If the Purva Mimamsa school believes that the Earth and humans are eternal, then how does it explain ancestral Gotras, such descendants of Bharadwaj, Angirasa, etc?
The Purva Mimamsa school seems to have had varied views regarding the nature of Gotras. In this excerpt from his commentary on the Purva Mimamsa Sutras, the Purva Mimamsa philosopher Shabara Swami apparently recognizes the notion that Gotra indicates the Rishi from whom you’re descended:
Nor are sages entitled to perform of sacrifices, because they have no Gotra; Bhrigu and other sages cannot belong to the Gotra of the same Bhrigu and others.
The idea is that your Gotra Rishi has to be your ancestor, and a Rishi isn’t the ancestor of himself, so Rishis who are founders of Gotras do not themselves have Gotras and thus are not eligible to perform Yagnas.
But the Purva Mimamsa philosopher Medhatithi gives a different view of Gotras, in this section of his Manubhashya. He says that Gotras are beginningless divisions of Brahmanas:
It is by [names like Bhṛgu’, ‘Garga’ and ‘Gālaya.] that the gotras of Brāhmaṇas should be known; as these are the principal gotras; the name ‘gotra’ applying to them by convention, and not by virtue of their fulfilling the condition mentioned in the above-mentioned definition that ‘the first progenitor, imparting his name to the family, is its gotra;’ in fact, ‘Bhṛgu,’ &c., have been known as ‘gotra’ from beginningless time, and their use in this sense is as much without beginning as the use of the caste-names, ‘Brāhmaṇa,’ and the rest. Prior to Parāśara’s birth, no Brāhmaṇas could have been named after him; hence, if the ‘gotra’ consisted in the first progenitor, &c., then the Veda (which makes mention of Parāśara as gotra) would have a beginning in time. Thus, then, since the use of the name ‘gotra’ is beginningless, it is this ‘gotra’ that is to be used in the pouring of water-libations. The persons imparting their names to families are not beginningless; they are modern; and at a Vedic rite, so long as one can make use of eternal names, there can be no justification for pronouncing names that have had a beginning in time. For these reasons, what the Brāhmaṇa should do when offering the libation of water, &c„ is to pronounce the appropriate ‘gotra- name’—such as ‘may this offering go to the Gargya,’ or ‘to one belonging to the Garga-gotra’—and then pronounce the name of the person. Among the Kṣatriya and the other castes, however, there is no such usage regarding gotra. These latter do not retain the memory of their ‘gotra’ in the same manner as the Brāhmaṇas do. Hence, for them, the ‘gotra’ must be something pertaining to this world; and it is for these that the ‘gotra’ consists of the ‘first progenitor, the most renowned, who imparts his name to the family:’ hence it is that they are referred to in Śrāddha, etc., by this gotra-name, even though it is one that has had a beginning in time. These, Kṣatriya, etc., are not worthy of being called by such titles as ‘Havirbhuk,’ and the like.
This is a species of the more general Purva Mimamsa argument that Vedas cannot contain the names of people or events in the past, because that would contradict their eternality. The Vedanta school’s refutation of this argument is that names and events recur in every age, because the Vedas are like the blueprint for the creation of the Universe. In any case, Medhatithi’s position is that given the Purva Mimamsa view that the Earth is eternal and humans are eternal on the Earth, your ancestors have eternally been the same Gotra as you going infinitely back in time.