Ideas about some kind of afterlife are commonly found in all religions. In the Bhagavad Gītā, Krishna explains that those who resort to him do not get ‘punarjanma’ (rebirth in this world). A precursor to ‘punarjanma’ is the idea of ‘punarmṛtyu’ or ‘re-death’, found in texts such as the Bṛhadāraṇyaka Upanishad. However, the Rig Veda presents its own distinctive ideas about a possible afterlife, in Maṇḍala 10 Sūktas 14 to 18.
What follows is my own translation of the the first eight verses of Rig Veda Maṇḍala 10 Sūkta 16 by Ṛṣi Damana Yāmāyana, which occur in the middle of this stretch. It would appear that these eight verses are spoken by the priest during the actual cremation of the dead body, and are addressed either to Agni (1,2,4,5,8) or the dead person himself (3,6,7 — in bold).
In making this translation, I again consulted the translations by Professors Stephanie Jamison and Joel Brereton and by Prof. Ralph Griffith, both of which I found helpful. Agni and Jātavedas are both names of fire, considered as a deity. The term ‘asunīti’ which I left untranslated seems to refer to the afterlife realm. Parts of verse five are rather unclear.
O Agni, do not burn this one; do not burn him up;
Do not destroy his skin; do not destroy his body.
O Jātavedas, when you would set him alight,
Then send him on to the ancestors.
O Jātavedas, when he is alight,
Then entrust this one to the ancestors.
When he goes to asunīti,
Then he will do the will of the gods.
Let your eye go to the sun, your breath to the wind.
Go to heaven and earth according to dharma.
Or go to the waters if you want.
With your body, dwell among the herbs.
The ram is your share; consume it with your heat.
Consume it with your burn; consume it with your flame.
O Jātavedas, with these which are your benevolent forms,
Carry this one to the virtuous world.
O Agni, send back to the ancestors
He who, offered to you, moves by his inherent power.
Entering into life, let him get the remainder;
Let him go across with his body, O Jātavedas.
That of yours which the black bird struck,
Or the ant, the snake, or the wild beast,
Let all-consuming Agni make it healthy,
And Soma which the Brahmins have obtained.
Wrap yourself up in leather as protection against Agni.
Keep it away with all kinds of fats.
So that the fierce one does not violently burn you up
With its flames clasped around you
O Agni, do not overturn this soma-spoon [camasa].
It is dear to the gods and to the Soma-lovers
This is the soma-spoon which the gods drink from.
This soma-spoon exhilarates the immortal gods.