[pixl97]

>Yes, the effective mass of a photon is determined by E=mc^2 even though it has no mass

I think for the photon itself it's the more complete equation of E = (square root) m^2c^4 + p^2c^2 that shows how much relativistic energy is in that photon.

[zackmorris]

Hey thanks you're absolutely right. Just to save people some time, here are some Google hits for that and its geometric representation:

https://www.britannica.com/video/185388/equation-theory-ener...

https://physics.stackexchange.com/questions/6202/does-e-mc2-...

So for massless particles your E=sqrt(m^2c^4 + p^2c^2) formula becomes E=sqrt(p^2c^2) which simplifies to the formula for photon momentum p=E/c (aka light pressure).

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HN wouldn't let me reply, but since it's Monday I might as well procrastinate:

If we assume that the light is columnated like a laser, then a 1 watt light absorbed by a 1 kilogram object for 1 second imparts 1 joule so p=E/c so that's 1/c or 3.34e-9 newton seconds of momentum. Note that the momentum imparted is twice this if the light is reflected, to account for absorption and reemission.

To see how much force is produced per watt it's:

https://www.physicsforums.com/threads/radiation-pressure-cau...

Power is just energy per unit time. If you really want forces, then note that force is momentum per unit time... so for light, the relation between power and force of a light beam is the same as the relation between energy and momentum of a light pulse.

  P<->F relation has same form as E<->p
  
  p=E/c
  p/t=(E/t)/c
  F=P/c
  
  Acceleration of an object of a certain mass produceed by light of a certain power:
  
  F=ma=P/c
  a=P/mc
  
  Power of light required to accelerate an object of a certain mass:
  
  P=amc

For example, the power needed to accelerate an object at Earth's gravity of 9.81 m/s^2 is P = 9.81 * m * c=m * 2.94e9 so that's about 3 GW per kg. A light jetpack for a 100 kg person to fly above Earth would take about 300 GW or about 150 Hoover dams or 250 DeLoreans from Back to the Future.

The power required to accelerate a 1 million kilogram nuclear interstellar spacecraft continuously at 1 Earth's gravity is 3 PW, or about 1000 times the 3.3 TW of power produced by all countries on Earth currently. So aliens have probably figured out a more efficient means of propellantless space travel.

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To procrastinate further:

If we go back to your mass-energy equivalence formula E = (square root) m^2c^4 + p^2c^2:

https://en.wikipedia.org/wiki/Mass–energy_equivalence

https://en.wikipedia.org/wiki/Equivalence_principle

The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same trajectories and landing at identical times. The extended form by Albert Einstein requires special relativity to also hold in free fall and requires the weak equivalence to be valid everywhere. This form was a critical input for the development of the theory of general relativity. The strong form requires Einstein's form to work for stellar objects. Highly precise experimental tests of the principle limit possible deviations from equivalence to be very small.

It's basically saying that gravitational and inertial acceleration are the same.

As a thought experiment, imagine being in a large spacecraft, feeling artifical gravity about the same as Earth's, and trying to determine if it's sitting on the ground or accelerating through space.

A way to do that is to measure the divergence of the acceleration with a (laser) tape measure and two plumb bobs. Hang the bobs on distant walls and measure the distance between the strings at floor and ceiling. In space, the strings will be parallel due to linear acceleration. But on Earth, the strings at floor will be slightly closer together due to the divergence of the gravitional field:

https://www.mathscinotes.com/2017/01/effect-of-earths-curvat...

In metric, that bridge is 1298 m long and 211 m tall with a 42.61 mm discrepancy. By similar triangles, a 1 km distance has h=211 * 1000/1298=163 m height and 42.61 * 1000/1298=32.83 mm discrepancy. So the discrepancy per meter of height is 0.03283/163=201.4 microns at 1 km.

Plumb bobs hung from the ceiling by a 3 meter tall wall will be about 600 microns or 0.6 mm closer at the floor for a room 1 km on a side.

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Where I'm going with this is that the divergence at the perimeter of galaxies is higher than it should be, because stars are orbiting faster there than can be accounted for by visible mass alone.

Since there's not enough mass (no attractor) to account for the faster orbits, then something else is curving space, which we currently call dark matter.

But it's possible that there's a missing term or dimension from your (Einstein's) E=sqrt(m^2c^4 + p^2c^2) formula, or that the Equivalence Principle doesn't always hold. Meaning that something else besides mass or energy might contribute to momentum, making objects appear to be moving faster than they should be, or that space is flowing around them to drag them along faster without an attractor.

Which is similar to how a warp drive would work. If we could figure out what that something is, and perhaps modulate it with energy, then we could modulate space and build a reactionless rocket. That would make the divergence of space higher in front of the rocket to accelerate it forward.

Unfortunately we can't detect dark matter, much less modulate it, so we either need a huge empirical discovery or new physics to build a warp drive.

[zackmorris]

Huh, the reply button is back, maybe a PEBKAC bug. I didn't realize this article was a dupe, but the comments on the other article are about black holes and MACHOs, and they got me thinking about the initial and steady state conditions of black holes, which seem to be rarely discussed.

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So might as well procrastinate some more now that it's Tuesday..

Imagine we have a neutron star one atom away from having enough mass to collapse into a black hole. It's so dense that time dilation makes it look red shifted and frozen to us, like how time passes slowly on the water planet on Interstellar. It would be so red shifted that it would look almost infinitely dark to us. Dropping an atom in would cause an event horizon to form at the surface.

Or would it? As soon as the atom fell in, one atom's worth of Hawking radiation would cause the event horizon to disappear again soon afterwards. So the neutron star would be perpetually on the brink of collapsing. I think that this is certain and there should be a way to state this argument formally.

Would the center of the neutron star be under enough pressure to exceed the neutron degeneracy pressure there and start pouring down a tear in space? Yes and no. The pressure would eventually increase, but we have to remember that the star is red shifted. It would take eons for the information that the new atom arrived at the surface to travel at the speed of light to the center of the star. By which time Hawking radiation would have already evaporated that additional mass. In a very real sense, the center of the star is frozen in time from our frame of reference and nearly unreachable. I think that this is also certain and that it can be formally stated.

If we could use a physics analog of mathematical induction, then we could show that no signal from outside ever reaches the center of a black hole, meaning that a singularity can never form there. Instead, time freezes at the center, or at least the relativistic distance to the center increases faster than neutron degeneracy pressure is exceeded. I can't come up with a formal argument for this, but I think whatever this is might help unify gravity and quantum mechanics - loosely that they haven't accounted for the propagation delay time where the infinities arise.

What if more mass fell in though, like a small asteroid that causes an evaporation time of years or more? We'd see the collapse, darkness, and then a lot of Hawking radiation, equivalent to the asteroid turning into energy (matter/antimatter pairs or energy from the virtual particles splitting at the horizon) over the duration of evaporation. But the neutron star would most likely slip back into view as the relativistic space to it shortens, it wouldn't turn completely into energy and explode in an instant (because the center is so time dilated as to appear frozen from our frame of reference). This mechanism might be similar to the naked singularity that Romulans use to generate energy from matter.

So if a singularity never forms initially, and doesn't exist at steady state, that has all sorts of ramifications for the interior of stars. Heat has a random distribution, so there will be regions in stars where the density is periodically hot enough to exceed the density of a black hole there. This is similar to how a water hammer uses the momentum of the water to exceed an elevation higher than the pressure of the water should be able to reach. That pressure would also be higher than that required for fusion. What if hydrogen atoms dip into these temporary black holes and get pushed out by Hawking radiation as helium? I don't know how to state this formally, but it could possibly link gravity with the strong force (which could explain its empirical value).

I posted a link to a black hole sun article in my submissions, which got flagged because its website might not be considered formal by the scientific community, but I think that there could be a large number of black holes popping in and out of existence at the center of the sun. So part of the pressure preventing collapse might come from Hawking radiation. Also, the bounce when a red giant collapses and rebounds as a supernova could be explained by this mechanism. Maybe it slips within a black hole temporarily and a portion of its mass comes back out as energy.

If we start looking at the interior of black holes as additional space instead of a tear in space, it opens up a lot of possibilities. It might be possible to dip into small black holes and escape again, hugely time shifted, so that a second inside might represent a year outside or something. Loosely the amount of time it takes the mass energy equivalent of the ship to be radiated by Hawking would be the duration of the time shift. It would basically be a time machine that exchanges mass energy for time. It might even be possible for the ship to survive if the orbit is high enough to survive the tidal forces. Does this count as information escaping the black hole? I don't know how to state that formally or what the limit might be - if it could be proven for something with structure above a nucleon (like an atom), then maybe a full relation could be stated.

The reason I wrote this out is that it got me thinking about neutrinos traveling at nearly the speed of light when they encounter black holes. Maybe they dip in temporarily before being radiated back out (when Hawking radiation shrinks the black hole radius below their orbital radius) possibly at a slower speed. If it takes a lightyear of lead to potentially stop a neutrino, then what happens if one orbits inside a neutron star on the brink of collapsing, potentially for lightyears? Could it come out at a slower speed from friction or a weak interaction with other neutrinos? I don't see why not. I also don't know how to state this formally - I don't think we have equations yet for how a large number of neutrinos in a small space might interact at subatomic distances via the weak force.

But if the neutrinos came out at a slower speed, perhaps beneath the escape velocity of the surrounding galaxy, then they would tend to collect in clouds around them, orbiting the supermassive black hole at the center. Which looks very much like the dark matter halos we infer from fast galaxy rotations, and maybe the filaments of mass around the universe's galactic voids.

I guess this negates what I was saying about maybe modulating the divergence of space with energy. Then again, I don't see why something like the NIF laser fusion reactor couldn't be scaled up from fusion densities to black hole densities. Then more matter could be added than pressure alone would fuse, and we'd effectively have the naked singularity that the Romulans use for generating much higher amounts of energy. Or maybe based on the derivation above, fusion density is already equivalent to a temporary black hole, and scaling is all that is needed. I'm pretty sure that there's no exclusion limit for photons (since they're bosons), so it might be more practical to focus smaller lasers on something approaching their wavelength (the size of an atom) rather than coming up with bigger or more powerful lasers. Which suggests that a sphere of small (LED?) lasers of a given radius and frequency produces fusion and even black hole pressures at the center. I don't know if the thermodynamic limit for concentrating heat hotter than the source applies here since the light is columnated, but it wouldn't surprise me if ultraviolet or x-ray lasers would make the device more practical. Looks like NIF uses infrared at 1053 nm instead of ultraviolet at 351 nm, I'm not sure why though.

An interstellar spacecraft like this wouldn't have to carry special fission or fusion fuel, it could just use the Hawking radiation as propellant directly, with a thrust somewhere between a photon and hydrogen ion engine, and a specific impulse (ISP) approaching the light pressure limit. It could even run on the hydrogen in the interstellar medium directly and not have to carry fuel, although it might run into the same friction and fuel scarcity limit as a Bussard ramjet.

Although technically if the hydrogen was focussed down to run through the singularity, it wouldn't encounter resistance there, it would just be converted to energy which could be collected at the sides or reflected for thrust. I'm imagining kind of an hourglass-shaped pair of parabolas with a magnetic collector as leader and a lead pusher-plate as follower, and a high number of lasers focussed on the focal point of the trailing parabola so that any matter that enters there gets converted to energy and thrust.

It should be possible to be formal about this and derive a formula for how large the singularity has to be (how long it lasts) to be able to pass breakeven for 5 atoms of hydrogen per cubic meter from rest up through the speed of light. Or the formula might show decreasing efficiency, meaning that the idea isn't scalable and that this is just a pipe dream, and that's fine too.

What I'm really looking for though is a way to use energy to modulate mass/space/time to control the divergence of space to build a warp drive. Going through this exercise, I don't feel any closer to finding that, but maybe something in the problem space of the missing formal statements could lead to an answer. If only we had AI to help..

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This is pretty esoteric I guess and hard to convey with words. But I think that so many people like me would gladly work toward answering these fundamental questions than work on endless mainstream tech projects to make rent, that it breaks my heart. I dream of working toward something like Uber for meaningful work, where people who won the internet lottery could pay stipends to problem solvers and move humanity forward toward the stars.