It's a fun, though confusing read, with some great nuances mixed in.

First, the fun parts. Action is very fluid and non-stop - the protagonist develops the dimension-hopping device, shows it to the professor, gets double-crossed, gets sent to another dimension, then connects with his dimensional counterpart and finds a way home. When I mean "non-stop", I mean it - even in the other dimension, where time runs slower, the protagonist is constantly in movement. It's impressive to see how the protagonist - as a stand-in for the reader - can describe their experience in this new dimension while still giving that sense of urgency.

I'm a sucker for both fantasy and sci-fi, and particularly stories that combine both elements. This story stands at the edge of both - the elements of the light-bulb can work in a fantastical frame as in a sci-fi frame. Since you didn't describe the mechanics behind how the light bulb works (other than it's apparently the preferred form for this kind of device), it doesn't get grounded into scientific fact - it can easily be a magical artifact by a scientifically-minded craftsman. It's not necessary to explain, of course - only to see it work, since that is what moves the story forward - but it's also fortuitous that it wasn't, because it'd shift the story towards one side or the other. There's plenty of sci-fi stories where the protagonists, wielding advanced technology, face challenges that would otherwise be considered fantastical (Stargate SG-1 comes to mind), which is why I say that this could easily be a sci-fi story - the resource of a secret portal to the first world doesn't necessarily push it into firm fantasy grounds. Again, it's a fun - and solid - element because it appeals to a wider audience.

Less "fun" but more "it works" is the description. It's concise, for sure - you don't bother giving a detailed physical description of the protagonist, which isn't necessary because, again, it's the reader's stand-in. It's also great that the professor doesn't get an extensive explanation other than having the protagonist notice very late that he wears a jacket even indoors. It suggests the protagonist - alongside the brief note about tidiness - is very focused in their job, to the extent they barely notice what's going around them. That's what I get from reading the protagonist's actions - the trip to the time-dilated world serves as both an example and a contrast to this trait, as they're forced for the first time to see their surroundings rather than devote mindlessly on their work. You place a lot of emphasis on the other dimension - its relative period in time, its inhabitants, the time-dilation effect that makes the protagonist absurdly faster than them - which is good because the idea is to explore this new dimension, given the first one is akin to our own.

I like that the professor has some nuance, but this is where I get to the "confusing" part. It's evident that the professor is a patron rather than a researcher, using the talents of others to achieve their purposes. The professor also has a motive for why they need the device - to search for his lost wife. However, I find the professor's insistence in shooting - therefore, attempting to kill the protagonist - senseless. Did the professor do so to leave no witnesses, making him a cruel and vicious patron? Or did the professor do it out of fear the protagonist would stop them? Just mentioning how he wants to seek for his lost wife gives the professor a valid motive to pursue and lead the protagonist into creating the device - suddenly drawing a gun feels completely unnecessary and pointlessly cruel. The professor, acting as a device within the story rather than a character, disappears too fast to justify their actions. I'll give you something - the transition between bullet to arrow as the protagonist shifts dimensions is an amazing visual, but the reason why it happens feels forced.

This ties to the second confusing part - the ending. The protagonist claims this is the beginning of their story, but as this is a short story, there's no follow-up. It leaves a question that, I presume, is meant for the reader to answer on their own, but when you tie it to the professor's (questionable) motives and how the protagonist returned, it weakens an otherwise good ending.

I feel this is the weakest part of the tale - while your instinct was spot-on at places where explanation would've been superfluous, it leaves others where more nuance could've been added. For example - if the professor knew about the device, why didn't he try it? Wouldn't it be better to demand the blueprint from the protagonist instead? This would strengthen the story a lot more - I mean, I could find a justification by saying that the professor was only interested in the device, but I feel the professor could've tried it on his own, if he was so desperate. The reason why I feel so conflicted is because of what you state through the professor's lips: the invention isn't nothing new, and the protagonist isn't the first one to succeed on it.

"You're not the first person to manage to create a device that can travel between dimensions. And you're also not the first to call the shape of a light bulb ideal for energy reasons."

If the professor knew this, then it's reasonable to consider he could've tried it. Perhaps I'm thinking too much about it, but it's because I find this disconnection a bit odd. There's a reasonable motive behind the professor's actions (seeking his lost wife), but it's the execution where it falters. Perhaps the professor isn't a professor at all, but passing as one - that would explain why he depended on the protagonist's inventive. Or, perhaps, the device can only be built once - adding a fantastical limitation - and the professor lost his wife when he created his own, which is why he's trying to replicate the device through someone else. The second bit is reinforcing why the professor would want to kill the protagonist. There's a throw-away line that suggests the professor is a psychopath ("I haven't killed in a long time. You would be suitable"), but rather than defining a secondary motive, it feels forced in. Considering the ending of the story, reinforcing this part would solidify the ending as a whole - if the professor is a psychopath obsessed with finding his wife and unstable enough to disturb other dimensions, it is the protagonist's purpose to stop them - hence, why it's the beginning of their story. As it stands, it makes for a surprise villain with a flimsy motive.

Another minor thing is the time frame. The reference to light bulbs and gym bags suggests anywhere between the 20th century and early 21st, but a clearer reference would be great - particularly since you make another temporal reference when first describing the other dimension, specifying 150 years. That would set the technological advancements of the second dimension circa 1750 - 1840, well within the Industrial Revolution, which would've rendered arrows obsolete. (The gym bag is what throws me off, because otherwise you could say it's early 20th century or perhaps late 19th century and get a more reasonable timeframe.) I say it's minor because the idea is to provide a frame of reference to the reader that they're on a different world, but it's part of a pivotal scene in the story and it acts as a dissonant distraction to the flow of action in it.

I mention this because, even though it's a contest entry and therefore it can justify some hastiness, it's a story that could use some polish. There's a few issues with grammar and syntax but I don't feel I'm the most qualified to speak about them, since they could be ascribed to differences in style. Reinforcing the description of the professor as a potential psychopath who uses a tragic event in his life as an excuse for his urges makes for a chilling villain with lots of nuances. The revelation that there's portals connecting two dimensions is also an interesting avenue to explore - the protagonist could've remained in that other dimension, attempting the light bulb experiment once more, only to end in another dimension - the portals render the experiment pointless to an extent, but luckily so as that means the professor can't use them to travel between dimensions at will. There's lots of themes introduced here that yearn to be explored - or perhaps left to the reader's imagination - but the antagonist definitely needs some reinforcement.

Lastly, I apologize if the review's too long. I tend to go on trains of thought when I write, but I hope it's detailed enough to address all the issues I have. Again - I like the idea of mixing fantasy and sci-fi (or at least, add the element of whimsy, a thoroughly fantastical element, to what could be considered early sci-fi), and the scenes are beautiful, but it falters on the development (or lack thereof) of the antagonist. It's not bad - it's great - but it could use some polish.