The Lorentz term which impacts time, length, and the energy required for acceleration.
This is known as γ or gamma.
As above, it’s better, equation wise, to discuss momentum and energy rather than mass.
The energy requirement increases without bound the closer you want to get to c.
That’s one of the things about ‘tachyons’ that’s kind of interesting. If a faster than light particle exists, it could never slow down to light speed, either, for the same reason of an infinite energy requirement.
ETA: In ‘conventional’ Newtonian physics, momentum can be written as mv and kinetic energy is 0.5mvv. But those go out the window at relativistic speeds unless you let ‘m’ increase to some sort of ‘relativistic mass’. So the idea of a relativistic mass is a not-good way to reconcile how we have taught ourselves things work in a Newtonian world with what we know at relativistic scales.
But that’s just a number in an equation. What are the real-world effects that would hinder a ship approaching the speed of light? “Mass increases” seemed pretty understandable.
If I am in a ship, who’s engine is capable of accelerating it 1 m per s per s, as I notice my speed increasing, what happens?
I should add that E= mc^2 is a special case. If you consider that Einstein was talking about rest mass and consider the longer form of:
E^2 = (mc^2)^2 + (pc)^2
You will see the momentum component in p
As momentum increases the E or energy increases. It is the “(pc)^2” portion that causes you problems.
You notice that maintaining that 1m per s per s requires more and more thrust from your engine as you go faster. And that eventually, it becomes impossible to maintain, even with infinite energy and that acceleration gradually gets lower and lower and you need more and more thrust just to eke out an infinitesimally small bit of additional speed.
Here is a minutephysics video which will probably help out with this intuition using simple math.
The universe around you changes. Remember everything is relative. It’s in the name of the theory.
As as your speed increases, the universe around you slows down. If you’ve somehow managed to invent an engine that will just keep accelerating you forever, as your speed increases the universe around you slows down sufficiently to make it take forever to reach light speed.
The best argument against relativistic mass at a more intuitive level, at least to me, is that it only works nicely for energy and momentum. If you try to apply it to F = ma you will have to divide it into transverse and longitudinal mass. Only forces working perpendicular to the direction of motion give behavior according to the same relativistic mass concept as energy and momentum. In the direction parallel to travel you need to multiply by the cube of gamma to get the longitudinal mass instead.
Only if you make the calculations for an external reference frame. In your own reference frame the energy requirements are the same.
Ok. But what has happened to you and your ship that causes the same amount of force to no longer provide the same amount of acceleration?
This seems like you are saying that it doesn’t take more energy to keep accelerating at 1 m per s per s.
Is that what you are saying?
That is a good point, remember that a photon simply does not experience time at all. For a photon being emitted and absorbed is a single event.
As a photon doesn’t even experience time how would you even calculate velocity as no time passes?
Time as a dimension is something that is experienced by us massive, slower than light objects.
Keep in mind the increasing amount of energy you’ll need as your payload approaches light speed. The increased value of your gold holdings will be cancelled out by your higher utilities bill.
If you are in a rocket and accelerating constantly, nothing causes the same amount of force to no longer provide the same amount of acceleration. Until you run out of fuel, which is why there are not yet five-day cruises to Saturn on offer. But, assuming you have enough magic antimatter or whatever to keep it up, you will soon enough find yourself in the Andromeda Galaxy and beyond; at the same time, an observer where you started out never sees you exceed the speed of light. In short, nothing is preventing you, it is all a matter of changing coordinates.
Your cargo of gold does not increase in mass either, as has been explained.
From the reference frame of the space ship doing the acceleration, yes, I believe that is the case. Otherwise speed is no longer relative.
But I wasn’t asking about an observer. I was asking about what happens in my ship as I am measuring my speed. As the speed I calculate gets closer to light speed, what happens? When the speed clicks over the SOL, and I say “Hey, we are going faster than light”, does someone say “No we are not, because the math on this paper says we can’t?”
Consider this - a spaceship approaches earth at 0.9c - as it gets near, its path is deflected due to the gravitational effect of the earth. However, the person on the spaceship sees it differently - to them, time is slower and the distance they are travelling toward earth appears shorter - as a result, the deflection due to earth’s gravity appears to be a greater angle. The only explanation for them is that the effect of the earth deflecting their path is due to a much higher mass - i.e. the earth appears on the spaceship to have the effect of a much higher mass.
Only for inertial frames of reference, for which there is no favored frame.
During acceleration, the ship is in a non-inertial frame of reference, and some frames are better than others.
The alternative is you see the earth (or whatever you are leaving behind) accelerating to beyond light speed behind you, which is not possible. There has to come a point when you see that the exertions of your engines no longer produce the same additional velocity for the same energy expenditure.
It is simply impossible to continuously accelerate at 1 m/s/s ad infinitum (assuming the energy was available, anyway). Otherwise, after 300 million seconds, you would observe the earth falling behind faster than the speed of light, which violates our known understanding of the universe.
Am I the only one who thinks that this guy is a wind-up merchant? This is the kind of discussion we used to have when I was a teenager.
Faster compared to what?
You can apply the same force, which will locally appear like you are accelerating but as you do your clock ticks slower and slower and slower compared to an initially co-moving observer.
To an initially co-moving frame, and compared to from your accelerating viewpoint to that same initially co-moving frame you will never reach the speed of light.
Speed is meaningless without another reference point and it will be very challenging to try and convey these effects with our intuitive assumptions.
Ignoring other matter it will still feel like you are being “accelerated” but at some point what still seems like an arbitrary segment of popper time or your local time will be so long in the initial co-moving frames time that you will exceed the heat death of the universe.
This proper acceleration is a measurable acceleration with an accelerometer, but the assumptions our minds try and make is to translate a four-dimensional motion into a three-dimensional motion that we have an intuitive understanding of.
When you get away from our slow speeds where our intuitions work our assumptions start to break down.
We simply want to use coordinate acceleration in our minds and that breaks down under GR as the universal internal frame assumption that is used in SR does not exist.
You can only use proper time, or the local clock time as coordinate time (inertial frame) simply does not hold.
Using Four-acceleration, as the proper time follows the world line there is no paradox here. The problem arises from our human mind’s belief that there is some universal clock but remember that the spacetime interval is the only invariant property in GR once you move past very local and slow interactions. The spacing between events and even the ordering of events is not invariant across observers. The sequence of events, timing of events and order of events is not universal and only causality is preserved.
Note this is an oversimplification and built with bad analogies.