Why do you assume that earth would be "split in half" by that beam? What particles btw?

I’m not 100% sure that 0.999 c is fast enough for any particles to do what you actually want, unless it’s less of a beam and more of a relativistically fast rod of heavy elements...

Pretty sure we have already had a question about splitting the earth into two equal halves. Just can't find it...

The beam slices the earth from top to bottom in 600 milisecond or 0.6 of a second. No, it does not. 40 meters diameter means 12 000 000 meters earth diameter split in 300 000 sections, each section must be cut through with approximate light speed. The equator section needs 12 000 km / 300 000 km/sec = 0.04 sec. So your 0.6 sec is only sufficient for 15 slices aka 600 meters. Your assumptions are wrong from the start.

The real time is slicing area / (beam diameter * light speed) ~ 9400 seconds ~ 2.6 hours.

Think of the earth as a big ball of molten iron with a thin skin, about the thickness of an orange peel (proportionaltely) surrounding it. What happens when you slice a globe of liquid? On the scale of the globe, not much, although the special effects would probably have George Lucas reaching for his copyright lawyers.

I think Alderaan's demise might provide some guidance here :)

May I ask, why are you asking what sliced the earth? the question was what would happen if the earth was sliced in half.

@ThorstenS. I didn't really put too much thought into the speed of the beam other than moving at relativistic speed so this information is appreciated but as Mr.J stated I am more interested in the effects of the Earth more so than the beam itself. Still, I wasn't aware of the intricacies and how it might impact the interaction between the earth and the beam itself so thanks for the information.

@ThorstenS. I don't think that's right. The speed of the particles from the beam itself is bound by the speed of light, but the effect of the beam (in this case, cutting) is not, just like a laser pointer dot can move faster than the speed of light.

You don't mention the density of the beam.

A back of the napkin calculation puts the energy of this event at around 1e30 Joules, or the total energy output of the sun for about 1 day. I think the answers below dramatically underestimate the effects of this event.

Will this beam cut from a fixed position? (Like you slice bread.) Or will it rotate around the earth (Like you cut on a late. Except there the material rotates, but you get the point.)

@EugenMezei My apologies but I do not fully understand your wording so I may not give a clear answer. I'm assuming your question on whether it's fixed or rotating means whether the beam emitter is at a fixed location and is merely changing the direction of the beam (sorta like a laser pointer) versus rotating which probably means the emitter is rotating around the earth. If so than the emitter is at a fixed location and merely moves the beam by changing where it is facing.

@MarchHo - yeah, that's because nothing is actually moving in that case...

You haven't specified what kind of particles are in the beam, but it almost doesn't matter. The beam simply doesn't do what you think it does. Particle beams are not magic and the rules that they follow are well known. You propose a $\beta$ (v/c) low enough to result in a Lorentz factor ($\gamma$) of only 22. Those speeds (a) won't preserve unstable particles through the travel time across the Earth's diameter and (b) lead to a beam energy not above $21\,\mathrm{GeV}$. The beam dumps it's energy over only a few tens of meters at most.

@MarchHo The problem is digging through the Earth, not only leaving a fire trace on Earth's surface. Imagine as thought experiment that the Earth is split by separating it into drill samples with thousands of kilometers length and 40 meters diameter. To achieve the effect of separating, you need to remove as much matter as putting all drill samples in a row which is a distance of 2.8 billion km. Your particle beam can only travel 0.6s and the effect is even much smaller because being a beam it means it has a much lower density as normal matter, removing only a fraction of matter.

@ThorstonS. I'm not following your argument. It seems to me that you missed the "in half" part of the question. I read the question as the beam is making a single cut, "from top to bottom" (presumably from north to south pole) parallel to a meridian, separating the earth into two hemispheres. The 40 meter width of the beam is just the "thickness of the knife" so to speak.

@MooseBoys: I stuck my finger in the air and pegged it above the binding energy of the Earth. That’s a hundred times greater than your estimate and I don’t think ‘the planet explodes’ is a dramatic underestimation of anything. I’m interested as to how you got your number though.

@JoeBloggs Link. Assumptions were that the particle beam has a densiy of 1g/cm^3, and that almost all energy is transferred to the earth.

@RandallStewart You are pointing the particle beam at the equator. Even if the particle beam proceeds with c, it needs at least 0.04 s to penetrate earth (In fact, it is much, much slower. Matter does not vanish, the gaseous/plasma needs to come out of the hole, scattering the beam.) So after 0.04 s you have a nice 40 m diameter hole in the earth (it would close immediately because of gravity, squirting magma out). The earth has now a diameter of 12 000 km, leaving 11 999. 960 km more to cut. Calculate for yourself the necessary time.

@dmckee Thanks for the informative post, if it's not too much trouble may I ask what speed is necessary for the beam in question to be able to slice the Earth all the way through if it's even possible?

Above $\gamma \approx 10$ no one but pop-sci writers talk about speed. instead physicists energy or Lorentz factor.

Do you accept micro black holes as "particles"? They come with the advantage that they don't fill the gap with complicated plasma that must be pushed out of the way.

Well, for one, the Earth would be split in half.