I Force due to acceleration and time flowing differently

[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.

 

[Dale]

I'm not sure, whether I understand what the question is.

Watch the video in post 132 at around 11:40. It seems like complete nonsense to me, and I have never read any professional scientific reference that says anything that I can remotely connect to this idea. Your recent posts are a little ambiguous, but seem to indicate that you recognize the concept.

The question is does it seem like nonsense to you, and if not then can you please cite some professional scientific literature that explains what you think the video is talking about?

 

[vanhees71]

I thinkt I've to agree with your assessment that this is "complete nonsense". I thought the question was about the equations of motion of a medium, i.e., something like fluid dynamics or dynamics of an elastic body, i.e., a continuum mechanical model, and there if you have an exact solution of the Einstein field equation, you also have the solution of the equations of motion for the dynamical system. An example is the FLRW solution, where you automatically get an ideal fluid for the cosmological medium. The various types usually put in (massive particles, radiation, cosmological constant/dark energy) just vary in their equations of state.

For me these somewhat Machian idea in this video has never been convincingly demonstrated from GR. The message of GR concerning the "equivalence principle" is that both "inertia" as well as "source of gravitation" are not (only) mass but all kinds of energy, momentum, and stress.

 

[HansH]

I thought the question was about the equations of motion of a medium,

The question is about mass being accelerated by an external force such as a rocket engine. the question now is if it could be that the energy delivered by the force to acellerate the mass is stored in the distortion of the spacetime around the mass. so if due to the acelleration spacetime if deformed different at the front of the mass compared to the back of the mass in the direction of the aceleration and so giving as result a force that we recognze as F=m*dv/dt.

 

[PeterDonis]

the question now is if it could be that the energy delivered by the force to acellerate the mass is stored in the distortion of the spacetime around the mass.

The answer to this is no for two reasons:

(1) The energy delivered by the force to accelerate the mass goes into the mass. If it didn't, it wouldn't be accelerating the mass, it would be doing something else.

(2) You can't store energy "in the distortion of spacetime" because there is no such thing; there is no tensor that describes "energy stored in the distortion of spacetime".

You can transport energy by gravitational waves, which can be thought of as changes in the curvature of spacetime that propagate. But there is no way to localize the energy that is being transported.

 

[HansH]

The answer to this is no for two reasons:

(1) The energy delivered by the force to accelerate the mass goes into the mass. If it didn't, it wouldn't be accelerating the mass, it would be doing something else.

(2) You can't store energy "in the distortion of spacetime" because there is no such thing; there is no tensor that describes "energy stored in the distortion of spacetime".

You can transport energy by gravitational waves, which can be thought of as changes in the curvature of spacetime that propagate. But there is no way to localize the energy that is being transported.

For the first point I would say that this sounds like a chicken and egg problem: you directly refer to the relation F=m*dv/dt and conclude that the energy should be stored in the mass. But the discussion is just about the question if it could be stored in something underlying different than directly the mass. So therefore I think you cannot use that argument that a different explanation cannot be true because the regular explanation must be true.

The second remark I cannot judge. however I could imagine that a mass curving the spacetime around it is a form of energy storage. Because if you would take away the mass instantly, the energy propagates away as gravitational waves. so then I would say that a moving mass can be interpretet as a a mass being taken away at one point and put back at another point and so then when spacetime curvature is a form of energy storage, then energy is taken away from spacetime around one point and stored into the spacetime around the next point similar to energy being stored in a magnetic inductor ad then converted to another inductor. If you do that slow, the energy stored in waves is very small so all energy is then transported. I think that is what is said in the movie I referred to in relation to mass and accelleration. and then the question was: how does this work if I have a mass following a geodesic and If I have a mass being accelerated by an external force. Unfortnately I do not have sufficient knowledge to write down the equations, so therefore I was hoping that a specialist would like to use these equations in relation to both situations to prove it or bust it.

 

[PeterDonis]

For the first point I would say that this sounds like a chicken and egg problem: you directly refer to the relation F=m*dv/dt and conclude that the energy should be stored in the mass.

Yes, because that is what $m \ dv/dt$ means -- it means you stored what the force transmitted in the kinetic energy of the mass. There is no "chicken and egg problem". This is just basic physics: a force does work, and when the work it does is to accelerate a mass, the energy transmitted by the work done goes into the kinetic energy of the mass.

I could imagine that a mass curving the spacetime around it is a form of energy storage

You should not be trying to "imagine" things like this. You should be learning how the physics actually works. When you do, you will find exactly what I said: that there is no tensor that represents "energy stored in spacetime", and in GR, physical things are represented by tensors.

the movie

Is, as you have already been told, nonsense. It is not a valid basis for PF discussion.

 

[HansH]

there is no tensor that represents "energy stored in spacetime"

Then is it stil unclear to me how we can explain that gravitational waves occur (representing enrgy flow) when you would instantly take away a mass. so where is the energy of the gravitational wave then coming from?

 

[Ibix]

Because if you would take away the mass instantly,

You can't, and attempting to describe doing this leads to a self-contradiction. This kind of thing (which seems a superficially reasonable idealisation, but actually invalidates all subsequent reasoning) is why generating plausible ideas in GR is hard.

 

[HansH]

Yes, because that is what $m \ dv/dt$ means -- it means you stored what the force transmitted in the kinetic energy of the mass. There is no "chicken and egg problem". This is just basic physics: a force does work, and when the work it does is to accelerate a mass, the energy transmitted by the work done goes into the kinetic energy of the mass.

I know that of course, but the fact that there is kinetic energy does on its own not say where this energy is actually stored I assume.

 

[HansH]

You can't, and attempting to describe doing this leads to a self-contradiction. This kind of thing (which seems a superficially reasonable idealisation, but actually invalidates all subsequent reasoning) is why generating plausible ideas in GR is hard.

bringing together 1 kg of matter and 1 kg of antimatter?