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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?

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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:

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  • \$\begingroup\$ Totally this. A lot of arrow hits could just as well be grazes that cut the flesh but don't stick in the target. Especially if the target has high hit points for its size and the arrow does low damage. \$\endgroup\$
    – shhalahr
    Jun 2 at 14:51
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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.

Arrow porcupine

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.
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  • \$\begingroup\$ @Medix2 my bad. \$\endgroup\$
    – Dale M
    Jun 3 at 4:48
  • \$\begingroup\$ Although hit points represent mental fortitude, surely an arrow hitting someone involves the mechanical reality of their being hit by an arrow? Don't think the 'abstract HP' rules override the weapon attack rules such that a 'hit' with a ranged weapon could be thought of as 'being nearly hit but a bit afraid because you were nearly hit.' \$\endgroup\$
    – Lovell
    Jun 3 at 12:48
  • \$\begingroup\$ @Lovell I do think that \$\endgroup\$
    – Dale M
    Jun 3 at 12:59
  • \$\begingroup\$ That's very counterintuitive - a hit with an arrow deals piercing damage (not psychological), and poisoned weapons poison the target on a hit. Clearly the intent of the rules is that 'hit' and 'miss' are interpreted through their most natural reading. \$\endgroup\$
    – Lovell
    Jun 3 at 13:05
  • \$\begingroup\$ Can you provide any concrete evidence for the start of your answer? The wording for the Rogue's Blindsight feature (PHB 96) says: "Starting at 14th level, if you are able to hear, you are aware of the location of any hidden or invisible creature within 10 feet of you." It would not be necessary to mention that you're aware of the location of an invisible creature if that was automatically the case if they haven't explicitly tried to hide from you. \$\endgroup\$ Jun 5 at 0:17
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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:

  1. Being hit by an arrow normally results in having an arrow sticking out of you
  2. 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.

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    \$\begingroup\$ Arrows can break upon impact with a target and arrows can pass through a target completely. Even smuggling in real world experience for this ruling doesn't support your conclusion that being hit by an arrow means it is sticking out of you. \$\endgroup\$ Jun 3 at 13:28
  • \$\begingroup\$ Yes true: apparently arrows passing all the way through is not uncommon \$\endgroup\$
    – Lovell
    Jun 3 at 13:37
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    \$\begingroup\$ Yeah, so it's a downvote from me as your answer depends upon an assertion that is not supported by the rules or the real world behavior of arrows. \$\endgroup\$ Jun 3 at 13:41
  • \$\begingroup\$ Well summarised. I've updated my answer into something longwinded and uncertain \$\endgroup\$
    – Lovell
    Jun 3 at 13:48

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