Is no one going to point out that what I’ve bolded above is pretty much the definition of not staying in sync? If you’ve typed them in correctly the difference between the two is 50% of the total gravitational time dilation for the potential at Earth’s surface, in other words, relatively speaking completely different.
Yep, if tomh were correct about the observed clock rate he would actually be proving the falsity of what he set out to prove. The problem is that when comapring inertial clocks the relationship between observed times is linear so Δt’ = γΔt where γ is a constant for a constant relative velocity.
The problem is though that when the original frame is inertial and the primed frame is non-inertial γ becomes a non-constant function of t, i.e. Δt’ = γ(t)Δt. Therefore γ can not be given as a single number as, say for example, a constant proper acceleration as it’s a function of t.
Of course, this still proves that he is incorrect in his original idea. Why is he incorrect? Becuase the relationship between a uniformly accelerated observer and inertial observer and the relationship between an observer held stationary in a static graviational field and an inertial observer at spatial conformal infinity is not the same. What differinates them is that in the latter case the spacetime between the two observers is curved. A better comparison for the former case would be the relationship between an observer being held stationary in a static gravitational field and an free-falling observer passing close by them.
The particular second we are talking about is by the reference clock described in Wikipedia “…tf is the coordinate time between events A and B for a fast-ticking observer at an arbitrarily large distance from the massive object (this assumes the fast-ticking observer is using Schwarzschild coordinates, a coordinate system where a clock at infinite distance from the massive sphere would tick at one second per second of coordinate time, while closer clocks would tick at less than that rate)…”
You are trying to muddy the waters in talking about “…t going to infinity…”, as that is not the scenario being discussed here. The formulae prove my original statement that clocks at 1G, whether accelerated or gravitational, run at the same rate to a high degree of precision.
The term for mass is :- m = m0 / sqrt( 1-(v/c)^2), and references can be found in chapter XV of Einstein’s book which I have already mentioned, and chapter 8 of Asimov, “The Stars In Their Courses” (and of course a myriad of others).
You’re still not understanding, I’m talking about the accelerated observer in Minkowski space (i.e. flat spacetime). The time dilation factor increases as t increases which means you cannot express it as a single number for acceleration = 1g as it is a function of time.
In comparision the proper time of a stationary observer in a static gravational field as compared to Schwarzchild coordinate time is constant with time and hence can be expressed as a single numebr (for a given r).
Is it that difficult to understand?
Look up “Lorentz transformation” on google. Do you see that equation as one of the equations that describes the Lorentz transformation? The answer is no. It’s a simple matter of fact/catergorization.
That’s not to say that the equation isn’t true when you define mass in a certain way of course, however as I’ve explained we don’t define mass in this way any more
Try reading this paper: http://www.stat.physik.uni-potsdam.de/~pikovsky/teaching/stud_seminar/einstein.pdf
It explores Eisntein’s understanding of the concept of mass (as well as some of the problems associated with the now defunct concept of relativstic mass)
::raises hand timidly from back of room::
Can gravity potential be likened, metaphorically, to potential (ie non kinetic) energy?
I am not comparing IFRs with non IFRs, I am concerned only with accelerated/gravitational FRs, and their relationship to the “standard” clock. The formula for gravitational time delay was known to me; what was new was the different formula for accelerated time dilation. I had assumed, correctly as it turns out (try running the figures yourself through the 2 formulae, and work to 8 decimal places), that under 1G, whether caused by gravity or acceleration, the clocks run at (to a high degree of precision) the same rate.
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You’re still not understanding, I’m talking about the accelerated observer in Minkowski space (i.e. flat spacetime). The time dilation factor increases as t increases which means you cannot express it as a single number for acceleration = 1g as it is a function of time.
In comparision the proper time of a stationary observer in a static gravational field as compared to Schwarzchild coordinate time is constant with time and hence can be expressed as a single numebr (for a given r).
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If Einstein’s equivalence principle is correct, you are wrong, if your quote above is correct, Einstein is wrong and there is no equivalence.
[quote=“tomh4040, post:106, topic:388750”]
I am not comparing IFRs with non IFRs, I am concerned only with accelerated/gravitational FRs, and their relationship to the “standard” clock. The formula for gravitational time delay was known to me; what was new was the different formula for accelerated time dilation. I had assumed, correctly as it turns out (try running the figures yourself through the 2 formulae, and work to 8 decimal places), that under 1G, whether caused by gravity or acceleration, the clocks run at (to a high degree of precision) the same rate.
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You’re still not understanding, I’m talking about the accelerated observer in Minkowski space (i.e. flat spacetime). The time dilation factor increases as t increases which means you cannot express it as a single number for acceleration = 1g as it is a function of time.
In comparision the proper time of a stationary observer in a static gravational field as compared to Schwarzchild coordinate time is constant with time and hence can be expressed as a single numebr (for a given r).
The paper I cited above has experimental results showing that (as Einstein predicted) two clocks experiencing the (almost precisely*) same gravity but at different potentials will have different dilations.
- the (per mass) gravity force are very slightly different - by a factor of 1 meter divided by the radius of the Earth cubed, but the difference in dilation is proportional to the difference in potential (proportional to g*dh)
Exactly what I asked for three times before (Yes three. Count them.), “an actual cite that gravitational time dilation depends on force”.
They are more authoritative than anything you’ve provided. It’s easy to sit back and say “Oh, they’re not authoritative enough”, without actually saying anything substantive about them, or providing any kind of cite yourself.
Perhaps you need a better text book on general relativity. I have Weinberg’s Gravitation and Cosmology. It has sections on time dilation for both special and general relativity. As we’ve seen, papers are written on the subject. Cites on time dilation in GR are available. It’s only cites that time dilation depends on force that don’t exist.
No, it’s not “arbitrary”. This is just bizarre.
I will try one last time. Here’s the time dilation for arbitrary motion in an arbitrary gravitational field from Weinberg:
dt/delta t = (-g[sub]uv[/sub] dx[sup]u[/sup]/dt dx[sup]v[/sup]/dt)[sup]-1[/sup] (Eq. (3.5.1))
No explicit force terms in sight. No second derivatives (the metric tensor is first derivatives), so no acceleration terms, so no force terms in hiding, expressed as acceleration. Thus, force does not contribute to gravitational time dilation.
QED.
I’m actually trying to argue its more complciated than that and that the very prescriptive statement you’ve made is not necessarily true. Note none of your cites actually support the prescritpive nature of your statement. Rather ask for cites for a psotion that I do not necessarily agree with (typing gravitional tiem dialtion and force in to Google will get you the cites you’ve asked for, but as I think these are equally if not more simplistic, there’s no point in me psoting them).
Rather than asking for cites, tackle my objections to the prescriptiveness of your statement directly. Posting cites that don’t actually disagree with what I’ve said doesn’t acheive this.
When dealing with the more theoretical side of the subject graviational time dialtion just isn’t that importnat,mainly for the reasons I describe below
Can you honestly not see what is wrong with what you’ve written?
Firstly, the fact that the equation uses the ordinary derivative should alert you as to its coordinate dependent nature.
Seocndly if what you’d said was actually true and there was a totally valid general method for non-arbitarily defining gravitational time dilation in an arbitary gravitational field, then the relationship between gravaitonal time dialtion and gravaitional potnetial must be very weak. This is as the gravaitional potnetial cannot be defined (satisfactorily) for an arbitary gravitational field.
The arbitariness in Weinberg’s equation lies in the fact that choices of the cooridinate basisare arbitary. If we had an exact prescription for deciding the components of the metric at an event then it wouldn’t be arbitary, but we don’t, so they are. Even if we limit the possible cooridnate bases by imposing restricitions so as to make them ‘physically sensible’ we can still end up with being able to define multiple different coorindate bases or even worst none at all! E.g. Schwarzchild cooridnates are only a cooridnate patch in Schwarzchild spacetime as opposed to a global coorinate system as they fail to cover the region of spacetime bounded by the event horizon. So you cannot use Schwarzchild compoents to define the time dilation factor of an object inside the event horizon (this may seem sensible, but on the other hand using a cooridnate system thta does cover the inside of the event horizon you could).
Consequently different chocies of the cooridnate bases in the above equation will result in different time dilation factors.
Of course gravaitonal time dialtion is also an empirical phenemona, but this comes from being able to compare a particular spacetime or region of spacetime with a flat sapcetime either because it is asymptoically flat or a region of it can be viewed as a peturbation of the flat metric.
If we’re assuming that we’ve defined the coordinate basis fields using physical prescriptions the ‘force’ (representing both gravaitional and four force) will appear in the connection of that coorinidate system and will help decide what the coordinate basis is.
As I’ve explained the don’t. Afetr a time you’d find the uniformly accelerated clock will appear to keep slowing down the more (inertial cooridnate) time passes, the clock in the graviational field will appear to run at slower, but constant rate.
Nope because Eisnetin’s equivalence principle only directly applies when comapring a free falling observer passing close by to the observer stationary in a gravtional field and only indriectly applies in the situation you describe.
Which is not what I disagree with, however if it were to truly be due to gravitational potential at the msot basic level, you’d have a hard time explaining why gravaitonal potnetial was only a relatively non-trivial concept only applicable in certain sitautions in the context of the theory of general relativity.
Einstein was certainly quite correct when he wrote:
It is a fully general result, derived from the Equivalence Principal. It applies to any coordinates. For any arbitrary coordinate system you like, there are no force terms. There’s the answer.
You can make all the hand-waving arguments you want, they don’t prove anything.
Smoke and mirrors. You are making incorrect assumptions, and extrapolating from those assumptions. “… a free falling observer passing close by to the observer in a gravitational field…” is pure fiction. It is not what Einstein said (WRT the man on Earth and the man in the chest). His observers and my observers are not in free fall, they are all experiencing 1G, and he made no mention of them being close.
More smoke and mirrors. You asked “Where in the equations representing the Lorentz transformation do you see a term for mass? Actually deriving so-called ‘mass increase’ from the Lorentz transformation is actually quite complicated. Also note that when you define mass in this manner you actually get two types of mass: transverse and longitudinal.”
You said that that their masses did not alter. L mass and T mass are old expressions which became relativistic mass, which definitely does alter with velocity when viewed from a different FR.
More smoke and mirrors. You said there was no Lorentz equation for mass increase, and yet in the next sentence said that deriving so-called mass increase fom the Lorentz equations is quite complicated. If there is no term for mass how can you derive it - complicated or not. Quote from Albert Einstein “It is not good to introduce the concept of the mass M = m/sqrt( 1-(v/c)^2)) of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the ’rest mass’ m.”
Quote from Wikepedia (which I assume that we all trust as we all quote it). “So, according to Lorentz’s theory no body can reach the speed of light because the mass becomes infinitely large at this velocity.”
Take the following two situations: 1) you are a stationary observer, while I am accelerating away from you at 1g; 2) you are an observer far away from the central mass in a Schwarzschild spacetime, while I stay put at a point where I experience a gravitational pull of 1g.
You seem to be saying that the equivalence principle implies that the time dilation between us in both cases is the same; but this isn’t so, as can easily be seen (and as These are my own pants has tried to point out). Obviously, in the second case, the dilation is constant, equal to t[sub]m[/sub] = t[sub]y[/sub]sqrt(1 - 2gr/c[sup]2[/sup]) = t[sub]y[/sub]sqrt(1 - 2GM/r*c[sup]2[/sup]), where t[sub]m[/sub] is my time, and t[sub]y[/sub] is yours – you observe my clock ticking slower by a fixed factor.
But in the first case, things are different – at any moment in time, my clock appears to you slowed by the factor sqrt(1 - v[sup]2[/sup]/c[sup]2[/sup]). However, I am accelerating, and thus, my speed is not constant – it increases. Thus, the factor by which you observe my clock ticking slower is not constant, either – it increases, as well.
The two situations thus are quite different, even though in both cases, I experience an acceleration of 1g; thus, it doesn’t suffice to state the acceleration to work out the time dilation.
Nevertheless, this situation is quite consistent with the equivalence principle. The reason is that it is only valid locally – I either in my spaceship or hovering in the Schwarzschild spacetime can’t tell the difference.
While it’s an intuitive concept, this is a bad explanation – it would, for instance, imply that a massless body can move at any speed, since its ‘mass’ doesn’t increase. As has been pointed out, the modern understanding of special relativity typically eschews the concept of relativistic mass; unlike invariant mass, it’s a concept that has no general validity, since you can make it have arbitrary values just by jumping to reference frames with different relative speeds, while invariant mass is the same in all frames of reference.
Perhaps a better, if also flawed, intuitive explanation is that you move at the speed of light all the time (the magnitude of the four-velocity is c), just that the bulk of this movement is through time; thus, as you move through space faster and faster, you move through time slower and slower, and going arbitrarily close to c means basically not moving through time at all.
I think your missing the point here. Coorindinate systems are generally speaking, arbitary and they don’t necessarily represent anything physical. Quite often we’ll start off with an observer and extend their local coorindate system outwards, but there’s multiple ways of extending an observer’s local coordinate system.
The actual force terms appear in the coordinate system itself, for example non-vanishing Christoffel symbols at an event will tell you at that point there’s an equivalnce between the coordinate basis and an accelerated observer’s local cooridnates.
As I pointed out earlier the equation that you posted results in a non-tensor quantity and so depends purely on the chocie of cooridnate systems and nothing else.
Blah blah blah. I’m done here.
…and with that departure, the tone of civility returns to polite.
I’m not familiar with situations where potential doesn’t apply (I’m more familiar with analyzing physical situations by treating potential energy as fundamental and forgetting about forces, frankly), but I’d like to hear more.
It’s been years since I studied general relativity, but I was introduced to the concept via examples that did not include forces, but did include potential (the elevator example for example), and while I find papers such as this one in Nature A precision measurement of the gravitational redshift by the interference of matter waves | Nature (which states “One of the central predictions of metric theories of gravity, such as general relativity, is that a clock in a gravitational potential U will run more slowly by a factor of 1 + U/c2, where c is the velocity of light, as compared to a similar clock outside the potential”) about the use of potential in gravitational time dilation calculations, I haven’t found anything helpful about force. Please give me some more information. Thanks.
I should say that I do understand that there is a problem with the setting of a zero for potential energy in GR (unlike in Newtonian gravity where the zero can be set anywhere), but I don’t think this problem doesn’t affect calculations of differences in potential, which is the quantity that affects gravitational potential energy.