I was thinking about the classic beanpole design for interstellar travel. Wouldn't the Whipple shield need to be at least twice as wide as the ship is long to fully protect it from collisions with interstellar matter? In my head, the scenario is as such. Assume the ship is moving at a speed approximately the speed of light and there is a dust particle traveling perpendicular to the path of the ship the Whipple shield would need to provide a 45-degree 'shadow'. Is there some relativistic effect of length contraction that I'm missing?
The shield width is a function of the ship's speed (more than its length, even if the length itself does enter into the calculation) and the inbound dust speed vector. A 45 degree angle seems right when both move at the same speed on trajectories at an angle of 90 degrees.
But I believe most of the interstellar dust would be almost stationary in comparison to a ship traveling at c-fractional speeds, so the shield wouldn't need to be much larger than the cross-section of the ship in the direction of travel.
First of all a Whipple shield is only good against dust. Larger objects would smash through the shield, generating a plasma burst and possibly cracking, and the resulting fragments would cascade the process, but still leave enough impacting mass to wreak considerable damage.
A ship one kilometer in length traveling at 10,000 km/s will cover its length in 0.1 milliseconds (L/V). A dust particle traveling at right angles at speed v, and barely escaping the Whipple shield will be dangerous if the shield "overhang" D (difference in section between it and the ship) is less than the distance covered by the particle in that time. This distance is time L/V multiplied by speed, so we need D > L(v/V).
(The dust particle, being stationary in the direction of travel, will impact with a relative speed of SQRT(10000^2+1000^2) = about 10050 km/s if it impacts at all. This calculation does not account for relativity and grows more and more incorrect the more the speeds approach c).
If the particle has a speed of 1000 km/s, D must be 100 meters or more (so if the ship is 200 m wide, the shield must be 100+200+100 = 400 m wide).
Interplanetary speeds for dust particles are in the 10-40 km/s range, though, so a Whipple shield would only need to be 2 to 8 meters wider than the ship.
As LSemi already stated, if you want a Whipple Shield just to shield from dust particles, than it only would need to be slightly bigger than the cross section of the ship itself, as dust would almost seem stationary at relativistic speeds
But you would also need the shield for other purposes, that would exceed the Whipple Shield definition of stopping dust particles. For example, if you move at relativistic speeds, the light that travels towards you would get blue shifted, until you basically run head into a stream of gamma ray, originally being the background radiation. So you would want a shield in front of you to stop that "light", just like a sun umbrella.
In this case, the shield would need to be bigger. Depending on the speed you are traveling and length of the ship, the light screen would need to be big enough to create a shade for the most dangerous radiation coming from straight ahead, so that every light the shade does not cover is depending on the angle and the speed relative towards that light would get absorbed by the normal ship hull.
Another use for these shields could be as a fuel scoop. Absorbing the hydrogen in the interstellar medium to use for e.g. your fusion drive. Then the shield should be big enough to scoop up enough material to sustain your drive, depending on the fuel density in interstellar space.
As you approach light speed, the length of your ship goes to 0
An object moving at relativistic speeds undergoes Length Contraction.
Length = Length at rest x (1-Velocity^2/Speed of light^2)^(1/2)
This means that as your speed increases, the size of shield you need becomes smaller and smaller. However, since it's probably not practical to decrease your shield size as you accelerate, you should probably just design it to protect you at non-relativistic speeds, so that you can be protected when you're only going 0.1c.
No, it wouldn't.
Let's look first at fast, but not relativistic travel with a speed maybe 10,000 kps. This is very fast compared with the lateral velocity of dust particles, so the shield is only needed directly in front of the ship. Making it a lot wider than the ship is pointless.
Think about a dust particle aimed directly at the ship from the side and moving at 100 kps (fast for dust). If the ship is moving at 10,000 kps, from the point of view of the ship the dust moves 100 meters towards the rear for ever meter it moves sideways. So it appears to come from about a degree to the side of straight ahead and the shield merely has to be a couple of degrees wide as viewed from the rear of the ship.
Now let's go relativistic. At circa 90% of lightspeed, the 100 kps dust particles appear to move 3000 meter back for every meter in and thus to come from even closer to straight ahead. Just like at 10,000 kps, only more so.
So as long as you're travelling fast, a Whipple shield need only be slightly wider than the ship.
Note that a Whipple shield would probably be pretty ineffective at 0.9c and useless at really high gamma.