Time dilation, relativistic mass and fuel consumption

[PainterGuy]

TL;DR Summary

I'm trying to understand the theories of relativity at a basic level and have some basic questions. I would really appreciate if you could help me.

Hi,

Could you please help me with the queries below?

Question 1:
A GPS satellite is moving faster than the earth, for every day on Earth the clock on the satellite shows one day minus 7 microseconds due to time dilation due to special relativity. However, since the Earth's gravitational pull is much stronger at the surface than at the altitude of the satellite (20000 km), the due to the effects of general relativity, one day in Earth would be measured in the satellite as one day plus 52 microseconds. The compounded effect is that the satellite clock gets ahead of the Earth clock by 45 (52-7) microseconds per day.

So, if the satellite clock is not synchronized with Earth's clock and the satellite keeps on orbiting Earth then it'd take 22222.222 days (approx. 61 Earth years) until the satellite clock is ahead of Earth's clock by 1 minute. After 61 years the satellite is taken down, its clock should show one minute difference from the Earth's clock. Do you agree?

Question 2:
In this question I understand that I'm making many generalizations but I'm only trying to basic understanding.

A hypothetical aircraft makes a round trip from Earth to Neptune at a significant speed but quite less than the speed of light. The trip distance is known and one can calculate the required amount of fuel. Once the aircraft gets back to earth, its fuel tank is checked. Wouldn't the aircraft have consumed more fuel than it should have? I'm thinking so because there was an increase in its relativistic mass and therefore it needed extra consumption of fuel to complete the trip. Do I make any sense?

Thanks a lot.

 

[Mister T]

Wouldn't the aircraft have consumed more fuel than it should have?

If by "should have" you mean the amount calculated using Newtonian physics, then yes.

The reason, though, is because Newtonian physics gives you erroneous results. Relativity is not something that's used to make corrections for Newtonian physics.

And, by the way, you don't need the concept of relativistic mass to do relativistic physics. It's best to not use it as it can lead to many misconceptions.

 

[PainterGuy]

Thank you very much!

If by "should have" you mean the amount calculated using Newtonian physics, then yes.

The reason, though, is because Newtonian physics gives you erroneous results.

But Newtonian physics is arguably more straightforward and makes more common sense to many. There is no doubt that theories of relativity are better in their own right.

Re Question 1
Do you think what I say in Question 1 is correct?

 

[DaveC426913]

They did actually do your experiment #1 in a commercial jet.
Yes, GR and SR time dilation mostly canceled out.
You'll have to Google it to get the deets.

Re: experiment 2: As MisterT points out, it depends on how you determine how much fuel it "should have"used.

Even if you didn't know about calculations for relativity, your empirical observations would tell you that the craft, the crew and the engines are all operating fractionally slower than they were when they left.

Because the engines are observed to be putting out less thrust, your calcs will determine that it should use less fuel than it would without this "slowing" effect. And indeed, when the craft returns, it will have used less fuel than originally expected.

That's simplistic. There's more to it than that.

 

[jbriggs444]

Summary:: I'm trying to understand the theories of relativity at a basic level and have some basic questions. I would really appreciate if you could help me.

22222.222 days (approx. 61 Earth years) until the satellite clock is ahead of Earth's clock by 1 minute.

45 microseconds per day times 22 thousand days is approximately one million microseconds. That's one second, not one minute.

 

[PeterDonis]

since the Earth's gravitational pull is much stronger at the surface

It's not a matter of "gravitational pull", it's a matter of gravitational potential, i.e., altitude. Clocks run faster at higher altitudes. The GPS satellites are at a high enough altitude that this effect outweighs the effect of their orbital speed.

if the satellite clock is not synchronized with Earth's clock and the satellite keeps on orbiting Earth then it'd take 22222.222 days (approx. 61 Earth years) until the satellite clock is ahead of Earth's clock by 1 minute. After 61 years the satellite is taken down, its clock should show one minute difference from the Earth's clock. Do you agree?

I have not checked your specific numbers [Edit: I see @jbriggs444 has and they are off] but the general idea is correct.

 

[PainterGuy]

Thanks a lot!

They did actually do your experiment #1 in a commercial jet.
Yes, GR and SR time dilation mostly canceled out.
You'll have to Google it to get the deets.

I believe you were referring to this experiment: https://en.wikipedia.org/wiki/Hafele–Keating_experiment

Re: experiment 2: As MisterT points out, it depends on how you determine how much fuel it "should have"used.

Even if you didn't know about calculations for relativity, your empirical observations would tell you that the craft, the crew and the engines are all operating fractionally slower than they were when they left.

Because the engines are observed to be putting out less thrust, your calcs will determine that it should use less fuel than it would without this "slowing" effect. And indeed, when the craft returns, it will have used less fuel than originally expected.

I'm sorry but to me it seems like that @Mister T was saying the opposite.

If by "should have" you mean the amount calculated using Newtonian physics, then yes.

Let me refine the original statement.

A hypothetical aircraft makes a round trip from Earth to Neptune at a significant speed but quite less than the speed of light. The trip distance is known and one can calculate the required amount of fuel. Once the aircraft gets back to earth, its fuel tank is checked for actual consumption. The aircraft completed the trip in scheduled time.

Wouldn't the aircraft have consumed more fuel than it should have? I'm thinking so because there was an increase in its relativistic mass and therefore it needed extra consumption of fuel to complete the trip in its scheduled time; it needed more thrust as a result of increase in its mass which resulted in more consumption. Do I make any sense? Assume that only Newtonian physics is used.

Thank you for your time!

 

[PeroK]

A hypothetical aircraft makes a round trip from Earth to Neptune at a significant speed but quite less than the speed of light. The trip distance is known and one can calculate the required amount of fuel. Once the aircraft gets back to earth, its fuel tank is checked for actual consumption. The aircraft completed the trip in scheduled time.

Wouldn't the aircraft have consumed more fuel than it should have? I'm thinking so because there was an increase in its relativistic mass and therefore it needed extra consumption of fuel to complete the trip in its scheduled time; it needed more thrust as a result of increase in its mass which resulted in more consumption. Do I make any sense? Assume that only Newtonian physics is used.

Thank you for your time!

You can study a problem using Newtonian physics. Or. you can study a problem using SR. The equations for KE are related by:$$KE = (\gamma - 1) mc^2 \approx \frac 1 2 m v^2 \ \ (v \ll c)$$

In any problem, if you calculate the kinetic energy of a particle accurately enough you find the relativistic equation. But, normally, for most practical purposes on Earth the Newtonian approximation is good enough. Similarly, to get an accurate enough calculation of the fuel consumption you would need the SR formula.

Note that there is a difference between the Newtonian and SR rocket equations for fuel consumption that you might look up.

But Newtonian physics is arguably more straightforward and makes more common sense to many.

Initially yes, but the more you learn about SR, the more it pulls things together in a more unified way. For example, in Newtonian physics we have independently the conservation of mass, energy and momentum. In SR these are unified into conservation of energy-momentum, which is a single four-vector.

Also, here's what Isaac Newton himself had to say about his own theory of gravity:

"That gravity should be innate inherent & essential to matter so that one body may act upon another at a distance through a vacuum without the mediation of any thing else by & through which their action or force may be conveyed from one to another is to me so great an absurdity that I believe no man who has in philosophical matters any competent faculty of thinking can ever fall into it."

Newton knew something was missing, which Einstein found 250 years later.

 

[jartsa]

A hypothetical aircraft makes a round trip from Earth to Neptune at a significant speed but quite less than the speed of light. The trip distance is known and one can calculate the required amount of fuel. Once the aircraft gets back to earth, its fuel tank is checked for actual consumption. The aircraft completed the trip in scheduled time.

Wouldn't the aircraft have consumed more fuel than it should have? I'm thinking so because there was an increase in its relativistic mass and therefore it needed extra consumption of fuel to complete the trip in its scheduled time; it needed more thrust as a result of increase in its mass which resulted in more consumption. Do I make any sense? Assume that only Newtonian physics is used.

Let's say some Newtonian people have build a spacecraft and written down a timetable for a trip from Earth to Neptune and back. A pilot adjusts the rocket motors so that the spacecraft proceeds according to the timetable. There's a gas pedal.

The pilot will press the gas pedal more than the Newtonians calculated. And the fuel consumption rate will be lower than what Newtonians calculate it to be given that gas pedal position.(The pilot does not have a clock onboard, but there are clocks floating in space, those clocks tell the pilot the time.)

 

[jbriggs444]

Are we seriously contemplating a relativistic treatment of an automobile on a highway in space using an internal combustion engine?

Before we start with the calculations, we need to figure out how this spacecraft operates.

And before we do that, we have to figure out why it matters? What understanding of relativistic physics is improved by deciding whether such a craft gets a slight boost, a slight penalty or no effect on its gas mileage?

 

[DaveC426913]

I'm sorry but to me it seems like that @Mister T was saying the opposite.

Apologies. I was referring to just the one sentence about determining the fuel.

 

[Mister T]

But Newtonian physics is arguably more straightforward and makes more common sense to many.

But it gives you the wrong answers unless the speed is very low compared to $c$.

Wouldn't the aircraft have consumed more fuel than it should have?

It consumed what it consumed. If you try to predict the amount of fuel it consumed using Newtonian physics you get the wrong answer. If you use relativistic physics to calculate the amount of fuel it consumed you get the right answer.

You are mixing up the phenomenon with the study of the phenomenon. Nature behaves in a certain way, in this case consuming fuel to make a trip in a space ship. That's the phenomenon. The purpose of physics is to provide an explanation of that phenomenon, by studying it. If you use Newtonian physics you get a result that doesn't match Nature's behavior, but if you use relativistic physics you get a result that does match Nature's behavior.

there was an increase in its relativistic mass and therefore it needed extra consumption of fuel to complete the trip in its scheduled time

What you are doing here is trying to explain what's going on by using Newton's 2nd Law, $\vec{F}=m\vec{a}$ and replacing $m$ with the relativistic mass. That will not work in general. It will work only if the direction of $\vec{F}$ is perpendicular to the direction of $\vec{a}$. That is why I made the warning in Post #2 about the use of relativistic mass. Of course, it is possible to do perfectly valid relativistic physics using the concept of relativistic mass, you just have to use it properly. But it is also possible to do perfectly valid relativistic physics without it, and that is the preferred method used by physicists when they're doing physics.

 

[vanhees71]

The relativistic mass is the most superfluous and confusing idea Einstein ever had. He himself didn't like it much since the math of his special relativity was sorted out once and for all by Minkowski. Today in all research using the theory of relativity nobody uses "relativistic mass" anymore but only the one and only sensible idea of mass or to emphasize the use of this one and only mass calling it sometimes "invariant mass", where "invariant" refers to the fact that mass is, as any intrinsic quantity of a physical object, a scalar under Poincare transformations by definition.

The closest which comes to something like Newton's 2nd law is the covariant equation
$$m \frac{\mathrm{d}^2 x^{\mu}}{\mathrm{d} \tau^2} = \frac{q}{c} F^{\mu \nu} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau},$$
as the equation of motion of a charged particle in an external electromagnetic field, which is valid only approximately since it neglects the very complicated (and finally ill-defined) problem of radiation reaction.

 

[PainterGuy]

Thanks a lot, everyone!

"That gravity should be innate inherent & essential to matter so that one body may act upon another at a distance through a vacuum without the mediation of any thing else by & through which their action or force may be conveyed from one to another is to me so great an absurdity that I believe no man who has in philosophical matters any competent faculty of thinking can ever fall into it."

Yes, you are right but I think Isaac Newton was more concerned about his 'instantaneous action at a distance' idea; it seems like he knew that there was something wrong with it. The speed of light at that time had not been definitely measured but it had been established in 1675 that it is not infinite. Newton published his theory of gravity around 1687. The idea of 'field' was also not known during Newton's time. I'd say that if Newton had know about the 'field' concept, he'd have been very much satisfied with his theory.

Let's say some Newtonian people have build a spacecraft and written down a timetable for a trip from Earth to Neptune and back. A pilot adjusts the rocket motors so that the spacecraft proceeds according to the timetable. There's a gas pedal.

The pilot will press the gas pedal more than the Newtonians calculated. And the fuel consumption rate will be lower than what Newtonians calculate it to be given that gas pedal position.

(The pilot does not have a clock onboard, but there are clocks floating in space, those clocks tell the pilot the time.)

Thank you for providing more clarity to my question. I hope you wouldn't mind if I use your exact wording.

Are we seriously contemplating a relativistic treatment of an automobile on a highway in space using an internal combustion engine?

Yes, Sir, it's a special kind of hypothetical automobile with an internal combustion engine traveling on a freeway from Earth to Neptune! :)

The purpose of physics is to provide an explanation of that phenomenon, by studying it. If you use Newtonian physics you get a result that doesn't match Nature's behavior, but if you use relativistic physics you get a result that does match Nature's behavior.

Agreed that the theories of relativity model the nature more correctly.

The relativistic mass is the most superfluous and confusing idea Einstein ever had. He himself didn't like it much since the math of his special relativity was sorted out once and for all by Minkowski.

You are right about it. I was trying to stick with this idea of 'relativistic mass' because it seemed to help me express myself better as a novice.

Coming back to the question:
Before the theory of general relativity, it was well known that the rate of Mercury's precession disagreed from that predicted from Newton's theory by 43″ (arc seconds), and hence it was clear to Newtonians that something was wrong. The answer was only found around 1915. I'm also trying to place my question in the same setting with the assumption that the theories of relativity are not known.

Let's say some Newtonian people have build a spacecraft and written down a timetable for a trip from Earth to Neptune and back. A pilot adjusts the rocket motors so that the spacecraft proceeds according to the timetable. There's a gas pedal. The pilot does not have a clock onboard, but there are clocks floating in space, those clocks tell the pilot the time; there are also distance markers floating in space. The pilot has to complete the trip on fixed schedule.

As the spacecraft is moving really, really fast so its mass increases and it'd require more thrust than was originally calculated. Let's say that it was originally calculated that the aircraft should consume 6 million gallons to complete the round trip.

When the trip is completed, would it be less than 6 million gallons or more? If it actually consumes 6 million gallons as was calculated then Newtonians should feel triumphant.

(I understand that I'm essentially asking the same question but it might be possible that I wasn't able to express myself properly. Thank you for the understanding.)

 

[PeroK]

When the trip is completed, would it be less than 6 million gallons or more? If it actually consumes 6 million gallons as was calculated then Newtonians should feel triumphant.

There are no Newtonians left. There isn't a debate. If you want to push the theory that Newtonian physics should be restored, you are in the wrong place.

 

[jbriggs444]

Yes, Sir, it's a special kind of hypothetical automobile with an internal combustion engine traveling on a freeway from Earth to Neptune! :)

So it is slurping up air from the vacuum, burning gasoline from the fuel tank, and rotating tires to produce friction against a non-existent roadbed. We are to evaluate this for efficiency.

An evaluation of efficiency calls for a detailed understanding of the mechanism. None has been forthcoming.

 

[jartsa]

When the trip is completed, would it be less than 6 million gallons or more? If it actually consumes 6 million gallons as was calculated then Newtonians should feel triumphant.

Let's consider what those Newtonians observe when they look at the passing spacecraft .

The Newtonians see that the area of the opening in the valve on the fuel line is gamma³ times larger than what they had calculated. And fuel flows 1/gamma² times slower than what they had calculated. And the fuel is gamma times denser than it normally is.

Now we can calculate the fuel consumption rate: It's gamma squared times larger than what they had calculated. (Let's say they measure the amount of fuel in moles, so that it's clear what they mean)

(Length contraction has no effect on the aforementioned opening, because of the way that the fuel line is oriented, only the gas pedal has an effect on it)

(The fuel is flowing parallel to the motion of the spacecraft , that is why it is flowing so slowly)

(The relativistic longitudinal mass of the spacecraft is gamma³ times the normal mass , that's why the pilot opens the valve as much as he does )

 

[jbriggs444]

The Newtonians see that the area of the opening in the valve on the fuel line is gamma³ times larger than what they had calculated.

What angle is the valve oriented at? Length contraction only works parallel to the direction of motion. That is the teeniest tiniest tip of the iceberg on the complexities here.

Though I am not sure how you got a gamma³ for an area measurement anyway.

Edit: A better way to model a complex engine is to shift to the frame of reference where the engine is at rest. Then you get power generation and fuel consumption rates directly from time dilation. But you notice that ingesting air and ejecting exhaust make analysis difficult. The next step is to shift to an idealized design where you carry the oxidizer with you, get rid of the highway and eject 100% of the burned fuel+oxidizer as reaction mass. Then visit Wikipedia.

 

[jartsa]

What angle is the valve oriented at? Length contraction only works parallel to the direction of motion. That is the teeniest tiniest tip of the iceberg on the complexities here.

Though I am not sure how you got a gamma³ for an area measurement anyway.

I added some explanations later. The orientation was one of the added things.

And the last line is supposed to explain that gamma³.

Perhaps this is better explanation:

Coordinate acceleration = Proper acceleration / gamma³

But the Newtonians think that coordinate acceleration = proper acceleration

So the pilot needs gamma³ times more proper acceleration than what the Newtonians calculated, in order to have some coordinate acceleration, so the pilot opens the throttle so much that the area of the opening is gamma³ times larger what Newtonians calculated it to be.

 

[PeroK]

Coordinate acceleration = Proper acceleration / gamma³

But the Newtonians think that coordinate acceleration = proper acceleration

So the pilot needs gamma³ times more proper acceleration than what the Newtonians calculated, in order to have some coordinate acceleration, so the pilot opens the throttle so much that the area of the opening is gamma³ times larger what Newtonians calculated it to be.

That begs the question of how fuel consumption is related to proper acceleration?

 

[SiennaTheGr8]

The relativistic mass is the most superfluous and confusing idea Einstein ever had. He himself didn't like it much since the math of his special relativity was sorted out once and for all by Minkowski. Today in all research using the theory of relativity nobody uses "relativistic mass" anymore but only the one and only sensible idea of mass or to emphasize the use of this one and only mass calling it sometimes "invariant mass", where "invariant" refers to the fact that mass is, as any intrinsic quantity of a physical object, a scalar under Poincare transformations by definition.

I agree, but I go a step further (perhaps a step too far?).

IMO, it's unfortunate that the most common convention is to use "mass" for the invariant/rest/proper quantity and "energy" for the frame-dependent quantity. This terminology obscures the true meaning of the mass–energy equivalence, which is that "invariant/rest/proper energy" and "mass" are precisely the same quantity, just expressed in different units. It's remarkable! The scalar $m$ that appears in Newton's second law (and indeed in his gravitation law) turns out to be nothing more than the conserved quantity $E$ when measured in the rest-frame.

Learning this should be a marvelous "aha!" moment! It synthesizes what were thought to be different fundamental concepts. Crucially, it should unify and simplify the physics! So why do we keep both terms around, when the whole point is that there's only one "thing" here? (The answer is convention, of course.) I'd wager that most beginners are "set back" in their understanding every time they see an equation like $E = \gamma mc^2$ or $E_k = (\gamma - 1) mc^2$. The relationship between "mass" (rest energy) and total energy is the same as the relationship between proper time ($\tau$ or $t_0$) and coordinate time ($t$), but for the latter pair the convention is to use the word "time" in both, express them in the same unit, and give them related symbols.

A common and compelling argument against "relativistic mass" is that it's just "total energy" in different units. Well, (invariant) "mass" and "rest energy" are redundant in the same way. The logical thing to do is to pick one—"mass" or "energy"—and stick with it. The natural choice is "energy," because it was already a velocity-dependent quantity ("kinetic mass" sounds terrible, and "total mass" would mislead people into thinking that it's the sum of rest-masses). Rest energy, not mass. $E = \gamma E_0$, as $t = \gamma t_0$.

(End of rant.)

The closest which comes to something like Newton's 2nd law is the covariant equation
$$m \frac{\mathrm{d}^2 x^{\mu}}{\mathrm{d} \tau^2} = \frac{q}{c} F^{\mu \nu} \frac{\mathrm{d} x^{\mu}}{\mathrm{d} \tau},$$
as the equation of motion of a charged particle in an external electromagnetic field, which is valid only approximately since it neglects the very complicated (and finally ill-defined) problem of radiation reaction.

More generally, aren't the four-force and four-acceleration related in a Newton-ish way (for constant rest energy mass)?

$$f^{\mu} = m \frac{\mathrm{d}^2 x^{\mu}}{\mathrm{d} \tau^2}$$

 

[vanhees71]

Yes, that's what I say: The mass is a scalar quantity, and energy and momentum build together a four-vector, i.e., an invariant object. The temporal component of this four-vector refers to a reference frame with its defining basis of Minkowski-orthonormal vectors.

Of course, because of
$$p^{\mu}=m \frac{\mathrm{d}^2 x^{\mu}}{\mathrm{d} \tau}$$
you have
$$p_{\mu} p^{\mu}=m^2 c^2$$
with $m$ the scalar invariant mass. Splitting into temporal and spatial components wrt. a Minkowski basis you have the energy-momentum relation
$$E=c \sqrt{m^2 c^2+\vec{p}^2}.$$
In the general equation of motion you quote, the $f^{\mu}$ are four-vector components but constrained due to
$$p_{\mu} \frac{\mathrm{d} p^{\mu}}{\mathrm{d} \tau}=p_{\mu} f^{\mu}=0.$$
That mass is a scalar but energy a frame-dependent quantity also "Newtonian" in the sense that also there mass is a scalar and energy a frame-dependent quantity, though there the relation between mass and the rest of the quantities is more complicated due to the different structure of the symmetry group of Galilei-Newton spacetime: While for the proper orthochronous Poincare group (and its Lie algebra) the mass (squared) is a Casimir operator, for the Galilei group it's a central charge. This makes a profound difference, particularly in quantum theory, but that's another story.

 

[PeterDonis]

it's a special kind of hypothetical automobile with an internal combustion engine traveling on a freeway from Earth to Neptune! :)

Nobody in this thread is actually analyzing any such thing. To the extent anyone is actually analyzing anything, they are analyzing a rocket. Even you talk about the pilot adjusting "rocket motors". So you might as well say "rocket" and stop talking about the hypothetical automobile; all the latter is doing is confusing people.

 

[PeterDonis]

"invariant/rest/proper energy"

Is not the quantity people are usually interested in when they talk about "energy". If you fall off a cliff, and you're worried about your chances of survival, you don't care about your invariant/rest/proper energy; you care about your kinetic energy in the frame in which the Earth's surface is at rest.

 

[SiennaTheGr8]

"invariant/rest/proper energy"

Is not the quantity people are usually interested in when they talk about "energy". If you fall off a cliff, and you're worried about your chances of survival, you don't care about your invariant/rest/proper energy; you care about your kinetic energy in the frame in which the Earth's surface is at rest.

Is your point that in some contexts the word "energy" is used as shorthand for "kinetic energy"?

 

[PeterDonis]

Is your point that in some contexts the word "energy" is used as shorthand for "kinetic energy"?

My point is that when people use the word "energy" they generally are not using it as shorthand for "invariant/rest/proper energy".

 

[Mister T]

Let's say some Newtonian people have build a spacecraft and written down a timetable for a trip from Earth to Neptune and back.

Those people will not have any affect on the way Nature behaves.

The pilot does not have a clock onboard, but there are clocks floating in space, those clocks tell the pilot the time;

There is no such thing as "the time". Those clocks could be ticking at different rates and be out of sync, depending on their speed relative to each other and to the pilot, and to the convention used to synchronize them. For example if two of those clocks were at rest relative to each other and synchronized in their rest frame they won't be synchronized according to the pilot who traverses the path between them. And if the pilot were to carry a clock with him he would find that those two clocks are also running slow compared to his clock.

(I understand that I'm essentially asking the same question but it might be possible that I wasn't able to express myself properly. Thank you for the understanding.)

I think what you're asking about can be answered by looking at the way particle accelerators work. It takes a lot more energy to accelerate the particles than what is predicted using Newtonian physics.

What I see is that you don't seem to be learning anything from the responses you're receiving. For example, you still seem to think that the mass of the spaceship changes with its speed, when there is in fact no way for observers aboard the ship to distinguish between a state of uniform motion and a state of rest.

 

[jartsa]

That begs the question of how fuel consumption is related to proper acceleration?

I'm not a rocket scientist, but if you have one rocket engine running, and you fire up another identical one, you get double (proper) force, double (proper) acceleration, and double (proper) fuel consumption rate.

(Actually now that I think about it, it also results in double coordinate force, double coordinate acceleration and double coordinate fuel consumption rate, I mean momentarily, as long as the velocity has not changed too much)

There is still the question is pressing the gas pedal of your spaceship so that the fuel consumption rate doubles equivalent to firing up another identical set of rocket engines. I would say that yes it is.

 

[PeterDonis]

There is still the question is pressing the gas pedal of your spaceship so that the fuel consumption rate doubles equivalent to firing up another identical set of rocket engines. I would say that yes it is.

For a real rocket, in general, it won't be, because the rocket's efficiency will not be constant over the entire range of thrust that it is capable of.

For purposes of this discussion, however, I think we can assume an idealized rocket that always has 100% efficiency. For such an idealized rocket, yes, you are correct.

 

[SiennaTheGr8]

My point is that when people use the word "energy" they generally are not using it as shorthand for "invariant/rest/proper energy".

Agreed—in fact, I don't think I've ever seen "energy" used as shorthand for "rest energy." Wasn't trying to suggest anything of the sort.

What I'm saying is that rest energy and mass are the same thing expressed in different units, that it's redundant and potentially confusing to keep them both around once this has been established (just as it's redundant and potentially confusing to introduce a "relativistic mass," which is just "total energy" in different units), and that "rest energy" is the better choice once we've kicked "relativistic mass" to the curb.

The equivalence of mass and rest energy reduces seemingly disparate concepts to one. Pairing "total energy" with "mass" instead of with "rest energy" is swimming against the tide—linguistically, symbolically, dimensionally, pedagogically. It's a bad convention IMO. Viva $E_0$!

Anyway, rather off-topic. Apologies.

 

[PeterDonis]

What I'm saying is that rest energy and mass are the same thing expressed in different units

If we interpret "mass" in the usual modern sense of "invariant mass", yes. (What's more, if we use "natural" units in which $c = 1$, they aren't even in different units.)

that it's redundant and potentially confusing to keep them both around once this has been established

Agreed.

and that "rest energy" is the better choice

I disagree. Your argument is basically that since we say "total energy" instead of "relativistic mass", we should say "rest energy" instead of "rest mass". But that argument assumes that "total energy" and "rest energy" are the same kind of thing, physically. They're not. Rest mass is an invariant; total energy is not.

Also, FWIW, physicists in general appear to use "rest mass" when talking about invariant mass. Using "rest energy" is extremely rare in the modern literature. So whatever your or my opinion might be, the physics community in general has settled on "rest mass" (or "invariant mass") being the standard usage.

 

[SiennaTheGr8]

Also, FWIW, physicists in general appear to use "rest mass" when talking about invariant mass. Using "rest energy" is extremely rare in the modern literature. So whatever your or my opinion might be, the physics community in general has settled on "rest mass" (or "invariant mass") being the standard usage.

Yes, and that's my point: I think it's a bad convention.

I disagree. Your argument is basically that since we say "total energy" instead of "relativistic mass", we should say "rest energy" instead of "rest mass".

That's part of my argument. There's also the question of dimensions/units (if not using $c = 1$) and the question of symbols (something $E$-ish like $\mathcal{E}$ or $E_0$ vs. $m$).

But that argument assumes that "total energy" and "rest energy" are the same kind of thing, physically. They're not. Rest mass is an invariant; total energy is not.

I don't think whether they're "the same kind of thing, physically" is such a straightforward question. Yes, one is the Lorentz-invariant norm of the four-vector that the other is the frame-dependent time-component of, but one is also the other when reckoned in the rest frame.

You could make the same argument about "coordinate time" and "proper time." And if we called "proper time" some other term that didn't have "time" in it and gave it a different unit and a symbol that wasn't $t$-ish, I'd likewise say that this isn't as learner-friendly as it could be.

Cheers, Peter.

 

[PeterDonis]

I don't think whether they're "the same kind of thing, physically" is such a straightforward question.

It is. They're not.

one is also the other when reckoned in the rest frame

Yes, in one particular frame. Which just highlights that total energy is frame-dependent and invariant mass is not.

You could make the same argument about "coordinate time" and "proper time."

Indeed you could. And to me, that's an argument for figuring out something besides "time" to attach the "coordinate" label to. Unfortunately physics terminology is not always logical.

if we called "proper time" some other term that didn't have "time" in it

No, if we called coordinate time some other term that didn't have "time" in it, to emphasize the fact that "time", the thing we actually experience and measure, the physical thing, is proper time.

I think doing that would be more learner friendly than what we do now. As evidence I give you the thousands of PF threads, including this one, that involve the mistaken belief that coordinate time is somehow physically meaningful, because it's called "time".

 

[PainterGuy]

Thank you, everyone, for your time.

There are no Newtonians left. There isn't a debate. If you want to push the theory that Newtonian physics should be restored, you are in the wrong place.

I did not say any such thing. I'm only trying to learn and each of us learn differently and each of us have different level of intelligence and ability to express themselves. I'm trying to learn the basic concepts in layman terms or you can say in 'pop science' fashion. I hope you don't mind it. Thanks!

So it is slurping up air from the vacuum, burning gasoline from the fuel tank, and rotating tires to produce friction against a non-existent roadbed. We are to evaluate this for efficiency.

An evaluation of efficiency calls for a detailed understanding of the mechanism. None has been forthcoming.

My apologies, Sir, that I couldn't frame the question properly but I made an honest attempt. But I hope now you understand what I was trying to say and the point I was trying to make.

Now we can calculate the fuel consumption rate: It's gamma squared times larger than what they had calculated.

So, finally, it's concluded that Newtonians would be see that more fuel has been consumed than was originally calculated. The mass increase during the trip was real. In their attempt to understand this difference, they will try to find an explanation. I wanted to clarify three more related points.

Question 1:
There is still one point which bothers me. During the trip, the pilot cannot know if local time has been dilated compared to that of the Earth and also he cannot find that mass of rocket has increased but he can definitely see the gauge and see that more fuel is being consumed than was calculated. Would the pilot be able to notice the difference?

Question 2:
Let's assume that there was a clock present on the rocket which wasn't used at all during the trip. Newtonians notice that the time on this clock lags behind the ones on earth. They carefully examine the engine and mass of the rocket and find no problem with it. In short, something has happened to the time and it's permanent but whatever happened to either rocket mass/engine was only true for the duration of trip.

Anyway, I stumbled upon the following text and this relates to what I'm saying above and what is confusing me, https://arxiv.org/ftp/arxiv/papers/0707/0707.2426.pdf, and the first para under 'Introduction' interested me.

"In this discussion, the speed of light c is assumed to be invariant. The physical reality is only time dilation
by velocity. The constancy of the speed of light causes the time dilation by motion. Mass m and length x are
also invariant though mass and length appear to be variant through the Lorentz transformation of reference
time. That is, Lorentz contraction of length and inertial mass increase can also be explained by the Lorentz
transformation of reference time".

In my view, if local time has been dilated then it would result into length contraction as well. In other words, length contraction results from the time dilation. For example, to an observer on the Earth, the muon travels at 0.950 c for 7.05 μs from the time it is produced until it decays. Thus it travels a distance L0 = vΔt = (0.950)(3.00 × 108 m/s)(7.05 × 10−6 s) = 2.01 km relative to the Earth. In the muon’s frame of reference, its lifetime is only 2.20 μs. It has enough time to travel only L0 = vΔt0 = (0.950)(3.00 × 108 m/s)(2.20 × 10−6 s) = 0.627 km. The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it. Source: https://courses.lumenlearning.com/physics/chapter/28-3-length-contraction/

I think it'd be appropriate to quote a passage from Wikipedia. I'm sorry that some of the math isn't appearing properly but you can see the original reference.

"Concepts that were similar to what nowadays is called "relativistic mass", were already developed before the advent of special relativity. For example, it was recognized by J. J. Thomson in 1881 that a charged body is harder to set in motion than an uncharged body, which was worked out in more detail by Oliver Heaviside (1889) and George Frederick Charles Searle (1897). So the electrostatic energy behaves as having some sort of electromagnetic mass {\displaystyle m_{em}=(4/3)E_{em}/c^{2}}

, which can increase the normal mechanical mass of the bodies.[9] [10]

Then, it was pointed out by Thomson and Searle that this electromagnetic mass also increases with velocity. This was further elaborated by Hendrik Lorentz (1899, 1904) in the framework of Lorentz ether theory. He defined mass as the ratio of force to acceleration, not as the ratio of momentum to velocity, so he needed to distinguish between the mass {\displaystyle m_{L}=\gamma ^{3}m}

parallel to the direction of motion and the mass {\displaystyle m_{T}=\gamma m}

perpendicular to the direction of motion (where {\displaystyle \gamma =1/{\sqrt {1-v^{2}/c^{2}}}}

is the Lorentz factor, v is the relative velocity between the aether and the object, and c is the speed of light). Only when the force is perpendicular to the velocity, Lorentz's mass is equal to what is now called "relativistic mass". Max Abraham (1902) called {\displaystyle m_{L}}

longitudinal mass and {\displaystyle m_{T}}

transverse mass (although Abraham used more complicated expressions than Lorentz's relativistic ones). So, according to Lorentz's theory no body can reach the speed of light because the mass becomes infinitely large at this velocity.[11] [12] [13]"
Source: https://en.wikipedia.org/wiki/Mass_in_special_relativity#Transverse_and_longitudinal_mass

Question 3:
To me, two different concepts are being mixed up by the term 'length contraction' here. When mass is moving really fast, it gets deformed and it results into the length contraction of mass. But at the same time when a mass is moving really fast, the distance traveled by it also gets shortened due to the time dilation so this is another type of length contraction involving 'distance traveled'.

"Length contraction was postulated by George FitzGerald (1889) and Hendrik Antoon Lorentz (1892) to explain the negative outcome of the Michelson–Morley experiment and to rescue the hypothesis of the stationary aether (Lorentz–FitzGerald contraction hypothesis).[2][3] Although both FitzGerald and Lorentz alluded to the fact that electrostatic fields in motion were deformed ("Heaviside-Ellipsoid" after Oliver Heaviside, who derived this deformation from electromagnetic theory in 1888), it was considered an ad hoc hypothesis, because at this time there was no sufficient reason to assume that intermolecular forces behave the same way as electromagnetic ones. In 1897 Joseph Larmor developed a model in which all forces are considered to be of electromagnetic origin, and length contraction appeared to be a direct consequence of this model."
Source: https://en.wikipedia.org/wiki/Length_contraction#History

I think what you're asking about can be answered by looking at the way particle accelerators work. It takes a lot more energy to accelerate the particles than what is predicted using Newtonian physics.

Yes, you have it right.

Thank you!

 

[PeroK]

Anyway, I stumbled upon the following text and this relates to what I'm saying above and what is confusing me, https://arxiv.org/ftp/arxiv/papers/0707/0707.2426.pdf, and the first para under 'Introduction' interested me.

1) The paper you link to is essentially nonsense. You are never going to learn SR by reading papers like the one above.

2) You still have fundamental misconceptions about SR. We can't really help you unless you willing to try to learn SR properly. For example:

In my view, if local time has been dilated then it would result into length contraction as well. In other words, length contraction results from the time dilation ...

This reveals a fundamental misunderstanding of SR.

The first chapter of Morin's undergraduate text on SR is available free online. I suggest you study it:

http://www.people.fas.harvard.edu/~djmorin/Relativity Chap 1.pdf

My advice is to start afresh with Morin. Try to put everything you think you know about SR to one side and try to learn it properly.

At the moment you are simply floundering in a sea of misconceptions.