BUG: `.str.split()` fails with pyarrow strings

### Pandas version checks

- [X] I have checked that this issue has not already been reported.

- [X] I have confirmed this bug exists on the [latest version](https://pandas.pydata.org/docs/whatsnew/index.html) of pandas.

- [ ] I have confirmed this bug exists on the [main branch](https://pandas.pydata.org/docs/dev/getting_started/install.html#installing-the-development-version-of-pandas) of pandas.


### Reproducible Example

```python
import pandas as pd
import io

data = '''name,x,y
matt b,10,20
fred c,20,1
jeff d,2,3
lisa e,5,7
adam f,23,1'''

df = pd.read_csv(io.StringIO(data), engine='pyarrow', dtype_backend='pyarrow')
df.name.str.split()
```


### Issue Description

In Pandas 1, you could split with no parameters (like python does with `'matt h'.split()`. This doesn't work with pyarrow types.

It gives a `TypeError`:

```
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
Cell In[47], line 12
      4 data = '''name,x,y
      5 matt a,10,20
      6 fred b,20,1
      7 jeff c,2,3
      8 lisa d,5,7
      9 adam e,23,1'''
     11 df = pd.read_csv(io.StringIO(data), engine='pyarrow', dtype_backend='pyarrow')
---> 12 df.name.str.split()

File ~/.envs/menv/lib/python3.10/site-packages/pandas/core/strings/accessor.py:136, in forbid_nonstring_types.<locals>._forbid_nonstring_types.<locals>.wrapper(self, *args, **kwargs)
    131     msg = (
    132         f"Cannot use .str.{func_name} with values of "
    133         f"inferred dtype '{self._inferred_dtype}'."
    134     )
    135     raise TypeError(msg)
--> 136 return func(self, *args, **kwargs)

File ~/.envs/menv/lib/python3.10/site-packages/pandas/core/strings/accessor.py:915, in StringMethods.split(self, pat, n, expand, regex)
    913 if is_re(pat):
    914     regex = True
--> 915 result = self._data.array._str_split(pat, n, expand, regex)
    916 return self._wrap_result(result, returns_string=expand, expand=expand)

File ~/.envs/menv/lib/python3.10/site-packages/pandas/core/arrays/arrow/array.py:2273, in ArrowExtensionArray._str_split(self, pat, n, expand, regex)
   2271 else:
   2272     split_func = pc.split_pattern
-> 2273 return type(self)(split_func(self._pa_array, pat, max_splits=n))

File ~/.envs/menv/lib/python3.10/site-packages/pyarrow/compute.py:258, in _make_generic_wrapper.<locals>.wrapper(memory_pool, options, *args, **kwargs)
    256 else:
    257     option_args = ()
--> 258 options = _handle_options(func_name, options_class, options,
    259                           option_args, kwargs)
    260 if args and isinstance(args[0], Expression):
    261     return Expression._call(func_name, list(args), options)

File ~/.envs/menv/lib/python3.10/site-packages/pyarrow/compute.py:221, in _handle_options(name, options_class, options, args, kwargs)
    216     if options is not None:
    217         raise TypeError(
    218             "Function {!r} called with both an 'options' argument "
    219             "and additional arguments"
    220             .format(name))
--> 221     return options_class(*args, **kwargs)
    223 if options is not None:
    224     if isinstance(options, dict):

File ~/.envs/menv/lib/python3.10/site-packages/pyarrow/_compute.pyx:1894, in pyarrow._compute.SplitPatternOptions.__init__()

File ~/.envs/menv/lib/python3.10/site-packages/pyarrow/_compute.pyx:1872, in pyarrow._compute._SplitPatternOptions._set_options()

File stringsource:15, in string.from_py.__pyx_convert_string_from_py_std__in_string()

TypeError: expected bytes, NoneType found
```

### Expected Behavior

I expect Pandas 1 behavior.

### Installed Versions

<details>

INSTALLED VERSIONS
------------------
commit              : e86ed377639948c64c429059127bcf5b359ab6be
python              : 3.10.13.final.0
python-bits         : 64
OS                  : Darwin
OS-release          : 21.6.0
Version             : Darwin Kernel Version 21.6.0: Wed Aug 10 14:28:23 PDT 2022; root:xnu-8020.141.5~2/RELEASE_ARM64_T6000
machine             : arm64
processor           : arm
byteorder           : little
LC_ALL              : en_US.UTF-8
LANG                : None
LOCALE              : en_US.UTF-8

pandas              : 2.1.1
numpy               : 1.23.5
pytz                : 2023.3
dateutil            : 2.8.2
setuptools          : 67.6.1
pip                 : 23.2.1
Cython              : 3.0.4
pytest              : 7.2.0
hypothesis          : 6.81.2
sphinx              : None
blosc               : None
feather             : None
xlsxwriter          : 3.1.2
lxml.etree          : 4.9.2
html5lib            : 1.1
pymysql             : None
psycopg2            : None
jinja2              : 3.1.2
IPython             : 8.8.0
pandas_datareader   : None
bs4                 : 4.11.1
bottleneck          : None
dataframe-api-compat: None
fastparquet         : None
fsspec              : 2023.3.0
gcsfs               : None
matplotlib          : 3.6.2
numba               : 0.56.4
numexpr             : 2.8.4
odfpy               : None
openpyxl            : 3.0.10
pandas_gbq          : None
pyarrow             : 13.0.0
pyreadstat          : None
pyxlsb              : None
s3fs                : None
scipy               : 1.10.0
sqlalchemy          : 2.0.21
tables              : None
tabulate            : 0.9.0
xarray              : None
xlrd                : 2.0.1
zstandard           : None
tzdata              : 2023.3
qtpy                : None
pyqt5               : None

</details>
