Spells like invisibility say the objects that the creature is carrying also become invisible, but does that mean the invisible object stops shedding light? Or will my embedded light arrow reveal their position by glowing?
There are no rules for embedded arrows, so it is up to the DM.
The rules simply lack the granularity required to make a definitive ruling, so it is going to be entirely up to your DM. There are no rules about arrows getting stuck inside of things. These two Q&A have some relevance for working toward a ruling here:
If I cast light on an arrow and hit a creature, then it becomes invisible, does it reveal where they are?
Let's deal with the misapprehension in the question.
Unless someone is Hidden, you know where they are. See What advantages does hiding have?
To be Hidden, they must have at some point in the past, take the Hide action, have made a Dexterity (Stealth) roll higher than your Wisdom (Perception) check (passive or active) and not have "come out of hiding". It's possible to be hidden from some creatures and not from others.
Being Invisible does a lot of cool things but it does not make your location unknown.
Having a light source while invisible
This has an answer: What happens when I'm invisible and something I'm wearing had Light cast on it?
Spoiler: it still sheds light or it doesn't shed light - answers conflict.
If someone Invisible and Hidden had a light source and the DM decides it sheds light, you could work out where they were, say with a successful Intelligence (Investigation) check (passive or active) or by allowing advantage on the Wisdom (Perception) check (passive or active) - they would then no longer be Hidden.
Does an arrow stick into someone who is hit by it? The rules don't say.
My ruling FWIW is no because:
- "Hit points represent a combination of physical and mental durability, the will to live, and luck." Only one of those 4 actually involve the actual arrow actually touching you - 3 times out of 4 the arrow has reduced your "mental durability, the will to live, and luck" without actually touching you.
- Misses can actually hit you. A character with a shield might have half a dozen arrows embedded in it. Ditto armour.
It depends on whether the arrow lodges into the creature or not
As others have pointed out, being invisible does not automatically mean that your position is unknown. However, I would rule that having a glowing arrow sticking out of you prevents you from hiding 'in plain sight' (ie. without visual cover) whilst invisible.
I initially answered this question by asserting that:
- Being hit by an arrow normally results in having an arrow sticking out of you
- Having an arrow sticking out of you is not the same as wearing an arrow or carrying it
However, some research into real world archery has shown me that it's not uncommon for arrows to pass all the way through living targets. Therefore, it is more accurate to assert that:
Being hit by an arrow can result in having an arrow sticking out of you
There are no specific rules about having arrows sticking out of you. However, being hit by an arrow involves receiving piercing damage from the arrow, and being pierced by the arrow implies that it has punctured your flesh and may be sticking out of you. An arrow could conceivably graze someone, and (as previously stated) it's not uncommon for an arrow to go all the way through someone.
Because there are no rules for whether or not the arrow sticks out of your target, it's up to your DM's ruling. Your DM might arbitrarily decide that it does or doesn't, or make a probability roll to see if it does - perhaps modified by the hardness of the target's armor.
Assuming the arrow does stick into your target, the most reasonable reading of the rules is that it remains visible and reveals their position, preventing them from hiding using invisibility alone.
Having an arrow sticking out of you is not the same as wearing or carrying the arrow
Wearing and carrying are not defined explicitly in the rules, so we interpret them as having the same meaning that they do in everyday speech. If I saw someone with an arrow sticking out of them, I wouldn't say "why are you wearing that arrow?" or "What are you carrying that arrow around for?" Since they are neither wearing nor carrying the arrow, it is not rendered invisible.
Bear in mind that removing an arrow from oneself would probably be a free object interaction, so it's not a strategy that's beyond countering.