Temperature is ill-defined in the presence of gravity because the concept of temperature is closely related to the notion of thermal equilibrium, where the distribution of particles and their energies are determined solely by their temperature. However, in the presence of a gravitational field, the situation becomes more complex due to the effects of general relativity.
In general relativity, gravity is not described as a force but rather as the curvature of spacetime caused by mass and energy. As a result, the properties of spacetime itself, such as its curvature, can influence the behavior of particles and their energies. This means that the gravitational field can affect the distribution of particles and their energies, which in turn affects the definition of temperature.
Specifically, in a gravitational field, the energy of a particle not only includes its kinetic energy and potential energy but also its gravitational potential energy. As a particle moves in a gravitational field, its potential energy changes, and this affects its thermal properties. This gravitational potential energy needs to be taken into account when defining the temperature of a system.
Moreover, in regions with strong gravitational fields, such as near black holes, the effects of gravity become significant. The intense gravitational field can cause significant time dilation and distort the local spacetime, leading to variations in temperature and other thermodynamic quantities.
Therefore, because gravity influences the distribution of particles and their energies, and because the concept of temperature relies on a well-defined equilibrium state, temperature becomes ill-defined in the presence of gravity. The traditional definition of temperature based on thermal equilibrium does not hold uniformly in gravitational systems, and alternative frameworks, such as the study of black hole thermodynamics, are required to describe temperature-like quantities in those contexts.