The specific formulae you quoted were for circular motion on an equipotential shell, I’m just pointing out you could easily replace the |v| term with an |F| term. Obviously the coordinate velcoity of a particle very much depends on the four force that it feels.
Delta tau would also be negligible.
I’m playing Devil’s advocate here somewhat. I’m saying it is too simplistic to view graviational time dilation as due to gravitational potential only and there are alternative ways of looking at it. The particualr formula is actually only a linearized approximation describing the low gravity situatiom. If you like the ‘force term’ appears in both the potential and the term representing the movement of the particle through the potential.
I’ll go back to my original point which is that 1) both grvaiational force nor gravitaional potential are merely emergent properties in general relativity, there are many situations were it makes no sense to talk about either. 2) the proper time experinced by an observer between two events, depends on their worldine between the two events, which itself is dependent on the spacetime metric and the four force they are subjected to along their worldine. 3) time dialtion is an effect of comparing clocks which in itself can be fairly arbitary.
Yes, because that’s what we’re talking about here. Looking at how poor tomh4040’s understanding of the situation is, do you really think bringing in the full exact GR solution would be helpful? I don’t.
In fact, I’d say your whole “Devil’s advocate” stance has been more obfuscatory than helpful to anyone. Are you just picking nits here, or do you really believe tomh4040’s initial statement that “two clocks which are experiencing 1G run at the same rate, and therefore stay in sync.”?
Provide an actual cite that gravitational time dilation depends on force.
No exact solution for the n-body problem is known and it’s doubtful that one even exists. However we must be aware of the limitations that these approximations bring. This particular approximation is for only when we are cofrtably enoguh inside the Newtonian limit for the Newtonian potential to be of use.
There’s actually a connection between the graviational force and the graviational potential, i.e. one is the gradient of the other. I may be wrong, as there may be a more adavnced method that I’m not aware of, but AFAIK the only generally used way to define graviational potential in general relativity is in terms of the force required to hold an object outside of a ‘matter containing sphere’ ‘at rest’ in a stationary asympotically flat spacetime.
No I am not agreeing with tomh, I’m just poitning out that actually it’s not quite as clear cut and straightforward as you think it is.
Where did this come from? This has nothing to do with this thread.
Which is what we’re talking about here. The time dilation due to gravity in the weak field limit.
Yes.
You can integrate the force from infinity (in the weak field limit, which, again, is what we’re talking about here). But this means you can have the same gravitational force at different gravitational potentials, and you can have different gravitational force at the same gravitational potential. Which is why it is important to understand that gravitational time delay is due to gravitational poitential, and not due to gravitational force. As the links I have provided all say.
And as all the links I’ve provided agree with. Provide an actual cite that gravitational time dilation depends on force.
"This is what I have been saying all along; please retract your statement that my assumption was wrong and yours was correct. The formula pointed to by “These are my own pants” is for gravitational time dilation for an observer held ‘stationery’ in the vacuum of a spherical gravitational field.
That link confirms my statement.
M & r can be arranged to bring the answer to a set value: ie for a given mass and a given distance (which then gives the same acceleration which I will arbitrarily set to 1G) time is in sync.
In other words, time on Earth runs at the same time as on the rocket undergoing an acceleration of 1G (see again my original example).
The part I disagree with is where you continue on to say that particles at the same gravitational force but at different gravitational potentials will have the same time dilation.
As I said above, I’m done trying to explain anything to you. But that’s OK, because These are my own pants has graciously volunteered to explain to you why time on Earth and on a rocket do not necessarily flow at the same rate. These are my own pants, you’re up.
repeating yourself doesnt make you more right, in this case it just makes you wrong over and over.
gravity does indeed have a dilating effect on time, but so does speed. so in your ever accelerating spaceship eventually its going to hit c at which point you will stop accelerating so you flip over and slow back down using the same 1g deceleration.
we promise space ship dudes clock will not even be close to the one back on earth.
I’m poitning out that that the fully general relatvistic version of the metric from which the equation in the wikipedia article would be a fully general relatvistic treatment of the n-body problem.
Actually asympotic flatness rather than using the weak field approximation is what allows us to ‘intergrate the force from infinity’ (thoguh asympotic flatness may bery well be a requirement of the weak field approximation, I don’t know, it’s not obvious if it is).
I’d also like to note that our starting off point for the formula for gravaitional time dilation was the Schwarzchild metric, where gravitational force and graviational potential are simply related to each other by M (i.e. we start off by working in a metric where knowing the graviational force allows us to calculate the graviational potential)
Alternatively we could view graviational time dilation as a result of the ‘weighted sum’ of gravitational forces acting on our synchronization signal.
One thing to note about using the gravitational potential (when it can be defined) is that it’s not a local property of the spacetime, it’s a global property. Two patches of spacetime could be equivalent (i.e. diffeomorphic), but have different gravitational potentials due to the difference in the spacetimes that they were embedded in. It’s possible to calculate a gravaitonal time dilation between two observes withotu even knowing what the gravitational potential is.
Provide a cite that it doesn’t. I’m not saying in this situation that it is not better to work in terms of potentials only. What I am saying is that to say that gravaiational time dilation depends on graviational potential and not gravaitional force is simply a too prescriptive statement
Do not think I necessarily agree with what tomh says, but actually what he says would only make sense if the proper acceleration was g and there’s no theoretical reason that you cannot have a proper acceleration of g indefinitely. what happens is that you asympotically approach c (in an inertial reference frame).
You’ve provided cites to say it depends on gravaitional potential. I’m saying that there’s alternative ways of looking at it. If view gravaiotional time dilation as a coming result of gravitational potential and graviational potential only how do you deal with the fact that in general relatvity graviational potential is a concept that only emerges and becomes useful when the spacetime itself has several properties.
You could argue legitmately that when gravaitional potential energy becomes harder to define/less useful than the concept of graviational time dilation also becomes harder to define and less useful. However viewing the graviational potential as the basic property behind gravaitional time dialtion belies the fact that it isn’t a basic property in general relativity.
I think you’re going to be a little bit more specific about what you want cites and equations for. I don’t disagree with the 3 cites you’ve cited, but two wikipedia articles (and may I point out the aerticle on gravaitional time dilation actually makes at least one very contentious statement on the equivalnce principle) and an amateur paper don’t in my mind consitute authortative cites. Part of the problem with providing cites is that gravaitional time dilation isn’t that big an issue in general relativity. I’ve got a 500 page text book on general relatvity next to me that doesn’t mention it once.
Basically speaking gravaitional time dialtion is a cooridnate artifact and so somewhat arbitary. You could argue against me in the weak field limit where the metric can be approximated as the Minkowski metric + some small petrubation, it does have physical significance as there’s a clear way of comparing the physics with the physics of Minkowski spacetime.
The reason I think force shouldn’t be ruled out is that four force ‘rotates’ four vectors and curvature rotates parallel transports of four vectors and its this rotation of four velocity and it’s parallel transports that can be seen as both due to ‘force’ and as responsible for the effect of graviational time dilation.
Thisa brings us to the second point that I am making. All experiments showing mass increase with velocity have been done by using an external force in a different FR, ie magnetism, to push a particle around a track. You cannot push something faster than the external push force you are using. In my example, the driving force, ie rocket motor, is in the same FR as the rocket ship, so using the Lorentz transformations for mass increase, v = 0, and the mass does not increase. See my original posting.
Mass is usually defined to be a Lorentz scalar (i.e. it’s value does not change under a Lorentz transformation). As I said earlier an object with a constant proper acceleration (the ‘g’ in the Wikipedia article on time dilation) will asympotically appraoch (i.e. get ever closer, but never reach) c from the point of view of an inertial observer.
This paper http://tf.boulder.nist.gov/general/pdf/2447.pdf contains the results of an experiment that measured gravitational time dilation with optical clock. The abstract states “Observers in relative motion or at different gravitational potentials measure disparate clock
rates. These predictions of relativity have previously been observed with atomic clocks at high
velocities and with large changes in elevation. We observed time dilation from relative speeds of less than 10 meters per second by comparing two optical atomic clocks connected by a 75-meter length of optical fiber. We can now also detect time dilation due to a change in height near Earth’s surface of less than 1 meter.”
It looks like I have got 2 points to answer on this posting.
Using the formulae on Wikipedia pointed to by “these are my own pants”, the time passage on the rocket accelerating at 1G is 0.9999999989802 of the clock which is well away from any gravitational effect and ticks at 1 second per second.
The time passage on Earth is 0.9999999993045 of this same clock. Therefore the 2 clocks stay in sync to within 2 milliseconds per year.
“zenbeam” said that the Earth and the rocket did not stay in sync.
“these are my own pants” stated that mass does not change
[quote]
“it’s value does not change under a Lorentz transformation”
According to Einstein’s SRT, this is quite wrong. The very reason cited in all textbooks on the subject is that mass increases with velocity (relative to an external observer), so as c is approached, mass increases without limit, so an infinite force is needed to push that mass to c. Of course, an infinite force does not exist.
Quote from “Asimov - The Stars In Their Courses” on this subject.
“…subatomic particles were detected speeding out from radioactive atomic nuclei and their velocities were sometimes considerable fractios of the velocity of light. Their masses could be measured quite accurately at different velocities and the Lorentz equation was found to hod with great precision. In fact, down to this moment, no violation of the Lorentz equation has ever been discovered for any body at any measured velocity.”
Isaac Asimov was a renowned and respected science and science fiction writer. He is probably best remembered for his fiction, but his science fact books are second to none.
Read the book referred to in my first posting “Relativity The Special And The General Theory” Methuen 1920
I’m sorry you’re wrong, the formula τ = t/c arcsinh (gt/c) is non-linear, so I’m struggling to see what the ‘0.9999999989802’ is supposed to represent. Whilst you get a nice linear relationship between Δt and Δt’ when comparing inertial reference frames in special relatrivity, the same is not true when comparing an inertial reference frame with a non-inertial reference frames in special relativity (infact it’s exceedingly simple to see why the relationship can never be linear).
When you PM’d and asked me for that formula I did warn you that unless you know what you’re doing accelerated reference frames in special relativity should be approached with caution.
So you’ve read all textbooks on the subject? Given that all textbooks written in the last 40 years or so pretty much define so as to be a Lorentz scalar, I must doubt that.
It’s a matter of how you define mass, however there are very good reasons why we choose to define mass so that it is a Lorentz invariant, it’s simply by far the most useful definition.
So let’s get this straight shall we?
A formula gives the time shown by a clock in FR1 in a gravitational field of (in this instance) 1G compared to the time of a reference clock. The reference clock ticks at 1 second per second, and the clock in FR1 ticks at 0.9999999993045 second per second of this reference.
A formula gives the time shown by a clock in FR2 undergoing an acceleration of 1G (compared to the same reference clock), and the clock in FR2 ticks at 0.99999999989802 second per second of this reference.
If the formulae are correct, whether linear or not, the 2 clocks stay in sync with the reference clock, and therefore with each other (or very nearly so).
As to mass increase. The Lorentz equations are all linked, and derive time dilation, length contraction, and mass increase. If you don’t agree with the mass increase formula, then you have to throw the lot out, you cannot cherry pick. Either Einstein is correct, or he is not correct - which is it?
It’s not linear, so your “seconds per second” depend on which particular second we’re talking about which is why the figure you’ve quoted makes no sense. Infact as t goes to infinity the “seconds per seconds” go to zero when comapring the proper time of a uniformaly accelerated observer and the coorindate time of an inertial observer. Try it for yourself (thoguh I noticed I quoted it incorrectly in my last post), i.e. try inserting large values for t into the formula and notice how tau/t goes to zero.
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.
Like I say it’s matter of definitions, but it may suprise you to learn that the subject of relativity has moved on a bit since 1905. Though Einstein was the first to define longitidnal and transverse mass, shortly after definign it and as the subject moved on, he preferred the invariant definition of mass. That’s not a suprise really when considering that he would’ve fully appreciated sticking to defining physical quanties in a Lorentz covariant manner in special relativity when formulating the general covariant theory of general relativity.