of space and time coordinates to specify the location of events. But the event is not defined by its location, which is dependent on observer, and the observer's choices of description. The event is defined by what actually (happened/might happen) at that location, which will be described the same way by all observers. e.g. "Bob picked up a can of green beans at the grocery store"
SN: Those that yield the same values for Lorentz invariant combinations of the coordinates are equivalent-or describe the same event.
I can't understand this sentence, but what WOULD describe the same event is if you are describing the same Bob, the same can of green beans, and the same grocery store shelf, and the same event of Bob picking up that can of green beans from that grocery store shelf. What makes two descriptions describe the same event is when they describe the same event.
SN: Those that yield different values describe different events.
If I say "Bob picked up a can of green beans at a grocery store two blocks east of here" and someone five blocks to my east says "Bob picked up a can of green beans at a grocery store three blocks west of here," we are still talking about the same event. It does not become a different event because we have defined "here" differently, or given the event different numeric labels for its coordinates.
SN: In general relativity even that's not enough, since the spacetime interval isn't invariant under general coordinate transformations-these describe what general relativity means.
This is more a question of "parallel transport" rather than "coordinate transformation". The spacetime interval between two events could be different if you follow two different paths between those two events. However, for either of the given paths, the proper time measured along that path is invariant under coordinate transformation.
SN: mc(gμν(x)(dxμ/dλ)(dxν/dλ)1/2
I don't disagree, exactly. However, I am confused by the presence of the "m" in the expression, for proper time.
But more importantly, I am also wondering whether you would agree with me that there are essentially 20 "blanks" in that equation. Sixteen of the blanks are represented by the 4x4 "metric" gμν with its sixteen undetermined coefficients.
The other four blanks are the parametric expression for the path xμ(λ). You have called this path "arbitrary".
If you mean "based on random choice or personal whim, rather than any reason or system" then I would ask, do you think you HAVE to select a random path, without any reason, or are you also allowed to choose the path of a clock.
My premise here is that you ARE allowed to choose the path of a clock to define xμ(λ). You are not required to base your choice "randomly, without any reason or system."
On the other hand, if you MUST choose a path randomly, what prevents you from choosing paths that start out timelike, go spacelike, and then go backwards in time, and then back forward again? My own opinion is that the choice of such paths should generally be made with a reason in mind.
“Proper time” is the integral of ds along a “worldine” (a "timelike" curve in a pseudo-Riemannian spacetime) connecting two “events”, where ds2 = gijdxidxj.
It is the elapsed time measured by a clock whose trajectory between the two events is described by the worldline.
It is an invariant – independent of the choice of coordinate system.
Whether we are dealing with GR or SR is irrelevant.
Geometrically, the only distinction between SR and GR is that in SR the curvature tensor is everywhere zero − the spacetime is “flat” (ie, Minkowskian) and we can then, for convenience, consider global coordinate systems for which gij = ηij = diag (1, −1, −1, −1) everywhere. But we don’t have to! In GR, of course, we cannot. The definition of “proper time” and its invariance under general coordinate transformations are unaffected.
Dear Sir E. Lord,
Thank you very much for your explanation about the proper time.Here,I want to put my views for your comments if it is wrong.
1-The proper time of the special rel . is different from the proper time of general rel.due to the presence of the grav. field in gen.rel .In the limit of the vanishing grav.field,the proper time of the gen.rel.reduces to the proper time of the special rel.
2-The proper time interval of gen. rel.may be determined completely whereas, the proper time of gen.rel.may be determined approximately.
3-Even,in co moving coordinates system of gen .rel. (e.g. the R.W.metric)the proper time of gen.rel. is different from the proper time of special rel.
Regard
Ramesh Chandra
Sir,
There is a mistake in the statement 2.It is as follows:
The proper time of special rel. is determined completely where as the proper time of gen.rel.may be determined approximately.
Regards
Ramesh Chandra
What proper time means depends on your model.
In some models universe proceeds with proper time steps. Coordinate time is a mixture of space and proper time. (It is not pure progression)
https://en.wikiversity.org/wiki/Hilbert_Book_Model_Project
Dear Ramesh ~
SR is a special case of GR. It isn’t a different (competing) theory.
If the metric field and the equations of a worldline are given, then the proper time of a worldline can in principle be determined completely. It’s just a matter of evaluating an integral. In practice, in complicated situations, one may have to resort to approximation methods.
That applies whether we are considering a GR problem or an SR problem. There is no fundamental distinction. I don’t see why you would say “The proper time of special rel. is determined completely whereas the proper time of gen.rel.may be determined approximately.”
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