If the Heat Metal spell is cast on a mounted character's armor, does it hurt their mount?

Question

The Heat Metal spell description says:

Choose a manufactured metal object, [...] Any creature in physical contact with the object takes 2d8 fire damage when you cast the spell. Until the spell ends, you can use a bonus action on each of your subsequent turns to cause this damage again.

[...] If it doesn’t drop the object, it has disadvantage on attack rolls and ability checks until the start of your next turn.

If a mounted character wearing metal armor has heat metal cast on it, would their mount be considered to be in contact with the rider's Plate armor (or any other armor such as Scale Mail)?

Scale Mail. This armor consists of a coat and leggings (and perhaps a separate skirt) of leather covered with overlapping pieces of metal, much like the scales of a fish. The suit includes gauntlets.

[...]

Plate. Plate consists of shaped, interlocking metal plates to cover the entire body. A suit of plate includes gauntlets, heavy leather boots, a visored helmet, and thick layers of padding underneath the armor. Buckles and straps distribute the weight over the body.

Answer 1

It depends on whether the armor covers your lower-body.

A Chain Shirt and a Breastplate is just upper-body.

Chain Shirt. This armor offers modest protection to the wearer's upper body

Breastplate. This armor consists of a fitted metal chest piece worn with supple leather.

Scale Mail, Half Plate, and all heavy-armor are full-body.

Scale Mail. This armor consists of a coat and leggings

Half Plate. Half plate consists of shaped metal plates that cover most of the wearer's body.

When riding, only your lower-body is in contact with your mount (often through a saddle). Depending on how realistic you want to be, your DM can have your mount take damage when you're wearing armor that extends to your lower-body.

More realistic tables can argue about saddles and other barriers offering protection. However, armor like Chain Shirt is worn between layers of clothing or leather and is expected to cause damage to the user. Isn't a saddle similar? This is up to DM fiat and table-style.

Interestingly, it doesn't matter how much contact there is to cause the damage. Even if it were a simple ring on your finger, you would take 2d8 Fire damage. So, a small part of the armor touching the mount causes the full damage as well.