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).

-

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.

-

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.

-

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.