Inertial Privilege: Reframing Physics for Accelerated Frames

[Ontophobe]

There are inertial reference frames and accelerated reference frames, and the laws of physics change depending on the frame through which you're observing them. The universe when viewed through an inertial frame won't let you go faster than light, but the very same universe when viewed through an accelerated frame will. Physics typically treats inertial frames as the default frames, thereby marginalizing accelerated frames. Meanwhile, with the exception of skydivers and astronauts, humans spend 99.999% of their lives in a genuine accelerated reference frame. So what would happen if we formulated the laws of physics as experienced through an accelerated frame instead of an inertial one? Imagine the Einstein of an alternate timeline. His life is identical to our Einstein's save one respect: he formulated his theory from the pov of an accelerated frame. What would his theory look like? We know, for example, that at 1g of acceleration you would reach the neighborhood of the speed of light in about a year, but (provided you had enough fuel) you would not suddenly stop accelerating. From the crew's reference frames, they would blow right past the speed of light. In 30 some years, they could cross a distance that takes 2,000,000 years for light to cross. So we know that this theory wouldn't regard 186,000 miles per second as a speed limit. What else would this alternate Einstein say?

 

[pervect]

There are inertial reference frames and accelerated reference frames, and the laws of physics change depending on the frame through which you're observing them.

I don't totally agree. The laws of physics can be written in a manner that doesn't depend on the observer (or their choice of coordinates). This has been known for approximately 100 years, using tensor formalisms. Of course, if one doesn't use the coordinate independent formalism (for instance, if one uses some more or less standard high school methods), the indepdendence of the laws of physics from the observer can be obscured. But the observer independence is still there, it just might not be obvious.

The universe when viewed through an inertial frame won't let you go faster than light, but the very same universe when viewed through an accelerated frame will.

If you, set up a short race course in a vacuum, and run a race between a light beam and some competitor, the laws of physics say that the light beam will always finish before the competitor can. This doesn't depend on the frame of reference at all. One imagines a referee at the end of the race course, who simply observes which contestant finishes first.

I should add for completeness , using the jargon of relativity, that it is assumed that "the competitor" is a massive object, i.e. one with non-zero rest mass.

So in this sense, using a fair test, nothing with mass outruns light. And it doesn't matter if one uses an inertial frame or a non-inertial one.

Important issues to make this claim true: the course must be fair, if there are multiple paths that the light beam could take to reach the finish (for instance through a wormhole or not), it's possible that the competitor could take a short-cut (through the wormhole, for example) while the light takes the long way around, and beat the light beam.

There's some theorems that say that in a small enough region of a manifold (the model of classical space time, the small region of the manifold being known as a local convex region of the manifold) there can't be multiple paths, so making the course short enough should solve the fairness issue.

Also, this applies to a vacuum situation. Light can be slowed by traveling through a media. There's some more interesting points that could be made here, but they'd be a bit of a digression so I'll avoid the temptation and just say that for our purposes, we're comparing speeds in a vacuum. If it is felt that the issue is significant for some reason, more can be said.

Possibly there are other caveats to add, but none come to mind at the moment. I believe these are the main two.

This doesn't explain why the high-school methods break down, but it does, hopefully , give some insight into the underlying physics, without getting into the mathematical details.

 

[Ontophobe]

Honestly, thank you millions for understanding that I was genuinely curious about this and for taking the question seriously and at face value. I was bracing myself for trolls.

 

[rkolter]

We know, for example, that at 1g of acceleration you would reach the neighborhood of the speed of light in about a year, but (provided you had enough fuel) you would not suddenly stop accelerating. From the crew's reference frames, they would blow right past the speed of light...

Well, no. No they wouldn't - I want to clear this bit up. You say "Provided you had enough fuel" but the amount of fuel to accelerate any mass to the speed of light is not really large. It's infinite. Even if I give you an engine that has the ability to instantly transport matter into itself from anywhere, convert that matter perfectly into energy and with perfect efficiency turn that energy into propulsion - you don't have infinite mass to feed the engine.

Your crew would never achieve the speed of light, nor "blow right past it". They would accelerate at 1g until their doomsday engine converted the entire universe into energy used to move their craft forward. Then... that would be it. The engine would power down and gravity would vanish (no acceleration). The battery, if they had one, would drain and the lights would turn off. If they wrote down their final speed before that happened, it would be very close to the speed of light.

In 30 some years, they could cross a distance that takes 2,000,000 years for light to cross. So we know that this theory wouldn't regard 186,000 miles per second as a speed limit. What else would this alternate Einstein say?

You have, at the core, a basic misunderstanding. Or, you appear to. The speed of light is not relative. In ANY frame of reference, it moves at the speed of light. If you are going 90% the speed of light and turn on a flashlight, the light spreads from the flashlight at the speed of light, not at only 10%, and not at 190%. But what IS relative is TIME and DISTANCE. If your crew was moving the speed of a neutrino (very close to light speed) As they passed through our solar system, the distance between the Sun and Earth would only be a few thousand kilometers, and the Earth would be less than 20 meters thick. They would say it took less than a second to pass the distance from the Sun to the Earth - but, they would also measure that distance to be much shorter, and so would still state they were moving under the speed of light. If they could watch the Earth as they approached, they would see clouds move - 8 minutes worth of "our" movement in less than a second of "their" time.

The logical but inescapable conclusion is that if they could accelerate to the speed of light and keep their frame of reference in existence until they hit light speed, that they would in that instant traverse an infinite amount of space while an infinite amount of time passes outside their window. This means, they would die.Having traversed all time, if the universe has an end, they would meet it. If it does not, they would die by flying at the speed of light into an infinitely hot, infinitely dense wall of matter made up of infinite virtual particles that would exist in the infinite space they traveled through.

 

[Battlemage!]

Well, no. No they wouldn't - I want to clear this bit up. You say "Provided you had enough fuel" but the amount of fuel to accelerate any mass to the speed of light is not really large. It's infinite. Even if I give you an engine that has the ability to instantly transport matter into itself from anywhere, convert that matter perfectly into energy and with perfect efficiency turn that energy into propulsion - you don't have infinite mass to feed the engine.

Your crew would never achieve the speed of light, nor "blow right past it". They would accelerate at 1g until their doomsday engine converted the entire universe into energy used to move their craft forward. Then... that would be it. The engine would power down and gravity would vanish (no acceleration). The battery, if they had one, would drain and the lights would turn off. If they wrote down their final speed before that happened, it would be very close to the speed of light.

Wait. Relative to what? Wouldn't their speed be zero relative to their own reference frame?

 

[rkolter]

Wait. Relative to what? Wouldn't their speed be zero relative to their own reference frame?

The speed of light is the speed of light. Measure it in the same medium in any frame of reference and it will be the same. Speed is not a vector - it is directionless and does not require an outside frame of reference to measure. The ship's speed could be calculated just by knowing the ship's acceleration and how long the ship had been accelerating.

Velocity on the other hand, requires a direction, and that means you need to measure it relative to something else. If the rocket was in a gravity well that light could not escape, then its speedometer would measure its speed, but its velocity would be at best, zero in the direction you want to travel. But, we've given the ship an engine that EATS THE UNIVERSE. So, imagining that it preferentially eats all the mass nearby the ship first and along its projected path next is not that much of a stretch.

Either way - uninhibited or in a gravity well, the engine will run out of fuel before the speedometer reaches light speed.

 

[Bandersnatch]

Well, no. No they wouldn't - I want to clear this bit up. You say "Provided you had enough fuel" but the amount of fuel to accelerate any mass to the speed of light is not really large. It's infinite. Even if I give you an engine that has the ability to instantly transport matter into itself from anywhere, convert that matter perfectly into energy and with perfect efficiency turn that energy into propulsion - you don't have infinite mass to feed the engine.

The OP also said 'neighbourhood of the speed of light', wchich is a perfectly valid, if ambiguous, statement. You're arguing against your own misreading of the question.

The ship's speed could be calculated just by knowing the ship's acceleration and how long the ship had been accelerating

and the direction of acceleration, and the initial value of speed, and the reference frame in which it is measured. Unless you can show that speed of a rocket that reverses acceleration half way through and as measured from a moving frame is the same as speed of a rocket that doesn't reverse and measured from a frame moving in a different direction (or stationary w/r to the ship, for that matter).
Speed is the legth of the velocity vector. Everything that influences the vector's legth affects speed.

 

[rkolter]

The OP also said 'neighbourhood of the speed of light', wchich is a perfectly valid, if ambiguous, statement. You're arguing against your own misreading of the question.

I'm not really here to pick fights - I think you picked that quote from the OP out of context. The OP said if they accelerated at 1 g that would get you to the neighborhood of the speed of light in about a year, but the statement went on to say that provided they had enough fuel they would not stop accelerating and from the crew's reference frame would "blow right past the speed of light". I have argued that this would not occur.

...and the direction of acceleration, and the initial value of speed, and the reference frame in which it is measured. Unless you can show that speed of a rocket that reverses acceleration half way through and as measured from a moving frame is the same as speed of a rocket that doesn't reverse and measured from a frame moving in a different direction (or stationary w/r to the ship, for that matter).
Speed is the legth of the velocity vector. Everything that influences the vector's legth affects speed.

Speed is a directionless quantity. Any given velocity has a speed associated with it. But we can, and regularly do, measure speeds without knowing velocities. Your car does this all the time - it has a speedometer that measures the car's speed by knowing the wheel radius and rotations per time period. A car traveling East or West at 70 miles an hour is going the same speed regardless. I suggested a similar device that measures the rocket's acceleration and time it has spent accelerating to determine the rocket's speed.

To sum it up - it takes more energy to accelerate an object, the faster the object is moving in that direction. To reach the speed of light in any direction will take infinite energy. How close the rocket gets to the speed of light depends on how close to infinite fuel you have. If you have less than infinite fuel, then any fuel spent accelerating in a different direction would impact your final top speed. If you HAVE infinite fuel, a lot of really awful things happen. Like becoming a singularity while smashing into an infinitely dense wall of virtual particles at the speed of light.

 

[Battlemage!]

The speed of light is the speed of light. Measure it in the same medium in any frame of reference and it will be the same. Speed is not a vector - it is directionless and does not require an outside frame of reference to measure. The ship's speed could be calculated just by knowing the ship's acceleration and how long the ship had been accelerating.

Velocity on the other hand, requires a direction, and that means you need to measure it relative to something else. If the rocket was in a gravity well that light could not escape, then its speedometer would measure its speed, but its velocity would be at best, zero in the direction you want to travel. But, we've given the ship an engine that EATS THE UNIVERSE. So, imagining that it preferentially eats all the mass nearby the ship first and along its projected path next is not that much of a stretch.

Either way - uninhibited or in a gravity well, the engine will run out of fuel before the speedometer reaches light speed.

Yes, but in their reference frame they aren't moving. They feel the acceleration, sure, but like the elevator thought experiment, they can claim to be at rest, right? (I'm assuming this might get into general relativity since it's not an inertial reference frame) In any event, we all hopefully agree they won't reach the speed of light according to any observer.

 

[rkolter]

Yes, but in their reference frame they aren't moving. They feel the acceleration, sure, but like the elevator thought experiment, they can claim to be at rest, right? (I'm assuming this might get into general relativity since it's not an inertial reference frame) In any event, we all hopefully agree they won't reach the speed of light according to any observer.

Right - nothing I've suggested changes the fact that the crew could equally measure themselves as moving or stationary. As they accelerate towards the speed of light, an observer outside the craft and an observer in the craft would both agree they never reach the speed of light. Regardless of the frame of reference, something with mass cannot reach the speed of light - that "something with mass" could equally well be the ship, or the universe outside the ship.

Apparently adding a speedometer on a vehicle that reaches relativistic speeds results in confusion. This is good to know. :)

 

[rbelli1]

From the crew's reference frames, they would blow right past the speed of light.

Only if you use Newtonian equations. Those are only approximations.

BoB