Time dilation, relativistic mass and fuel consumption

[PainterGuy]

TL;DR

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.

 

[Herman Trivilino]

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.

 

[Herman Trivilino]

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".

 

[Herman Trivilino]

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.