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Let's assume there are more than 3 spatial dimension, for example 4 (the 4th not beeing time, but spatial)

Would a common energy discharge, explosion etc. affect the 4th dimension?

I am not sure if the answer would be speculation, since the 4th spatial dimension is theory right now, or is it? Therefore I chose the "reality check" tag.

There is no known 4th spatial dimension, so anything we say is speculation. In fact, whether or not there are dimensions at all is a philosophical question. At the deepest level it is not clear if dimensions actually exist, or if we invented the concept to make sense of the world we experience.

Thus, if something were to be called the 4th spatial dimension, it would have to fit cleanly with what we have called the first 3 dimensions. Otherwise it would be given special treatment in a way we don't give the first 3 dimensions. If there was a 4th spatial dimension, we would expect explosions to propagate into it.

Warning to laymen: Advanced mathematics ahead!

Warning to mathematicians: Incredible oversimplification of advanced mathematics ahead!

Our spatial dimensions form a differentiable manifold. Manifold is a very fancy topology term for a connected space which looks euclidean locally. The surface of the Earth is a manifold. While the Earth as a whole is clearly curved (it's a sphere), if you look "locally" it looks like a flat plane. That's why Flat Earther's can get away with so many arguments: if you only look at a small section of the Earth, it is almost flat. Likewise, a ribbon can define a manifold. No matter how much you twist and turn it, locally there's always a plane made by the weft and the weave.

The second piece of our fancy topology term is differentiable. At the most technical level this means you can do calculus on it, but at a nice easy to understand level, this means things are "smooth" and we can talk about them changing over time. When your explosion goes off, it seems to be very sudden, but if you look at it on much smaller timescales, you see that the explosion grows very smoothly over time.

End advanced mathematics.

The equations of motion we use to describe things like explosions tend to have some random particle movements to them. When the explosion heats up air molecules near the explosive, they start to move fast and collide with slower moving air molecules outside of the explosion, transferring energy and thus raising the temperature. These collisions scatter the molecules in random directions in all spatial dimensions.

Thus, if we saw that there was a 4th dimension, we would expect these collisions to scatter in all 4 dimensions, not just 3, because there would be nothing privileged about this 4th dimension. Energy would propagate in it just the same.

As a result, you would find the power of the explosion diminishes faster than it does in our normal 3 space. We can think of the power of an explosion as being on the surface of a sphere as that sphere expands. The surface area of a sphere goes up by the square of the radius, meaning if the sphere expands by 10 fold in radius, it's surface area expands to 100 fold. The amount of energy remains constant (we only set off so many explosives), so that means the energy that hits any part of the sphere goes down by 100 fold. This is why an explosion is deadly up close but becomes survivable at a distance.

If we had 4 dimensions, the explosion would expand on a 4-d hypersphere rather than a 3-d sphere. In this case, the surface area of the hypersphere would go down by the cube of the radius. This falls off much faster, so explosives would be less useful at a distance. Note that we do not observe this in empirical testing, which is why we do not believe there is a 4th spatial dimension (or at least, if there is one, it's complicated and counter-intuitive)

Ways this can change

There's a few corner cases worth considering. One is the case where everything we are is actually a big rod that extends into the 4th dimension. This means that the 4-d space will act as-though it is a 3d space. However, this is likely not the case. It requires tremendous assumptions about symmetry and determinism. It certainly doesn't fit well with the uncertainty of QM. If there's anything about the rod's behavior which is not perfectly symmetric, the rod will act as a 4-d object, and you won't see simple 3d behaviors.

The second corner case to consider is a shaped charge. Shaped charges are designed to focus the energy of the explosion in a shape other than a sphere. A tank-buster may shape it to be as close to a line as possible, like a spear skewering the tank. Demolitions groups often use shaped charges to create a plane which cuts concrete pillars cleanly without excessive amounts of explosives. If our universe happened to be structured like a shaped charge, we might only see 3d propagation of explosives.

No, common explosions and energy discharges only affect our 3 dimensions.

If energy slipped into other dimensions, the law of conservation of energy wouldn't be supported by observable evidence. The law states that the total energy of an isolated system (our detectable 3-dimensional universe, for example) remains constant over time. If energy were escaping into other dimensions, we wouldn't be able to detect that energy anymore, so it would appear that in some cases energy just vanished. We've never witnessed that; all the energy from explosions and energy discharges stays in the 3 dimensions we live in.

It would not affect the 4th spatial dimension unless you allow us to observe less energy output in the 3 spatial dimensions we can observe.

For a simple example, let's take a pulse of light radiating out from a point source. As the light travels, it spreads itself out in a spherical shell (a 2-sphere) that increases its surface area as it grows. The surface area of a sphere is proportional to the radius squared. So, the energy drops off from the source proportionally to 1/r^2.

Now, if you have a 4th spatial dimension, a pulse of light radiating out from a point source will spread itself out in a 3-dimensional volume instead of a 2-dimensional surface. As the radius increases, the volume increases proportionally to the radius cubed instead of the radius squared. Energy would drop off proportionally to 1/r^3.

What does all this mean? To start, the energy of any explosion would dissipate much faster through 4 dimensions than 3. The same explosion in 4D would be much smaller and weaker than its 3D equivalent.
But...if you actually had light propagate through 4 dimensions instead of 3, there would be a whole host of implications. Stars and galaxies (and maybe even the Sun?) would be completely invisible. Not to mention if you had gravity and other forces do this as well. (If you have gravity propagate across 4 dimensions, you can't have stable elliptical orbits for your planets!)

TL;DR: unless you treat your 4th spatial dimension as special in some way and restrict the interactions between it and the other dimensions, it will make things very messy!

[Edit: BrettFromLA's answer scooped mine. But I put in other details, so I'll leave it up!]

Whatever are the dimension of your particular space-time, any object existing in it will use all of them.

Including explosives and energy dischargers.

Therefore in a 4 spatial dimensions space explosions would be 4 dimensional, too.