Force due to acceleration and time flowing differently

[PeterDonis]

The point is that mass curves spacetime and mass also resists against a change in speed.

These are two different concepts of "mass". The precise term for the first one is "stress-energy", and the mathematical entity that embodies it in relativity is the stress-energy tensor.

The precise term for the second one (which actually is not properly described as "change in speed"--see below) is "invariant mass", which is a scalar, the norm of an object's 4-momentum vector.

In some particular cases there are relationships between these two things, but they're not the same.

a change in speed is acceleration

No. Proper acceleration is not a "change in speed", it's path curvature of a worldline.

The "change in speed" concept of acceleration is called "coordinate acceleration" in relativity, and it is not an invariant (whereas proper acceleration is). IMO this concept should simply be ignored when trying to learn relativity; it causes far more problems than it helps to solve. This is an instance of the general rule that frame-dependent quantities in relativity don't have any physical meaning and should not be focused on. They can be useful as conveniences to simplify calculations but they should not be focused on conceptually to try to understand what is happening.

The rest of your post just builds on the above misconceptions.

 

[PeterDonis]

I talked about acceleration during a certain time interval. so not ony the acceleration but also the duration of the accelleration. so it is about the combination of acceleration during a certain time interval (causing a speed difference between the 2 rockets) and the building up rate of the difference between the clocks.

Your approach here is based on some fundamental misconceptions (see my post #106 just now). As I have already commented multiple times now, you might be able to get something like this to work in certain particular scenarios, but nothing you learn from that will generalize. IMO you would be far better served by dropping this entirely and focusing on the geometric approach.

 

[Vanadium 50]

@HansH, you are not on the path to success.

You complain you don't want to spend years studying this. Fair enough, but this will limit what you will be able to learn - as Euclid said 2300 years ago, "There is no royal road to understanding mathematics".

You also want to start in the middle. OK, but there is a reason that normally people start at the beginning. You need to lay a proper foundation. If you tried to build the top floor of your house building the lower floors what is going to happen?

You need to seriously think about the replies you are getting. I can tell you aren't doing this because of your speed in responding. Fully half of your messages post in under 5 minutes since the message you are responding to. The most common case? Zero minutes - you post a reply in the same minute that the message you are replying to is posted.

If you want to learn, you need to do things differently than you are - especially more thinking about what people are saying to you. You have to put in the work. We can't do it for you.

 

[HansH]

in#105 I am asked what I am trying to accomplish.
I give a fair and open answer of my way of thinking and already indicated several times that I am not a professional physicist but eager to get answers on things that bother me and then as result I am given twice a shot in the back.
I appreciate your opinion and understand that you want everyone to learn physics, but in this way you block people to ask any question and show what keeps them busy
sorry for that, but this gives me a really bad taste in my mouth. Next time I am only asking a question after studying the full theory. Then I am at least prepared to understand al your reactions. If this is what (the way of commuinicating within) physics is about, then I am really happy I made the right decision in my study.

 

[PeterDonis]

I am given twice a shot in the back.

No, that's not what you have been given. What you have been given is an honest response telling you that what you say you want is not possible. What you say you want to accomplish in #105 cannot be done. If you don't want to accept that answer, that's up to you, but to characterize it as "a shot in the back" is wrong. We can't help it if the actual physics doesn't allow you to do what you want to do.

 

[HansH]

No, that's not what you have been given. What you have been given is an honest response telling you that what you say you want is not possible. What you say you want to accomplish in #105 cannot be done. If you don't want to accept that answer, that's up to you, but to characterize it as "a shot in the back" is wrong. We can't help it if the actual physics doesn't allow you to do what you want to do.

I think you still do not understand my point. of course I can agree on your conclusion that things are not possible and that there is not such relation as I brought into the discussion. You are the expert here. But when you are saying I react within 5 minutes and keep a stopwatch to see how fast I respond then I feel that as not respectfull and intimidating. and if I dare to ask a question that comes into my mind because I am a creative person in life helping technology forward with 75+ patents on my name helping my company to be the world leader, then you probably also can imagine that I have enaugh esperience to not letting anyone intinidate me. also not if I ask a question that does not fit into the logical sequence how normally a physics study is organized and get that back as my fault.

 

[PeterDonis]

when you are saying I react within 5 minutes and keep a stopwatch to see how fast I respond

If that specifically is what you are concerned about as being "a shot in the back", you should reference it specifically. (You can also use the Report button to report posts that you think violate PF rules.)

You might also want to consider that the criticism might be making a valid point. Bear in mind that everyone here who is an expert in this was once in the same position that you are in. Nobody is born knowing these things. We all have to learn them. And those of us who have learned them took years to do it, and made plenty of false starts and went down plenty of blind alleys in the process. You might want to consider that that experience gives us knowledge not just about the physics you are asking about, but about what works and what doesn't when trying to learn it.

if I dare to ask a question that comes into my mind

Nobody is saying you can't ask questions that come into your mind. But in this thread, you have repeatedly been asking questions that are based on fundamental misconceptions of yours, and the answers you have gotten are trying to tell you that--which amounts to telling you that in order to learn what you say you want to learn, you are first going to have to unlearn things that you think you already know. It's not clear that you've grasped that and considered its implications.

also not if I ask a question that does not fit into the logical sequence how normally a physics study is organized and get that back as my fault.

It's not a matter of "fault". It's a matter of what works and what doesn't in learning these things. See my comments above.

 

[HansH]

ok I will make one final remark about this. I think I already indicated several times that my knowledge level is not sufficient at this specific topic. Therefore I am not going into discussions about if something what an expert says is true or not, but instead I ask additional questions to get the picture more clear and to better prepare the picture what I still need to do. That probalby is the point of asking 5 questions in 1 minute.get things clear and get the way forward clear. That there are misconceptions seems locical, because that is the reason why I involve this forum becausing I want to test my thoughts with experts that can rapidly hit the essentials. I also indicated somewhere that I thought I was loading your team too mucht and already indicated on my own initiative that perhaps I should do more study first. And then I still get 10 reactions back that challenge me to use that to come further. I also asked advice what topics to study and at least Dale gave good advice there I think. but that is not a matter of a few days work.

You can discuss about the technical things being true of not, but you can't discuss about how someone should experience the way of communicating. When that is experienced as intimidating it is what it is. posting that should ring a bell at the people concerned at the level of intelligence we have here. so using the report button to me sounds more like being anonimous not daring to mention the things as they are and hiring other to solve that. That is not how I am.

for me this topic is sufficiently answered to be able to continue for a while with the input I got. so probably I will come back later with some more knowledge. (If I am allowed after mentioning things by name as I just did)

 

[PeterDonis]

for me this topic is sufficiently answered to be able to continue for a while with the input I got

Ok, good. That's the ultimate goal.

so probably I will come back later with some more knowledge.

Good.

(If I am allowed after mentioning things by name as I just did)

That's not an issue. By all means come back with further questions once you have continued for a while on your own.

 

[Sagittarius A-Star]

thanks I will keep in mind.

Here you can find good lecture videos from Leonard Susskind as introduction:
https://theoreticalminimum.com/courses

You can find there a lecture video about the Lagrangian for classical mechanics under

• "Classical Mechanics": Lecture 3

You can find lecture videos about Special Relativity (including relativistic laws of motion) under

• "Special Relativity and Electrodynamics": Lectures 1 to 3

Related textbooks:

• https://www.amazon.de/dp/0465075681/
• https://www.amazon.de/dp/0141985011/

 

[HansH]

Here you can find good lecture videos from Leonard Susskind as introduction:
https://theoreticalminimum.com/courses

You can find there a lecture video about the Lagrangian for classical mechanics under

• "Classical Mechanics": Lecture 3

You can find lecture videos about Special Relativity (including relativistic laws of motion) under

• "Special Relativity and Electrodynamics": Lectures 1 to 3

Related textbooks:

• https://www.amazon.de/dp/0465075681/
• https://www.amazon.de/dp/0141985011/

Thanks. 1 or 2 years ago I already followed the lecture video's about special relativity and also a part of the lectures about general relatvity. alt a certaim monent I lost the overview over the general relativity part and after some time started from scratch again. Then I saw that whe I viewed it for the second time I could follow more than the first time. so it seems difficult to to get this clear from video's alone. I think however more exersices would be needed as following a video only doesn't give the problems one runs into when doing yourself. so that is a thing I need to solve in some way also.

 

[malawi_glenn]

@HansH Lenny is releasing GR book end of this year/ beginning of next :) same series "the theoretical minimum"

 

[vanhees71]

Thanks. 1 or 2 years ago I already followed the lecture video's about special relativity and also a part of the lectures about general relatvity. alt a certaim monent I lost the overview over the general relativity part and after some time started from scratch again. Then I saw that whe I viewed it for the second time I could follow more than the first time. so it seems difficult to to get this clear from video's alone. I think however more exersices would be needed as following a video only doesn't give the problems one runs into when doing yourself. so that is a thing I need to solve in some way also.

The problem with such online-movies is that you tend to just watch the movies and think you have understood the material. This is particularly true for well-presented movies like the Susskind movies. The important point is that you have to sit down and do something actively with the material to see, whether you can work with the material, particularly solve problems for yourself. This holds, of course, true also for textbooks, manuscripts, and papers.

 

[HansH]

be comfortable using the Lagrangian for classical physics.

2 questions at this stage:
-From the given reference to the lectures of Leonard Susskind I understand that this method allows you to optimize something to a minimum value such as calculate the shortest trajectory between 2 points. Is that the reason why it is important for relativity?
-I also understood from one the reactions in this topic that the time dilatation is defined by the length difference between 2 worldslines. So then I assume that that length is the sum of ds over such trajectory. but for calculating ds pythagoras cannot be applied as we need to take the difference of the squares of the time and the space component. But from the lecture of Susskind I also understand that the Lagrangian method includes a part where pythagoras is used. So how should I interpret that?

 

[vanhees71]

The main reason to use the "principle of least action" is that it admits an elegant formulation of symmetry principles that underly all of modern physics. E.g., the form of the equations of motion in Newtonian mechanics look the way they look, because of the symmetry of the underlying Galilei-Newtonian spacetime model. As well the relativistic field equations of motion (in the classical realm that means classical electrodynamics) look the way they do in a similar way because of the symmetry of the spacetime model underlying special relativity (Minkowski spacetime). In its "Hamiltonian formulation" in phase space it provides one of the most elegant tools to heuristically "quantize" classical theories, known as "canonical quantization". To make this heuristical hand-waving a bit more rigorous you can refer to the representation theory of Lie groups and algebras, which tell you why the quantum theoretical equations look the way they look too, e.g., it explains why in a wave-mechanical formulation (1st quantization formalism of non-relativistic QM) the self-adjoint operator describing the momentum of a particle is $\hat{\vec{p}}=-\mathrm{i} \hbar \vec{\nabla}$: It's because momentum is the infinitesimal generator for spatial translations, and infinitesimal spatial translations are described by the Nabla-operator in field theory.

 

[Sagittarius A-Star]

-From the given reference to the lectures of Leonard Susskind I understand that this method allows you to optimize something to a minimum value such as calculate the shortest trajectory between 2 points. Is that the reason why it is important for relativity?

In the Susskind movie "Special Relativity and Electrodynamics", Lecture 3, about relativistic laws of motion, the Lagrangian and the Hamiltonian are used to derive in a systematic way the 4-momentum and $E_0 = mc^2$.

 

[Nugatory]

I also understood from one the reactions in this topic that the time dilatation is defined by the length difference between 2 worldslines.

You want to be careful here. The phrase “time dilation” and statements about time passing at different rates are often applied to two different situations with different explanations:
a) You are moving relative to me; I find that your clock is running slow compared to mine.
b) You and I synchronize our clocks together and then one or both of us travels somewhere else; when we get back together and compare we find that less time has passed for one of us.

The arc length thing applies to #b but not #a. #a is about relativity of simultaneity instead.

 

[Dale]

Is that the reason why it is important for relativity?

That is a reason, but there are many others. It can also be used to find conserved quantities, fictitious forces, include the effects of other forces besides gravity, and of course find the equations of motion. (and probably others I missed). It also unifies relativity, quantum mechanics, and classical mechanics under the same framework.

-I also understood from one the reactions in this topic that the time dilatation is defined by the length difference between 2 worldslines. So then I assume that that length is the sum of ds over such trajectory. but for calculating ds pythagoras cannot be applied as we need to take the difference of the squares of the time and the space component. But from the lecture of Susskind I also understand that the Lagrangian method includes a part where pythagoras is used. So how should I interpret that?

The Minkowski metric can be written $ds^2=-c^2 dt^2 + dx^2 + dy^2 + dz^2$. The Euclidean metric is $ds^2= dx^2 + dy^2 + dz^2$ which is basically just the Pythagorean theorem. So yes, the Minkowski metric includes a part where Pythagoras is used. Spacetime includes both space and time so we shouldn't be surprised to see that part of the spacetime metric looks like an ordinary space metric.

 

[vanhees71]

That's part of the fundamental assumptions underlying the special-relativistic spacetime model: There's a class of preferred frames of reference, called "inertial frames", and any inertial observer describes space as a 3D Euclidean affine manifold. Together with Einstein's 2nd postulate (independence of the speed of light of the velocity of the light source as well as the detector wrt. any inertial frame) you get Minkowski space as the spacetime model.

It's is crucial to understant that Minkowski space is a pseudo-Euclidean (not a Euclidean) affine space, i.e., the importance of the minus sign in its fundamental form (unfortunately often called a "metric") although this form is not positive definite but of the signature (3,1) (in the here used east-coast convention of signs). It's crucial, because in contradistinction from a Euclidean affine space it allows for "time ordering" and thus to establish a "causal ordering".

 

[Sagittarius A-Star]

mass also resists against a change in speed. and a change in speed is acceleration

Proper acceleration $\alpha$ (measured by the accelerated object with an attached accelerometer) is different from coordinate-acceleration $\vec {a} = d\vec {v}/dt$ with reference to an inertial frame. If the force points into the direction of the velocity with reference to an inertial frame, then
$F = m\alpha = m\gamma^3 a = (\frac{1}{\sqrt{1-v^2/c^2}})^3 ma$.

Source:
https://en.wikipedia.org/wiki/Proper_acceleration#Acceleration_in_(1+1)D

 

[Sagittarius A-Star]

what I mean is then probably 2 rockets in space with an observer in each rocket going with relative speed =0 close to each other. Then they both synchronze their clocks and then one rocket accelerates for a certain amount of time with say 1g. so this gives a force of 1 Newton per kg mass at that mass during this amount of time. then after some time the same rocket accelerates for (I assume) a same amount of time with -1g so that after that there is no speed difference between the 2 rockets but only a certain distance in between. now they come together again by slow acelleration en de-aceleration and after that compare their watches.

This scenario sounds like the twin-"paradox". Here you can find a calculation of the age difference at event $F$:
https://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_spacetime.html

 

[HansH]

@Sagitarius A-star: not sure if I get the point you want to make with #125 and #126. From 126 I conclude that Terence is in rest and Stella undergoes some acceleration. when they meet again the watch of Stella is behind compared to that of Terence. so for me clear that there is no paradox because Stella undergoes some (coordinate?) acceleration while Terence did not. so their situation is different. so that is the difference in this case that causes [1] the difference between the watches. (assuming they are both in flat spacetime, so coordinate-acceleration is the only possibility to be able to follow a different trajectory in spacetime)
Not sure what to conclude from #125 (I am not familiar (yet) with the theory in the link). I think I understand the difference between Proper acceleration and coordinate-acceleration but not sure what point you want to make here.

[1] (not sure I Iam allowed to conclude that, because then I again would connect coordinate-acceleration to difference between watches that was already concluded to be not allowed, at least not in general)

 

[PeterDonis]

From 126 I conclude that Terence is in rest and Stella undergoes some acceleration

No. "At rest" is frame dependent and has no invariant meaning.

The correct statement is that Terence is moving inertially (in free fall, zero proper acceleration) for the entire scenario and Stella is not; she has nonzero proper acceleration for at least some portion of the trip (her turnaround). And since the scenario is set in flat spacetime, Terence's clock will show more elapsed time between their two meetings, because of simple geometry: given any two events in flat spacetime which are timelike separated, the longest timelike path between them is the one corresponding to free-fall motion.

 

[PeterDonis]

Stella undergoes some (coordinate?) acceleration

Coordinate acceleration is frame-dependent. Your best bet is to forget you ever heard of the concept.

The important thing is, as noted in my previous post just now, that Stella has nonzero proper acceleration during her turnaround.

assuming they are both in flat spacetime

Yes.

so coordinate-acceleration is the only possibility to be able to follow a different trajectory in spacetime

No. Proper acceleration is the only way they can meet twice. Coordinate acceleration is frame-dependent; it is perfectly possible to construct a (non-inertial) frame in which Stella has zero coordinate acceleration (and Terence has nonzero coordinate acceleration). But their respective proper accelerations are unchanged, since proper acceleration is an invariant.

 

[Sagittarius A-Star]

@Sagitarius A-star: not sure if I get the point you want to make with #125 and #126.

I added mathematical formulas to the descriptions in text, which I cited from you. They may help you to do related calculations if you want.

(I am not familiar (yet) with the theory in the link).

You can derive the formula in posting #125 from the "relativistic velocity addition" formula for two inertial frames, of which one is momentarily co-moving with the accelerated object.

I think I understand the difference between Proper acceleration and coordinate-acceleration but not sure what point you want to make here.

Sorry, I was not 100% sure, if you understood this difference.

 

[Sagittarius A-Star]

-From the given reference to the lectures of Leonard Susskind I understand that this method allows you to optimize something to a minimum value such as calculate the shortest trajectory between 2 points. Is that the reason why it is important for relativity?

What I find also interesting is, that the principle of least action, which was first found for classical mechanics, to determine the trajectory of an object between two events, is equivalent to the principle of maximum elapsed proper time in SR. This can be seen when dividing the Lagrangian by the constant and invariant term $-mc^2$.

Source (unfortunately, the audio track is not in English):

 

[HansH]

related to this topic I was triggered by this video:

at t=11:40 the video discusses sometjing interesting:
when an object flows through spacetime without a force, the mass is wharping spacetime in the direction of movement in the same way as it unwharps wharpes spacetime in the direction it is leaving, giving a nett force on the mass of 0.

If you apply a force on the mass it wharpes space more in the direction of movement than it unwharpes space in the direction it is leaving. this gives an energy being delivered to the wharping of spacetime so explains a force. so I assume that could be the thing i was looking for in this topic: Is there a relation between the fact that accellerating a mass requires a force while mass wharps spacetime. so this could explain a relation between accelleration and force at the one hand and curvature of spacetime by a mass at the other hand. What do you think of this?

 

[Dale]

What do you think of this?

I think it is complete handwaving. I have never seen that concept in any professional scientific source. I have no idea what mathematical concepts he could be referencing by the verbal description, so the description is probably not based on any sound concept.

 

[member 728827]

In special relativity we have moving reference frames resulting in a different flow of time in each reference frame. This we can explain because we use the fact that the speed of light is the same in all reference frames, leading to the Lorenz transformation giving the amount of slowdown of time.

Between two inertial reference frames at relative motion, there's no "different flow of time", each observer sees the time in the other to run slower by the Lorentz factor, is a perspective issue. In its own frame, the observers don't notice anything special.

If you start with a second reference frame having the same speed as the first frame then there is no difference in flow of time. So the fact that the second frame is accelerated and the first frame is not should be the reason for time to flow slower in the second frame after that acceleration.

The reason in SR is always the same: the invariance of speed of light, even with accelerated frames. I don't know in GR. And the force is usually due to Nature expecting you to move in a certain way, but then something disturbs this natural motion.

 

[vanhees71]

In GR everything is as in SR as far as local concepts are concerned, i.e., the global Poincare symmetry of SR is "gauged" to be only local in GR. In more physical terms it means that in GR you always can introduce an inertial frame of reference at any point in spacetime, and it's realized by a reference frame fixed at a point in free fall and a non-rotating tetrad along the corresponding time-like geodesic. Then in a not too large region around these points along the time-like geodesic this is an inertial frame of reference, and there are only tidal gravitational forces on extended objects.

 

[HansH]

I think it is complete handwaving. I have never seen that concept in any professional scientific source. I have no idea what mathematical concepts he could be referencing by the verbal description, so the description is probably not based on any sound concept.

but I assume it would be possible to check it as we (we in general,however not I) can calculate (or at least simulate) how a moving mass distorts spacetime and if that requires energy you should be able to derive the relation between mass, acceleration and force from such deeper underlying reason. but if no one has has ever seen that concept in any professional scientific source, it could also mean that no one ever investigated that, or at least also not proven that it is wrong because then that should be somewhre in literature too.

 

[vanhees71]

In GR the equations of motion for mechanical systems follow from the Einstein field equation through a Bianchi identity of the underlying gauge symmetry, i.e., from
$$G^{\mu \nu} = -\kappa T^{\mu \nu}$$
you necessarily have
$$\nabla_{\mu} T^{\mu \nu}=0,$$
and for mechanical systems (fluids or elastic bodies) that's the basic equation of motion.

 

[HansH]

In GR the equations of motion for mechanical systems follow from the Einstein field equation through a Bianchi identity of the underlying gauge symmetry, i.e., from
$$G^{\mu \nu} = -\kappa T^{\mu \nu}$$
you necessarily have
$$\nabla_{\mu} T^{\mu \nu}=0,$$
and for mechanical systems (fluids or elastic bodies) that's the basic equation of motion.

is this the answer on #136? (unfortunately I am not that familiar with the equations) so does it mean that you conclude that accelleration does not store energy in spacetime curvature?

 

[Dale]

I assume it would be possible to check it as we (we in general,however not I) can calculate (or at least simulate) how a moving mass distorts spacetime

He isn’t just claiming that mass distorts spacetime, which we can calculate, but that it produces specific pushing and pulling forces that cancel out. He didn’t explain how to calculate such forces, and I have never seen such a calculation in the literature. So how exactly could it be checked?

This appears to be a “not even wrong” idea. I couldn’t tell from @vanhees71 if he has seen this somewhere. He certainly knows this literature more than I do

 

[vanhees71]

I'm not sure, whether I understand what the question is. How to interpret gravitational field energy is a notorious problem from day 1 of Einstein's final version of the theory, and it lead to Emmy Noether's famous paper on symmetries and conservation laws. Energy is in general not conserved in GR.