Carlo Rovelli’s “Analysis of the Distinct Meanings of the Notion of “Time” in Different Physical Theories” (Il Nuovo Cimento B, Jan 1995, Vol 110, No 1, pp 81–93) describes ten distinct versions of the concept of time, which he arranges hierarchically. Here are excerpts from his article:

We find ten distinct versions of the concept of time, all used in the natural sciences, characterized by different properties, or attributes, ascribed to time. We propose a general terminology to express these differences. p.81

… our aim is to emphasize the general fact that a single, pure and sacred notion of “Time” does not exist in physics. p.82

The real line is a traditional metaphor for the idea of time. Time is frequently represented by a variable *t* in *R*. The structure of *R* corresponds to an ensemble of attributes that we naturally associate to the notion of time. These are the following:

a) The existence of a topology on the set of the time instants, namely the existence of a notion of two time instants being close to each other, and the characteristic “one dimensionality” of time;

b) The existence of a metric. Namely the possibility of stating that two distinct time intervals are equal in magnitude. We denote this possibility as metricity of time.

c) The existence of an ordering relation between time instants. Namely, the possibility of distinguishing the past direction from the future direction;

d) The existence of a preferred time instant, namely the present, the “now”. p.83

In the natural language, when we use the concept of time we generally assume that time is one-dimensional, metrical, external, spatially global, temporally global, unique, directed, that it implies a present, and that it allows memory and expectations. The concept of time used in Newtonian physics is one-dimensional, metrical, external, spatially global, temporally global, unique, but it is not directed and it does not have a present. In thermodynamics, time has the additional property of being directed. Proper time along world line in general relativity is one-dimensional, metrical, temporally global but it is not external, not spatially global, not unique; on the other side, the time determined by a matter clock is one-dimensional, metrical, but not temporally global, an so on. p.87

… the notion of present, of the “now” is completely absent from the description of the world in physical terms. This notion of time can be described by the structure of an affine line *A*. p.88

… our list does not include the possibility of considering a non-metric but directional notion of time. p.89

Table I. [without the fourth column]

Time concept |
Attributes |
Example |

time of natural language |
memory |
brain |

time with a present |
present |
biology |

thermodynamical time |
directional |
thermodynamics |

Newtonian time |
uniqueness |
Newton mechanics |

special relativistic time |
being external |
special relativity |

cosmological time |
space global |
proper time in cosmology |

proper time |
time global |
world line proper time |

clock time |
metricity |
clocks in general relativity |

parameter time |
1-dimensional |
coordinate time |

no time |
none |
quantum gravity |

… our hypothesis concerning time is that *the concepts of time with more attributes are higher-level concepts that have no meaning at lower levels.* p.91

If this hypothesis is correct, then we should deduce from it that most features of time are genuinely meaningless for general systems. p.91

… we suggest that *the very notion of time, with any minimal characterization, is likely to disappear in a consistent theory that includes relativistic quantum-gravitational systems*. p.91

… the concept of time, with all its attributes, is not a fundamental concept in nature, but rather that time is a progressively more specialized concept that makes sense only for progressively more special systems. p.92